6.2 Adaptive Model-based Tracking Control
6.2.2 Adaptive Control Implementation
6.2.2.3 Parameter Projection
In addition to the aforementioned stability constraint, the parameter estimates could be bounded by a priori knowledge of the servo-hydraulic system. The gradient algorithm with projection is computed by minimizing the cost function with a convex subset of constraints S defined as:
{
n|( )
0}
boundary of S. To ensure that the estimated parameters are always bounded within S, the gradient adaptive algorithm in Eq. (6-28) can be modified as:( )
parameter space. Then, the resulting parameters are always bounded at each time step.Figure 6-31 illustrates the control scheme in this study. By utilizing this adaptive law, the adaptive control scheme is applied to the model-based feedforward-feedfack tracking control.
6.2.3 Numerical Simulation
This section investigates the feasibility of the proposed adaptive control algorithms by using Simulink. Four cases of actuator compensation schemes are used: (a) feedforward controller (FF), (b) feedforward controller and feedback LQG controller (FF+LQG), (c) feedforward controller with adaptive control algorithm (FF+Adaptive),
controller (FF+Adaptive+LQG). The feedforward controller is derived from the three-pole model with the CDM and linear acceleration extrapolation. In this analysis, the tracking performance among these four schemes is evaluated and discussed.
The feedforward controller is expected to yield good performance; therefore, the control authority of the LQG feedback controller is designed to be moderate. In real-time hybrid testing, amplifying the magnitude over the high frequency range appropriately can extend the tracking capability; however, these high-frequency components may introduce instability to the system due to the measurement noise and modeling errors. Meanwhile, a LQG controller with a high control authority can significantly increase the high-frequency components to the system. These components would generate a lead phase over the high frequency range and lead to an unstable system. As a result, the LQG feedback controller with a moderate control authority is used in this study.
The transfer function of a servo-hydraulic system in Carrion and Spencer (2007) and a perturbed system are adopted to investigate the feasibility of the adaptive control law through the analytical studies. In the analyses, the transfer function of the servo-hydraulic system is given by:
6
p 3 2 6
6.12 10
( ) 384 73770 6.12 10
G s s s s
= ⋅
+ + + ⋅ (6-31)
The three poles of the identified plant are -162.24 ,and -110.88±159.46i, where i2=-1.
It indiates that a3=1.634·10-7, a2=6.275·10-5, a1=0.01204, and a0=1. To consider a modeling error between the identified model and the nominal plant, the nominal plant is assumed as:
6
p 3 2 6
5.2 10
( ) 360 64500 5.2 10
G s s s s
= ⋅
+ + + ⋅ (6-32)
The three poles of the nominal plant are -160, and -110±150i. It shows that a3=1.923·
10-7, a2=6.923·10-5, a1=0.0124, and a0=1. The upper and lower bounds for the parameters a3, a2, and a1 can be determined by the priori knowledge of the identified
transfer function. Subsequently, the parameters can be assumed being bounded within
The parameter projection algorithms can be finally obtained by combining the constraints from Eq.(6-23) and Eq.(6-30) as: unknown parameters. In this simulation, the chosen input is given by
1
( ) sin(2 ) 2cos( ), unit: mm
r t = π ft + πft (6-35)
exciting (PE) condition is hold as the input is SR of order 4. All the numerical simulations are conducted with a time step size of 0.005 second. The MATLAB ode5 (Dormand-Prince) solver is used to do the integration. The adaptive gains for each case are obtained by trial and error.
Table 6-5 shows the RMS error of the tracking performance in each case. It is found that the feedforward controller with the adaptive laws performs better than the feedforward controller only. In addition, the feedback LQG controller further improves the tracking performance of both the feedforward controller with and without the adaptive laws. The effects of adaptive gains are also investigated by trying different values of the adaptive gains. Figure 6-32 shows the estimated a of the ˆ3 simulation when f=5Hz. It is apparent that the parameter projection algorithm can effectively constrain the estimate within the predetermined boundary conditions. In addition, larger adaptive gain results to faster converging time but more severe oscillation. Therefore, the adaptive gains should be tuned carefully depending on the control purpose, i.e., to be conservative or progressive. In the simulations for four cases of actuator compensation schemes, a set of progressive gains is examined. As shown in Figure 6-33, the estimated parameters converge within a small period. This figure shows the parameter estimates from the case (d) with the 30Hz excitation. Each parameter oscillates in the very beginning and then converges to the exact given value.
In practice, the adaptive gains must be carefully tuned to prevent the system from oscillating and damaging the specimen.
In summary, the adaptive control law is shown to improve the tracking performance of the model-based feedforward and feedback control scheme in the numerical simulations.
6.3 Experimental Validation
The experimental validation of the proposed adaptive model-based tracking control strategy for real-time hybrid testing is conducted at the Smart Structures Technology Laboratory (SSTL) at the University of Illinois at Urbana-Champaign (UIUC). The real-time hybrid testing facilities at SSTL has been developed to investigate the structural responses containing velocity-dependent components. In this test setup, a
second generation, large-scale 200 kN MR damper is attached with a 556 kN hydraulic actuator (Phillips and Spencer, 2012), as shown in Fig. 6-34. The MR damper is manufactured by the Lord Corporation with a stroke of ±292 mm. The input currents to the MR damper are operated by a pulse-width modulator, which is comprised of an Advanced Motion Controls model PS2x300W unregulated power supply providing 80 VDC to an Advanced Motion Controls model analog servo-drive.
The input currents are between 0 Amp and 2.5 Amps in this study. The overall setup allows researchers to explore semi-actively controlled structures using a single MR damper.