## 4 Application Scenario: Step-Down Converter
### Comparison with a Taylor Model-Based Solution Approach
For the comparison with the proposed solution approach described in Section 3, the outcome of verifyode is investigated for the following solver settings:  
1. with default settings (order 12) for all options [43] except for the step size parameters and the choice of a QR preconditioning \[\text{verifyodeset('h0', 1e-4, 'h_min', 1e-6, 'precondition', 1);}\]
2. with order 10, manually specified small tolerances, and QR preconditioning \[\text{verifyodeset('order', 10,'shrinkwrap', 0, 'precondition', 1, 'blunting', 0,... 'h0', 1e-4, 'h_min', 1e-6, 'loc_err_tol', 1e-11,'sparsity_tol', 1e-20);}\]
3. with order 30, manually specified small tolerances, and QR preconditioning \[\text{verifyodeset('order', 30,'shrinkwrap', 0, 'precondition', 1, 'blunting', 0,... 'h0', 1e-4, 'h_min', 1e-6, 'loc_err_tol', 1e-11,'sparsity_tol', 1e-20);}\]
4. with order 10, manually specified small tolerances, and curvilinear preconditioning \[\text{verifyodeset('order', 10,'shrinkwrap', 0, 'precondition', 3, 'blunting', 0,... 'h0', 1e-4, 'h_min', 1e-6, 'loc_err_tol', 1e-11,'sparsity_tol', 1e-20);}\]
5. with order 30, manually specified small tolerances, and curvilinear preconditioning \[\text{verifyodeset('order', 30,'shrinkwrap', 0, 'precondition', 3, 'blunting', 0,... 'h0', 1e-4, 'h_min', 1e-6, 'loc_err_tol', 1e-11,'sparsity_tol', 1e-20);}\]  
The simulations have shown that the **cases 1-3** complete successfully up to the point \(t=75\,\text{ms}\) (the same final time instant as in Figures 3 and 4, comprising 15 full duty cycles), while the state enclosures in the **cases 4-5**, in which the QR preconditioning was replaced by the curvilinear one blow up and break down shortly after \(t=25\,\text{ms}\) because the integration step size falls below the specified threshold. Setting the step size parameters to the default values as described in [43] does not resolve this issue. Activating the options  
proposed approach but that the interval bounds for the capacitor's voltage \(u_{\text{C}}\) are much wider if the Taylor model solver is employed. Especially the fact that this simulation is practically not able to predict the correct sign of the voltage \(u_{\text{C}}\) is counterproductive for practical applications where the simulation results may be employed to forecast the magnitude and direction of power flows towards a consumer (represented by the resistance \(R_{\text{S}}\) in the considered scenario). The new solution approach (without counting the offline parameter splitting) is faster by a factor of at least 100 than the alternative solver despite the use of multiple parameter intervals in the simulation if both are executed on the same computer using Matlab 2019b. **Remark 3**. : _Due to the linearity of the point-valued bounding systems produced by the new Algorithms 1-4, it is even possible (in the case when the dimension \(n\) is sufficiently small) to compute the worst-case state enclosures efficiently in closed form by using techniques for symbolic formula manipulation. The existence of such symbolic expressions can be exploited in future work to design fast model-predictive control strategies for uncertain dynamic systems._  
**Remark 4**. : _Due to cooperativity of the transformed system models--which are obtained with the approach presented in this paper--they are less affected by overestimation resulting from the wrapping effect, even if simulations were carried out for the corresponding interval boxes of the uncertain parameter and not for the decoupled bounding systems as suggested. Therefore, the obtained changes of coordinates could be employed also by general-purpose verified ODE solvers. This represents a further means to reduce overestimation in addition to the typically employed options such as QR preconditioning [11; 34]._