Consider a linear classically damped structural system whose response x(t) is de- scribed by the matrix differential equation
M x00(t) +Cx0(t) +Kx(t) =g(t) ; x(0) = 0, x0(0) = 0, (2.1) where M is the n by n positive definite symmetric mass matrix, C is the n by n symmetric damping matrix, and K is the n by n positive definitive symmetric stiffness matrix. The forcing function is given by then-vectorg(t).
Since the system is classically damped, it can be transformed to the diagonal system
z00(t) + Ξz0(t) + Λz(t) =TTM−1/2g(t) ; z(0) = 0, z0(0) = 0, (2.2) where Ξ = diag(2ξ1,2ξ2, . . . ,2ξn), Λ = diag(λ21, λ22, . . . , λ2n), and the matrix T = [tij] is the orthogonal matrix of eigenvectors ofM−1/2KM−1/2. Taking the Laplace transform of (2.2) and solving, we get
˜
x(s) =M−1/2TΞT M−1/2g(s)˜ , (2.3) where the wiggles indicate the transformed functions, and Ξ is given by
Ξ = diag((s2+ 2ξ1s+λ21)−1,(s2+ 2ξ2s+λ22)−1, . . . ,(s2+ 2ξns+λ2n)−1). The open loop poles are given by the zeros of the equations
s2+ 2ξqs+λ2q = (s−γ+q)(s−γ−q) = 0, q= 1,2, . . . , n . (2.4) The poles have been denoted by γ±q, the sign indicating the sign in front of the radical of the quadratic equations given in (2.4). In this paper we assume the system to be generic and all poles to have multiplicity one (no repeated poles).
The feedback control uses a linear combination of p responses xsk(t), k = 1,2, . . . , p, which are fed to a controller. In general, the responses are time-delayed byTsk(t). The actuator will apply a force to the system, affecting the j-th equa- tion of (2.1). When j ∈ {sk : k = 1,2, . . . , p} the sensors and the actuator are collocated, and ifj /∈ {sk : k= 1,2, . . . , p}the sensors and the actuator are non- collocated (dislocated). The control methodology applied to a shear frame building structure is shown in Figure 2.1.
Denoting the non-negative control gain byàand the controller transfer function byàτc(s), the closed loop system poles are given by
A(s)˜˜ x(s) = [M s2+Cs+K]˜x(s) = ˜g(s)−àτc(s) Xp
k=1
askx˜sk(s) exp[−sTsk]ej, (2.5) whereej is the unit vector with unity in thej−th element and zeros elsewhere. The numbers ask are the coefficients of the linear combination of the responses fed to the controller. Moving the last term on the right of (2.5) to the left gives
A˜1(s)˜x(s) = ˜g(s), (2.6) where A˜1(s) is obtained by adding àτc(s)askexp[−sTsk] to the (j,sk)-th element of ˜A(s), fork= 1,2, . . . , p.
Figure 2.1 Shear frame building structure and control methodology.
The closed loop poles are then given by the relation det[ ˜A1(s)] = det[ ˜A(s)]
(
1 +àτc(s) Xp
k=1
askexp[−sTsk]˜x(δ)sk,j(s) )
, (2.7)
where ˜x(δ)sk,j(s) is the Laplace transform of the open loop response to an impulsive force applied at nodejat timet= 0. The open loop response to the impulsive force is given by ˜x(δ)sk,j(s) =
Xn
i=1
t(Msk,j)t(M)i,j
s2+ 2ξis+λ2i, wheret(M)sk,r = Xn
u=1
m−1/2sk,utu,r, with m−1/2i,j being the (i,j)-th element ofM−1/2andT(M)=M−1/2T = [t(Mi,j)].
The following set of conditions will be referred to as condition set C1. Given that the open loop poles of the system areγ±q, we have
(1) τc(γ±q)6= 0 for q= 1,2, . . . , n , (2)
Xp
k=1
askexp[−γ±qTsk]t(M)sk,q6= 0 for q= 1,2, . . . , n , (3) t(Mj,q)6= 0 for q= 1,2, . . . , n .
The first condition means that the open loop poles of the system are not also zeros of the controller transfer function. The second condition is a generalized observabil- ity condition which requires that all mode shapes are observable from the summed,
time-delayed sensor measurements. The third condition is a controllability condi- tion which requires that the controller cannot be located at any node of any mode of the system. Observe that if any of the three conditions are not satisfied for one open loop pole γq, then by (2.7) the open loop poleγq is also a closed loop pole.
However, if C1 is satisfied, we have the following result.
Result 2.1: When the open loop system has distinct poles and condition set C1 is satisfied, the open loop and the closed loop systems have no poles in common.
If condition set C1 is satisfied, then by Result 2.1 and (2.7), the closed loop poles of the system are given by the values ofsthat satisfy the equation
1 +àτc(s) Xp
k=1
Xn
i=1
askexp[−sTsk]
t(M)sk,jt(M)i,j s2+ 2ξis+λ2i
= 0. (2.8)
In general, equation (2.8) may have an infinite number of zeros due to the time delay term. As the parameter àis varied, we obtain the root locus of the closed loop poles. The poles that are on a root locus that starts at an open loop pole of the structural system will be called system poles. The poles that do not originate at an open loop pole of the system will be called non-system poles. To simplify matters, some of the following results deal with the system poles only. The simplification of dealing with the system poles only, allows us to obtain bounds on the gain and the time delay to guarantee stability. These results should be viewed with caution since in general there are an infinite number of poles, and as is shown in Section 4, a non-system pole may determine what is the maximum gain for stability for some systems. It will be made clear when our results apply to all the poles considered, and when only to the system poles.
The following result and all of its consequences apply to the case when only system poles are considered. Multiplying (2.8) bys2+2ξrs+λ2r, then differentiating with respect toàand letting s→γ±r=−ξr±i(λ2r−ξr2)1/2 andν→0, we obtain
ds dà
¯¯
¯¯ à→0
s→γ±r
=− τc(γ±r) 2±i(λ2r−ξ2r)1/2
" p X
k=1
askexp[−γ±rTsk]t(Msk,r)
#
t(M)j,r . (2.9)
Result 2.2: A sufficient condition for the closed loop system to remain stable for infinitesimal gains is that
Re
ds dà
¯¯
¯¯ à→0
s→γ±r
<0, r= 1,2, . . . , n . (2.10) Again, Result 2.2 applies only when system poles are considered, and in general the result may not be true when non-system poles are also considered.
We now particularize the controller to be of the proportional, integral and deriva- tive (PID) form. The transfer function of the controller is then given by
τc(s) =K0+K1s+K2
s , with K0, K1, K2≥0.
The term K0 corresponds to proportional control, the term K1s corresponds to derivative control and the term K2
s corresponds to integral control. Next we specialize results for the undamped case.