The feedforward control principles

Consider the block diagram shown in figure 4.1 describing the feedforward control strategy, where Hf f is the feedforward controller, the velocity vector vm denotes the

measured velocities in the cost function and is calculated from the mobilityYmd which

is related to the vector of disturbance forces fd, and from the mobility Ymc which

related to the vector of control forces fc in the frequency domain as

vm(ω) = Ymd(ω)fd(ω) +Ymc(ω)fc(ω) (4.1)

where,ω is the angular frequency. This explicit dependence onω has been dropped in the following formulation for simplicity. The vector vm has dimension nm×1, where

nm is the number of degrees of freedom to be controlled. The vector fd has dimension

nd×1, wherendis the number of disturbance forces; the vectorfc has dimensionnc×1,

where nc is the number of control forces. The mobility Ymd has dimension nm ×nd

and the mobilityYmc has dimensionnm×nc. Equation 4.1 can be written in the form

vm =vmd+Ymcfc (4.2)

where, vmd is the contribution to the vector of velocities vm due to the disturbance

forces. This vector can be calculated as

vmd = nd X

k=1

Chapter Four 4.1. Introduction to feedforward control where,k = 1, . . . , ndis the number of disturbance forces,ymd(k) is a vector of mobilities

relating each element of the vector vmd to the disturbance force fd(k). The vector

ymd(k) has dimension nm×1 and each column of the matrix Ymd if formed from the

vectors ymd(k) which is calculated from

ymd(k) =jω BmD−1d(k)

for k = 1, . . . , nd (4.4)

where, D is the dynamic stiffness matrix of the whole system that can be calculated from the methods described in chapter 2. d(k) is a distribution vector, where all elements are equal to zero, except the element corresponding to the degree of freedom where the disturbance force (or moment) is applied in the system. The non-zero element of d(k) has value 1. This lead to a normalization of the vector ymd(k) expressed in

metres per second per unit force (m/Ns). Bm is a Boolean matrix that maps the

degrees of freedom m corresponding to each element of the vector vm with all degrees

of freedom in the structure. Bm has dimension nm×ndof, wherenodf is the number

of degrees of freedom in the system. Considering the lattice structure with 33 joints where 6 degrees of freedom are used to describe the motion at each joint, the structure has a total of 6×33 = 198dofs and thedof number of the the linear velocities at joints 31, 32 and 33 are given in table 4.5, where, the dof number for the linear coordinate

x,y and z for a joint is given by (jn×6)−5, (jn×6)−4 and (jn×6)−3, respectively,

where jn in the joint number. The matrix Bm is given by

                        dof 1 ··· 181 182 183 ··· 187 188 189 ··· 193 194 195 vm(1) 0 · · · 1 0 0 · · · 0 0 0 · · · 0 0 0 vm(2) 0 · · · 0 1 0 · · · 0 0 0 · · · 0 0 0 vm(3) 0 · · · 0 0 1 · · · 0 0 0 · · · 0 0 0 vm(4) 0 · · · 0 0 0 · · · 1 0 0 · · · 0 0 0 vm(5) 0 · · · 0 0 0 · · · 0 1 0 · · · 0 0 0 vm(6) 0 · · · 0 0 0 · · · 0 0 1 · · · 0 0 0 vm(7) 0 · · · 0 0 0 · · · 0 0 0 · · · 1 0 0 vm(8) 0 · · · 0 0 0 · · · 0 0 0 · · · 0 1 0 0 · · · 0 0 0 · · · 0 0 0 · · · 0 0 1                         (4.5)

Chapter Four 4.1. Introduction to feedforward control On the right hand side of equation 4.2, Ymc can be written in expanded form as

Ymc =

h

ymc(1) . . . ymc(k) . . . ymc(nc)

i

(4.6) where the vectors ymc(k), for k = 1, . . . , nc, forming each column of Ymc are determi-

nated individually for each one of the control forces fc(k) as

ymc(k) =jω BmD−1

BT_c(k)TT(k)bT

(4.7) where, the term BT

c(k)TT(k)bT

is used in order to take into account the way the actuator applies forces in the structure. An example of an actuator in the lattice structures is shown in figure 4.2. The applications of feedforward control in lattice structures usually make use of piezoactuators which replace the structural members (or part of them). The structural members which before were passive elements are replaced by active members. The passive modifications introduced into the system dynamics can be considered in the system model by adding the dynamic stiffness matrices of the

active members to the dynamic stiffness matrix of the whole system D (according to

the principles of compatibility and force equilibrium), and removing the contribution of the replaced members. These actuators act by reaction in the structure which are represented by two point forces applied at the ends of the actuator in the longitudinal directions. For low frequencies these two forces can be considered equal in magnitude, however, they are applied in opposite directions as shown in figure 4.2. Therefore the use of the vector b given by

b =h −1 0 0 0 0 0 1 0 0 0 0 0

i

(4.8) where the non-zero elements of the vectorbcorrespond to the longitudinal coordinates at the ends of a actuator. The matrix T(k) is the coordinate transformation matrix for the actuator k and Bc(k) is a Boolean matrix to map the ends of the actuator

according to the joint numbering scheme of the structure (which has been discussed in chapter 2). Bm is the same matrix of equation 4.5.

Chapter Four 4.1. Introduction to feedforward control