In most of cases, the coupling force depends mainly on the displacement and the calculated displacement can be used as the input of the data model. If the coupling force depends mainly on acceleration (for example, a specimen with a large mass under a dynamic loading situation), the input of estimation may be the computed acceleration. However, this is not a different case because the compensation does not use any assumption on physical quantity and various signals can be chosen for the input. Any signal that has a strongly dynamical relation to the coupling force can be used as the input of the error fore compensation. It is important that the input of the data model be a computed quantity without any noise.
With the methodology described above, the model of the estimation for the error force compensation can be briefly described as follows: The estimation is based on ARMAX model (as Eqs. (9.1) - (9.4)) and recursive estimation to estimate the compensating force ∆fi+1.
1 1 =ϕ ( +1)θ
∆fi+ T i (9.1)
−∆ −∆ −∆ + + − − −
=
+1) − ... − ( 1) () ... ( 1 ) () ( 1) ... ( ) (i fi fi1 finu xi xi xi nu ei ei ei nu
ϕ T (9.2)
=
u u
u n n
n b b b c c c
a a
a1 2 ... 1 2 ... 1 2 ...
θ1 T (9.2)
ui
i
x( +1)= (9.4)
where x(i+1) and ∆fi+1 are relatively the input and the output of the data model at step (i+1), ϕ(i+1) is the vector of regression variables, nu is the order of the data model, θ1is the vector of adaptive parameters. It is worth noticing that the input x(i+1) can be
also set as the explicit displacement u0i+1, which is known at the beginning of the next step.
Both recursive least square and recursive pseudo-linear regressions can be used to estimate parameter of the data model. It is not much different between two recursive methods, exceptionally a small difference on the terms p and R and a slight difference on the convergence speeds (Ljung 1999).
In this implementation, the recursive algorithm as Eqs. (6.84) - (6.88) with forgetting factor λ is used to estimate the parameters. Some discussions on the initial condition and the forgetting factor will be expressed in section 9.1.2.
Figure 9.1: Algorithm for substructure control with error force compensation
The compensation mechanism is embedded in the sub structure algorithm as depicted in Figure 9.1. In order to start initial variables of the recursive mechanism, the error
Check and calculate θ1i +1
Check and estimate ∆f i+1
i = n ?
Calculate variables of at new step
j = 1
Apply new displacement to actuator
Measure coupling force fci+1, j
j = ksub?
Calculate velocity, acceleration, error at the end of step
j = j+1
i = i+1 Initations
i = 0 begin
end i ≥ i0 ?
calculate θ1i +1
Calculate displacement at sub step ui+1, j i > i0 ?
estimate ∆f i + 1
∆f i + 1 = 0
Check and estimate ∆fi+1
Check and calculate θ1i+1
no no
no
no
force compensation will not work at its initial state with at i0 first steps. At the end of the step i0, the estimator starts to estimate parameter. At beginning of each next step (when i >i0), the estimator calculates the compensating force ∆fi+1 to feed into the substructure algorithm. When the step (i+1) is finished, based on the current parameter
i
θ1 and the residual value e(i+1) at step i, the new parameter θ1i+1 is estimated by the recursive estimator.
9.1.2 Discussion on accuracy and convergence
The accuracy of error force compensation depends on accuracy of the data model and the convergence of the system estimation. In order to obtain highly accurate result, the data model should be chosen properly.
The input of data model should be chosen in order to have a strong relationship between input and output data. Since the output is the compensating force at the coupler, the input could be the computed displacement, acceleration or velocity at the coupler. Among these possible inputs, the term that has most strong sensitivity to the coupling force should be chosen as the input. For example, when the coupler is a spring, a displacement input should be used. If the coupler is stiffly connected between two masses, an acceleration input should be used.
The order of the data model should be large enough but it must not exceed the maximum order of the data system. When the order of model is too low, the data model cannot represent well the dynamic system. If the order of data model is close to the maximum order, the data model can represent well the dynamic of the system. If the order exceeds the maximum order, a parameter of the model can be specified by more than one state of the system, thus the system identification cannot work properly.
Regarding stability, the convergence of recursive algorithm is the important point on convergence and stability of the method in RTST.
As long as the estimator is well convergent, the error force will be compensated effectively. As the result, the substructure solution of the substructure test will be more stable and more accurate than that of a test without error force compensation. Thus, the convergence is only the critical condition for the stability of the compensation method in RTST. According to Ljung (1999), the convergence condition of the recursive estimation for the ARMAX model is specified as Eq. (9.5).
ω
∀
>
−
1 1 0
Re (9.5)
where Co is the polynomial associated with noise of the true system.
Because the real noise is unknown, the convergence condition cannot be guaranteed generally. However, in the error force compensation, the condition Eq. (9.5) means that contribution of the real noise into the output (as the compensating force) should be restricted as a small level. From this, the following guideline should be considered in RTST with the error force compensation.
With an existing measuring system, the noise of force measurement can be evaluated and the sub-step time interval could be chosen appropriately. When the sub-step time interval is too small, the compensating force will be very small and the convergence condition may not be satisfied. However, substructure tests with large numbers of sub steps usually have very good stability event without any force compensation.
In some substructure tests under earthquake loads, when the test is starting with very low level of load, the values of compensating force are small. At this testing state, the estimation does not work well because of the noise effect. That is the reason why the test in section 9.4.3 uses this force estimation only after starting the test about 1.5 second.
One concerning issue is the convergence speed. The convergence speed depends on the forgetting factor and the initial condition.
The forgetting factor λ is usually chosen from 0.96 to 0.99 in most of application (Sửderstrửm et al. 1989). When the forgetting is 1, the result of recursive identification will be in the same with that of off-line identification for the whole data history. With forgetting factor λ < 1, the parameter is estimated based on the recent data. For viewing of the effectiveness of forgetting factor λ, the contribution value (measured by λn) of a data at n steps backward in the estimation can be seen in Figure 9.2. For a substructure test of a fast time-varying system, a small value of λ is recommended.
The initial value for p (or R) of the recursive identification can be set as Eq. (6.90) or Eq. (6.96). Eqs. (6.87) and (6.88) show that if the value of p is small, the convergence speed will be slow and the estimation will be less sensitive to noise. In the contrary, if the value of R is small, the convergence speed will be fast (see Eq. 6.94). It is recommended that small values of p should be used first; then, if the convergence is too slow, the value for p can be increased (Sửderstrửm et al. 1989).
Figure 9.2: Contribution of a datum in the past to the cost function of the estimation
9.1.3 Discussion on features and applications
Some features relating to application of the error force compensation based on estimation are given in the following.
Firstly, the compensation uses data model and on-line system identification to estimate parameter of the data model and estimate the compensating force. This is an advantage because it is not needed to perform any separate test for system identification. Unlike the other force estimations based on physical assumptions, this estimation does not use any physical assumption. Therefore, there is no restriction on what kind of substructure is being tested with this force compensation method.
Secondly, the method has an advanced mechanism to estimate the compensating force and minimize the error force. The residual error including error of the estimation and noise is fed into the data model and the dynamic of noise can be handled. With this property, noise that does not correlate to the dynamics of the coupling fore can be removed effectively. However, special noises that have strongly correlation with the coupling force could not be removed effectively.
Thirdly, the compensation has adaptive capacity for time-varying systems and nonlinear systems. The compensation is based on online system identification and the parameter of the data model is updated from the current state of the test system. With on-line system identification using a time weight as the forgetting factor λ, the parameter of the data model can be adjusted and the dynamic of the data model can approach the dynamic of the test system. In addition, certain nonlinear systems can be treated by using nonlinear model such NAMARX, NAMAX, etc. and the regression
0.0 1.0
0 5 10 15 20
The numbe of step back to the past
Contribution of sample
0.999 0.99 0.98 0.96 0.95
λn
λ
The number of steps backward n