**12.2.5. Online Adaptation of Mode Transition Controllers
**

In this section, an adaptation scheme is proposed for the online customization

of mode transition controllers designed offline via the method of

blending local mode controllers. The control objective is to adapt the

blending matrices such that the plant output vector tracks the output vector

of a desired transition model. In order to apply the discrete-time adaptation

scheme to the continuous-time system, it is assumed that the sample rate has

been appropriately selected. Figure 12.2 shows the configuration for indirect

adaptive mode transition control. The adaptation scheme is composed of five

components: the *desired transition model, the active plant model, the plant*

adaptation mechanism, the active controller model, and the controller adaptation

mechanism.

| is incorporated into the consequent part of the fuzzy neural |

model. Afterwards, the *active plant model *is adapted online via the plant

adaptation mechanism.

**Plant Adaptation Mechanism**

The active plant is adapted online to account for plant variations on a

real-time basis. At time instant *t*_{k}, the adaptation of the active plant model is

accomplished by performing structurerparameter learning on the basis of the

current inputroutput data {(*x*_{}_{pq}(*t*_{k}),*u*_{pq} (*t*_{k})) → *x*_{}_{pq} (*t*_{k + 1})}. Since a desired

output is not known ahead of time when performing structure learning on the

incoming input (*x*_{}_{pq}(*t*_{k}),*u*_{pq}(*t*_{k}), the strongest fired rule’s consequence is

used. Likewise, the strongest fired rule’s consequent parameters are used to

initialize the consequent parameters of the newly formed rule since the

required linear model information is not known ahead of time.

**Active Controller Model**

The active controller model is the mode_{p} to mode_{q} controller. Given *x*_{d}_{pq}(*t*_{k})

and *k*_{pq}(*t*_{k}) for *k* = 0, . . . , *N*_{1} where *t*_{k + 1} - *t*_{k} = (*t*_{f} - *t*_{0})*N*_{1}, a fuzzy neural

model of the mapping* x*_{d}_{pq} (*t*_{k}) → *k*_{pq}(*t*_{k}) for *k* = 0, . . . , *N*_{1} is determined by

offline training as suggested in Section 2.4. Afterward, the blending weights

of the active controller model are adapted online using the *controller adaptation*

mechanism.

**Controller Adaptation Mechanism**

Let ACM and APM denote the active controller model and the active plant

model, respectively. Let *u*_{pq}(*t*_{k}) be the currently developed control input by

the ACM which corresponds to *x*_{pq}(*t*_{k}). Suppose that *x*_{d}_{pq}(*t*_{k}) represents the

desired trajectory at *t*_{k} provided by the desired transition model. Let *u'*_{pq}(*t*_{k})

denote the control that is to be determined such that it is the weighted

least-square (WLS) optimal control value at *t*_{k}. The plant output vector

corresponding to *u'*_{pq}(*t*_{k}) is denoted as *x'*_{pq}(*t*_{k}).

The optimal control input increments *u'*_{pq}(*t*_{k}) are determined such that the

following performance index is minimized:

The following steps implement the controller adaptation algorithm:

- Apply ACM to
*x*_{pq}(*t*_{k}) and produce the current initial estimate of the

control input to *u*_{pq}(*t*_{k}).. Since it is possible for the fuzzy neural model

of the blending weights not to be sufficiently activated by *x*_{pq}(*t*_{k}),

structure learning with local model information is performed at this

stage. - Input
*u*_{pq}(*t*_{k}) and *x*_{pq}(*t*_{k}) to APM and produce *x*_{pq}(*t*_{k +}_{ 1}). Calculate

*x*_{pq}(*t*_{k +}_{ 1}) using the predictive one-step-ahead output *x*_{pq}(*t*_{k +}_{ 1}) in place

of the unavailable output *x*_{pq}(*t*_{k +}_{ 1}). - The true control sensitivity matrix
*D*(*x*_{pq}(*t*_{k})*u*_{pq}(*t*_{k})) is approximated

via the APM’s incremental control matrix. When the APM is not to be

sufficiently activated by (*x*_{pq}(*t*_{k})*u*_{pq}(*t*_{k})), the control sensitivity information

contained in the strongest fired rule’s consequence is used. - Compute the adjusted control law,
*u′*_{pq}(*t*_{k}) = *u*_{pq}(*t*_{k}) + [*D*^{T}·*Q*·*D*]^{-1}*D*ˆ·

*Q* ·*x*_{d}_{pq}(*t*_{k + 1}). Afterwards, calculate the desired blending weights *k′*_{pq}(*t*_{k}). - Train ACM to capture desired blending weights
*k′*_{pq}(*t*_{k}) given current

input *x*_{pq}(*t*_{k}). Note that parameter learning with local model information

is used to train the ACM. - Put
*t*_{k} ← *t*_{k + 1} and perform the same procedure at the next time *t*_{k + 1}.

**12.2.5. Online Adaptation of Mode Transition Controllers
**

In this section, an adaptation scheme is proposed for the online customization

of mode transition controllers designed offline via the method of

blending local mode controllers. The control objective is to adapt the

blending matrices such that the plant output vector tracks the output vector

of a desired transition model. In order to apply the discrete-time adaptation

scheme to the continuous-time system, it is assumed that the sample rate has

been appropriately selected. Figure 12.2 shows the configuration for indirect

adaptive mode transition control. The adaptation scheme is composed of five

components: the *desired transition model, the active plant model, the plant*

adaptation mechanism, the active controller model, and the controller adaptation

mechanism.

| is incorporated into the consequent part of the fuzzy neural |

model. Afterwards, the *active plant model *is adapted online via the plant

adaptation mechanism.

**Plant Adaptation Mechanism**

The active plant is adapted online to account for plant variations on a

real-time basis. At time instant *t*_{k}, the adaptation of the active plant model is

accomplished by performing structurerparameter learning on the basis of the

current inputroutput data {(*x*_{}_{pq}(*t*_{k}),*u*_{pq} (...

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