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and the estimated value is calculated by

where F(p) = 1/Tfp +1) is a low pass filter with a time constant Tf and p is the time differentiation operator.

Therefore, for the reference acceleration of vehicle i>0ef, the control input with disturbance compensation becomes

Ub = MBBvr0ei + fL, and the configuration of the basic disturbance compensation is shown in Figure 3(a).

(a) Basic disturbance compensation

(b) Modified disturbance compensation Fig. 3. Configuration of disturbance compensation

Next, the basic disturbance compensation is modified. For Mbm , NBB, NBM and fB in Equation (21) the nominal models using constant values of added mass, added moment of inertia and drag coefficient are defined as Mbm , N BB, NBM and fB, respectively. Then the vehicle control input with these nominal models and the reference acceleration aref becomes

Ub = mbbVoef + ft where ft = MBM0ref + NBBVo + NBM0 + fB .

From Equations (23) and (26) the modelling error with respect to the hydrodynamics can be defined as fE = uB - MBBV0 - ft , and the estimated value is calculated by fE = F(P)(uB - MBBV0 - ft ) ■

Therefore, the control input with disturbance compensation becomes

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