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_^dp^q(t_k)
and k_pq(t_k) for k = 0, . . . , N₁ where t_{k + 1} - t_k = (t_f - t₀)N₁, a fuzzy neural
model of the mapping x_^dp^q (t_k) → k_pq(t_k) for k = 0, . . . , N₁ 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_^dp^q(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:

1. 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.
2. 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}).
3. 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.
4. Compute the adjusted control law, u′_pq(t_k) = u_pq(t_k) + [D^T·Q·D]^-1Dˆ·
   Q ·x_^dp^q(t_{k + 1}). Afterwards, calculate the desired blending weights k′_pq(t_k).
5. 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.
6. Put t_k ← t_{k + 1} and perform the same procedure at the next time t_{k + 1}.