# Comparison of Disturbance Compensators for a Discrete-Time System with Parameter Uncertainty
## 4 Comparison and Results
### Simulation Results and Comparison  
Figure 1a-r exhibits the simulation results of the disturbance compensators, parameter uncertainties and compensation errors, which are generated by the N-steps delay estimation (NSDE), the one-step delay estimation (OSDE), the two-step delay estimation (TSDE), the SMC disturbance compensator (SDC), the decoupled disturbance compensator (DDC),and the discrete-time disturbance observer (DTDO). The switching functions \(s(k)\) are illustrated in Figure 2a-n utilizing the seven different DSMC controllers, i.e., \(u_{i}(k)\), \(i=1,2,\,\ldots,7\). It is observed from Figures 1a-c and 2a,b that the DSMC system is unlikely to be stable no matter how the control parameters of \(u_{1}(k)\) are adjusted. The reason why the NSDE and \(u_{1}(k)\) generate such poor results is mainly due to the unsmooth and fast varying parameter uncertainty \(\Delta\Phi x(k)\). Since \(N=5\) in \(u_{d1}(k)\), NSDE requires the fifth-order difference of parameter uncertainty to estimate the current value of \(\Delta\Phi x(k)\). A chattering phenomenon exists in \(x(k)\). Hence, Assumption 1 cannot hold, and the fifth-order difference of parameter uncertainty may be quite large to cause system instability. When \(N\) decreases, the requirement for the smoothness of the disturbance signal also is reduced. Comparing Figures 1d-i and 2c-f, it is seen that the OSDE and \(u_{2}(k)\) achieve a smaller compensation error and a smaller switching function, i.e., \(\max\{|s(k)|\}=0.011\), in the steady state than the TSDE and \(u_{3}(k)\), which is attributed to the lower difference order, i.e., \(N=1\). Regarding the unsmooth and fast varying parameter uncertainty, a larger \(N\) can cause system oscillation and even instability. Additionally, it is observed from Figure 1d-f,j-l and Figure 2c,d,g,h that the results of SDC and \(u_{4}(k)\) are the same as those of the OSDE and \(u_{2}(k)\), i.e., \(\max\{|s(k)|\}=0.011\), which proves that the SDC \(u_{d4}(k)\) is equivalent to the OSDE \(u_{d2}(k)\). The simulation results of DDC/\(u_{5}(k)\) and DTDO/\(u_{6}(k)\) are depicted in Figure 1m-o, Figure 2i,j and Figure 1p-r, Figure 2k,l, respectively. It is found that the two methods further reduce the compensation error and the switching function, i.e., \(\max\{|s(k)|\}=0.0095\), in the steady state than in the OSDE and \(u_{2}(k)\). This is mainly owing to the additional control parameters. It is notable, however, that the DSMC controller without disturbance compensators, i.e., \(u_{7}(k)\), achieves the smallest switching function, i.e., \(\max\{|s(k)|\}=0.008\) in Figure 2m,n, in the steady state than the other six methods. Hence, it is reasonable to deduce that these disturbance compensators have negative effects on the DSMC system with parameter uncertainty. Plotted in Figures 1 and 2, the disturbance compensators, i.e., OSDE, SDC, DDC, and DTDO, estimate the real parameter uncertainty by its one step-delayed value. To achieve smooth and slowly varying disturbance, the impact of one step-delay on the estimation can be ignored and the compensation error may be small. Regarding unsmooth and fast varying parameter uncertainty \(\Delta\Phi x(k)\), however, one step-delay has a large impact on the estimation and the compensation error is quite large. Due to the existence of chattering, the estimated values of these disturbance compensators are opposite to the values of the actual parameter uncertainty in the steady state. This is the reason why these disturbance compensators have negative effects on a DSMC system with parameter uncertainty.