9. A SIMPLIFIED APPROACH TO THE SYNTHESIS OF DUAL CONTROLLERS
9.1. Modification of Certainty-Equivalence Adaptive Controllers
Consider the linear discrete-time system in state-space representation described by eq. (8.1). In addition to the assumptions described in section 8.1 for the given system, it is also assumed that the adaptive controller has already been designed, using the CE as- sumption
) ), ˆ( ( )
( CE
CE k =u p k ℑk
u , (9.1)
which is a vector function of the available observations ℑk [eq. (8.2)] and the parameter estimate pˆ(k)=E{ }pℑk at timek.
N.M. Filatov and H. Unbehauen: Adaptive Dual Control, LNCIS 302, pp. 105–118, 2004.
© Springer-Verlag Berlin Heidelberg 2004
Following the bicritirial approach, again the two cost functions for control optimi- zation are introduced
} )]
1 ˆ ( ) 1 ( [ )]
1 ˆ ( ) 1 ( [
E{ T
c n n k
k k k k k
J = x + −x + W x + −x + ℑ , (9.2)
} )]
1 ˆ( ) 1 ( [ )]
1 ˆ( ) 1 ( [
E{ T
a k
k k k k k k k
J =− x + −x + W x + −x + ℑ , (9.3)
where xˆ(k+1k) is determined by eq. (8.10), W andWare positive semi-definite sym- metric weighting matrices and xˆn(k+1) is an estimate of the nominal output of the sys- tem. A cost function according to eq. (9.2) is introduced to minimize the derivation of the system state from the nominal state xn(k+1), which is the response of the system to the nominal control signal un(k), generated by the unknown desired controller (Filatov and Unbehauen, 1994; Filatov et al., 1995; see also Chapter 6). It is clear that in the case of exact estimates pˆ(k)= p the CE controller, eq. (9.1), generates the nominal control signal. Therefore, the nominal control signal can be determined only after finishing the adaptation when the exact values of the controller or plant parameters are available. The second cost function, according to eq. (9.3), is introduced for the acceleration of the pa- rameter estimation as in eq. (8.9). Minimization of the cost functional in eq. (9.3) in- creases the expectation of the squared one-step-ahead prediction error, which is used in the estimation algorithm [recursive least squares (RLS) and similar algorithms] for up- dating the parameter estimation. To derive a general adaptive dual controller with simpli- fications, the nominal state is defined here using the CE assumption
) ( )) ˆ( ( ) ( )) ˆ( ( ) 1
ˆn(k A p k x k B p k uCE k
x + = + . (9.4)
In the system based on the CE assumption, all random variables are assumed to be equal to their expectations. In the present approach, the CE assumption is applied only to cal- culate the estimate of the nominal system output according to eq. (9.4).
Adopting the bicriterial design method, the dual control law is obtained after the minimization of the cost functions in eqs. (9.2) and (9.3) in the form of eqs. (8.15) and (8.16) with constraints described by eq. (8.11). The non-negative scalar function fk of the covariance matrix can be defined, for example, in one of the forms of eq. (8.13) or (8.14). Therefore, the magnitude of the excitation depends on the scalar parameterηand the function fk of the covariance matrix, that is, the magnitude of the excitation depends on the uncertainty measure, and the size of the domain Ωk, according to eq. (8.11).
Substituting eqs. (8.1) and (9.4) into eq. (9.2) gives
{ ( ) ( ) ( ) ( ) 2 ( ) ( ) (ˆ( )) ( )
E T T T T CE
c k k k k k
Jk = u B pWB p u − u B pWB p u
[ ( ) (ˆ( ))] ( ) } ( )
) ( ) (
2 T k T p p − k k ℑk +c1 k
+ u B W A A p x , (9.5)
where c1(k) is independent ofu(k). As in eqs. (8.21)-(8.23), the covariance matrices
{ ( ) ( ) } (ˆ( )) (ˆ( ))
E )
( T T
W
A k A pWA p k A p k WA p k
P = ℑ − , (9.6)
{ ( ) ( ) } (ˆ( )) (ˆ( ))
E )
( T T
W
B k B pWB p k B p k WB p k
P = ℑ − , (9.7)
9.1. Modification of Certainty-Equivalence Adaptive Controllers 107
and PBWA(k)=E{BT(p)WA(p)ℑk}−BT(pˆ(k))WA(pˆ(k)) (9.8) are introduced. These matrices can be calculated by the covariance matrix of the pa- rameter vector according to eq. (8.12). Then taking the expectation of eq. (9.5), similar as in Appendix D, we obtain
) ( )]
( )) ˆ( ( )) ˆ( ( [ )
( T BW
T
c k k k k k
Jk =u B p WB p +P u
) ( )) ˆ( ( )) ˆ( ( ) (
2uT k BT p k WB p k uCE k
− +2uT(k)PBWAx(k)+c1(k). (9.9) The minimum of the last equation with respect to the control value is provided by the cautious control law
W 1 T B
c(k)=[B (pˆ(k)WB(pˆ(k))+P (k)]− u
)]
( ) ( )
( )) ˆ( ( ) ˆ( (
[BT p k WB p k uCE k −PBWA k x k
⋅ . (9.10)
The second performance index according to eq. (9.3) after the substituting of eq. (9.1) and taking the expectation can be represented in the form
] } tr{
) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( ) (
[ T A T B T BA
a =− x k P k x k +u k P k u k + u k P k xk + WQξ
Jk ,
(9.11) where the covariance matrices PA, PB and PBA are defined as in eqs. (8.21) to (8.23).
The ellipsoid, which defines the amplitude constraint of the excitation according to eq. (8.11), has the form given in eq. (8.24). Taking into account the convexity of the constraint and concavity of the performance index according to eq. (9.11), the necessary conditions for minimization of the second performance index can be written as in eqs. (8.25) and (8.26). Substituting eqs. (9.11) and (8.24) in eq. (8.25), we have analog to eq. (8.27)
0
=
−
− +
−λkPB(k)u(k) Wau(k) Wauc(k) λkPBA(k)x(k) . (9.12) After solving the last equation, the dual control law is finally obtained as
)]
( ) ( )
( [ )]
( [
)
(k Wa kPB k 1 Wauc k kPBA k x k
u = −λ − +λ . (9.13)
The parameter λk has to be chosen such that
0 )) ( ( )]
( ) ( [ )]
( ) (
[u k −uc k TWa u k −uc k − fk P k = , (9.14)
and
0
>
− B( )
a kP k
W λ (9.15)
are satisfied.
Therefore, the dual controller can be computed from the CE control using eqs. (9.10), (9.13), (9.14) and (9.15). The additional parameter ηand weighting matrix Wa should be selected such that the optimal value of the excitation ua(k) is obtained.
These parameters may be determined using computer simulations of the control process.
It should be noted that in the cases of SIMO and SISO systems only the scalar parameter ηhas to be selected because the matrixWa may be assumed as the constant 1 and an on- line solution of eqs. (9.14) and (9.15) to determine the optimal λk is not required. This is shown in detail in the next chapter. The dual controller is derived here independently of the structure of the CE adaptive systems; therefore, it can be used as an additional unit, which transforms the CE control signal to a dual control one. The suggested dual unit can modify and improve the performance of various CE adaptive controllers, for example, the LQG, pole-placement, predictive, generalized minimum-variance, self-tuning controllers, etc. The standard structures of an adaptive control system and an adaptive dual control system (with a dual control unit) are portrayed in Fig. 9.1 and Fig. 9.2, respectively.
CE controller Controller design
algorithm
Plant Estimation
algorithm
u x
) ˆ(k p
Fig. 9.1.Standard adaptive control system based on the CE assumption
CE controller Dual controller Plant
Controller design
algorithm Estimation
algorithm ) ˆ(k ) p
(k P
u x
uCE
Fig. 9.2a.Adaptive dual control system (modification of the CE adaptive system)