# Transformation of Uncertain Linear Systems with Real Eigenvalues into Cooperative Form: The Case of Constant and Time-Varying Bounded Parameters

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## Abstract

**:**

## 1. Introduction

`verifyode`[34]. The application scenario considered in Section 4 is an electric step-down converter with a predefined duty cycle. Finally, conclusions and an outlook on future work are given in Section 5.

## 2. Cooperativity-Enforcing Similarity Transformations

#### 2.1. Special Case: Linear and Quasi-Linear Systems with Bounded Parameters

#### 2.2. Illustrating Example

#### 2.3. Purely Real Eigenvalues

**Remark**

**1.**

#### 2.4. Mixed and Conjugate-Complex Eigenvalues

- mapping the uncertainty directly into the location of the eigenvalues turns the transformation into an interval-valued expression.

`verifyeig`) are included in the Matlab toolbox IntLab [40].

**Remark**

**2.**

## 3. Novel Subdivision-Based State-Space Transformation

Algorithm 1: Splitting-based transformation |

Algorithm 2: Stability analysis |

Algorithm 3: Subdivision procedure A |

Algorithm 4: Subdivision procedure B |

## 4. Application Scenario: Step-Down Converter

`ode23`with standard tolerance setting and the maximum step size ${10}^{-5}$ was used. This solver is sufficiently accurate and more effective than, for example,

`ode45`for the moderately stiff system model under consideration. For this application, the resulting approximation errors are also several orders of magnitude smaller than the absolute values of the corresponding states. This was checked by a computation of the corresponding solutions with the help of symbolic formula manipulation in Matlab. In addition to the presentation of the transformation approaches introduced in this paper, a comparison with the verified ODE solver

`verifyode`is illustrated. This solver is also included in IntLab.

#### 4.1. Modeling

#### 4.2. Simulation-Based Comparison of Two Cooperativity-Enforcing Similarity Transformations

`verifyeig`routine:

#### 4.3. Comparison with a Taylor Model-Based Solution Approach

`verifyode`[34,43] is employed to obtain a representative comparison.

`verifyode`does not at all succeed in computing guaranteed state bounds. This results from a division by zero when substituting the Taylor model representations of both uncertain parameters into the system model (46). This problem is independent whether identical Taylor model orders are chosen for all four state variables or if the Taylor model order for the interval parameters is reduced to its minimum value 1 as described in [43].

`intval`). Then, manually specifying the initial step size of the solver as ${h}_{0}={10}^{-4}$, the minimum step size as ${h}_{min}={10}^{-6}$, the Taylor model orders as either 10, 12 (the default setting), or 30, performing either a QR preconditioning or a curvilinear preconditioning with or without blunting (a strategy of widening the angles between the column vectors of an ill-conditioned preconditioning matrix to reduce overestimation [44]) and/or shrink wrapping (aiming at an elimination of additive error intervals and incorporating them in the Taylor model coefficients [45]), it is possible to simulate the system model.

`verifyode`is investigated for the following solver settings:

- case 1
- with default settings (order 12) for all options [43] except for the step size parameters and the choice of a QR preconditioning
`verifyodeset(’h0’, 1e-4, ’h_min’, 1e-6, ’precondition’, 1);` - case 2
- with order 10, manually specified small tolerances, and QR preconditioning
`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);` - case 3
- with order 30, manually specified small tolerances, and QR preconditioning
`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);` - case 4
- with order 10, manually specified small tolerances, and curvilinear preconditioning
`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);` - case 5
- with order 30, manually specified small tolerances, and curvilinear preconditioning
`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);`

**cases 1–3**complete successfully up to the point $t=75\phantom{\rule{0.166667em}{0ex}}\mathrm{ms}$ (the same final time instant as in Figure 3 and Figure 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 $t=25\phantom{\rule{0.166667em}{0ex}}\mathrm{ms}$ 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 for blunting and/or shrink wrapping did not influence the solutions for the considered application. Hence, they are not investigated further in this paper. Therefore, the following graphical comparison in Figure 6, Figure 7 and Figure 8 will be restricted to the

**cases 1–3**.

`verifyode`are tighter than those of the proposed approach but that the interval bounds for the capacitor’s voltage ${u}_{\mathrm{C}}$ 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 ${u}_{\mathrm{C}}$ 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 ${R}_{\mathrm{S}}$ 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.**

**Remark**

**4.**

## 5. Conclusions and Future Work

`verifyeig`, as long as the assumption on limited parameter variabilities, detailed in this paper, are satisfied.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Raïssi, T.; Efimov, D.; Zolghadri, A. Interval State Estimation for a Class of Nonlinear Systems. IEEE Trans. Autom. Control
**2012**, 57, 260–265. [Google Scholar] [CrossRef] - Efimov, D.; Raïssi, T.; Chebotarev, S.; Zolghadri, A. Interval State Observer for Nonlinear Time Varying Systems. Automatica
**2013**, 49, 200–205. [Google Scholar] [CrossRef][Green Version] - Mazenc, F.; Bernard, O. Asymptotically Stable Interval Observers for Planar Systems With Complex Poles. IEEE Trans. Autom. Control
**2010**, 55, 523–527. [Google Scholar] [CrossRef] - Angeli, D.; Sontag, E. Monotone Control Systems. IEEE Trans. Autom. Control
**2003**, 48, 1684–1698. [Google Scholar] [CrossRef][Green Version] - Smith, H.L. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems; Mathematical Surveys and Monographs, American Mathematical Soc.: Providence, RI, USA, 1995; Volume 41. [Google Scholar]
- Hirsch, M. On the Nonchaotic Nature of Monotone Dynamical Systems. Eur. J. Pure Appl. Math.
**2019**, 12, 680–688. [Google Scholar] [CrossRef] - Rauh, A.; Kersten, J.; Aschemann, H. Interval and Linear Matrix Inequality Techniques for Reliable Control of Linear Continuous-Time Cooperative Systems with Applications to Heat Transfer. Int. J. Control
**2020**, 93, 2771–2788. [Google Scholar] [CrossRef] - Raïssi, T.; Efimov, D. Some Recent Results on the Design and Implementation of Interval Observers for Uncertain Systems. at-Automatisierungstechnik
**2018**, 66, 213–224. [Google Scholar] [CrossRef] - Nedialkov, N.S. Interval Tools for ODEs and DAEs. In Proceedings of the 12th GAMM-IMACS Intl. Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006, Duisburg, Germany, 26–29 September 2006; IEEE Computer Society: Duisburg, Germany, 2007. [Google Scholar]
- Lohner, R. On the Ubiquity of the Wrapping Effect in the Computation of the Error Bounds. In Perspectives on Enclosure Methods; Kulisch, U., Lohner, R., Facius, A., Eds.; Springer–Verlag: Wien, NY, USA, 2001; pp. 201–217. [Google Scholar]
- Lohner, R. Enclosing the Solutions of Ordinary Initial and Boundary Value Problems. In Computer Arithmetic: Scientific Computation and Programming Languages; Kaucher, E.W., Kulisch, U.W., Ullrich, C., Eds.; Wiley-Teubner Series in Computer Science: Stuttgart, Germany, 1987; pp. 255–286. [Google Scholar]
- Kapela, T.; Mrozek, M.; Wilczak, D.; Zgliczynski, P. CAPD::DynSys: A Flexible C++ Toolbox for Rigorous Numerical Analysis of Dynamical Systems. Commun. Nonlinear Sci. Numer. Simul.
**2020**, 105578. [Google Scholar] [CrossRef] - Berz, M.; Makino, K. COSY INFINITY Version 8.1. User’s Guide and Reference Manual; Technical Report MSU HEP 20704; Michigan State University: East Lansing, MI, USA, 2002. [Google Scholar]
- Hoefkens, J. Rigorous Numerical Analysis with High-Order Taylor Models. Ph.D. Thesis, Michigan State University, East Lansing, MI, USA, 2001. Available online: http://www.bt.pa.msu.edu/cgi-bin/display.pl?name=hoefkensphd (accessed on 14 January 2021).
- Alexandre dit Sandretto, J.; Chapoutot, A. Validated Explicit and Implicit Runge–Kutta Methods. Reliab. Comput.
**2016**, 22, 79–103. [Google Scholar] - Mullier, O.; Chapoutot, A.; Alexandre dit Sandretto, J. Validated Computation of the Local Truncation Error of Runge–Kutta Methods with Automatic Differentiation. Optim. Methods Softw.
**2018**, 33, 718–728. [Google Scholar] [CrossRef][Green Version] - Rauh, A.; Auer, E.; Hofer, E.P. ValEncIA-IVP: A Comparison with Other Initial Value Problem Solvers. In Proceedings of the 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006, Duisburg, Germany, 26–29 September 2006; IEEE Computer Society: Duisburg, Germany, 2007. [Google Scholar]
- Auer, E.; Rauh, A.; Hofer, E.P.; Luther, W. Validated Modeling of Mechanical Systems with SmartMOBILE: Improvement of Performance by ValEncIA-IVP. In Proceedings of the Dagstuhl Seminar 06021: Reliable Implementation of Real Number Algorithms: Theory and Practice; Lecture Notes in Computer Science; Springer–Verlag: Berlin, Heidelberg, Germany, 2008; pp. 1–27. [Google Scholar]
- Rauh, A.; Westphal, R.; Aschemann, H.; Auer, E. Exponential Enclosure Techniques for Initial Value Problems with Multiple Conjugate Complex Eigenvalues. In Proceedings of 16th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN2014; Lecture Notes in Computer Science; Springer: Cham, Switzerland, 2016; Volume 9553, pp. 87–122. [Google Scholar]
- Rauh, A.; Westphal, R.; Auer, E.; Aschemann, H. Exponential Enclosure Techniques for the Computation of Guaranteed State Enclosures in ValEncIA-IVP. 2013. Available online: http://interval.louisiana.edu/reliable-computing-journal/volume-19/reliable-computing-19-pp-066-090.pdf (accessed on 13 January 2021).
- Rauh, A.; Westphal, R.; Aschemann, H. Verified Simulation of Control Systems with Interval Parameters Using an Exponential State Enclosure Technique. In Proceedings of the IEEE 2013 18th International Conference on Methods and Models in Automation and Robotics MMAR, Miedzyzdroje, Poland, 26–29 August 2013. [Google Scholar]
- Nedialkov, N.S. Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation. Ph.D. Thesis, Graduate Department of Computer Science, University of Toronto, Toronto, ON, Canada, 1999. [Google Scholar]
- Nedialkov, N.S. Implementing a Rigorous ODE Solver through Literate Programming. In Modeling, Design, and Simulation of Systems with Uncertainties; Rauh, A., Auer, E., Eds.; Mathematical Engineering, Springer: Berlin/Heidenberg, Germany, 2011; pp. 3–19. [Google Scholar]
- Lin, Y.; Stadtherr, M.A. Validated Solutions of Initial Value Problems for Parametric ODEs. Appl. Numer. Math.
**2007**, 57, 1145–1162. [Google Scholar] [CrossRef][Green Version] - Kaczorek, T. Positive 1D and 2D Systems; Springer–Verlag: London, UK, 2002. [Google Scholar]
- Gennat, M.; Tibken, B. Computing Guaranteed Bounds for Uncertain Cooperative and Monotone Nonlinear Systems. IFAC Proc. Vol.
**2008**, 41, 4846–4851. [Google Scholar] [CrossRef][Green Version] - Aschemann, H.; Rauh, A.; Kletting, M.; Hofer, E.; Gennat, M.; Tibken, B. Interval Analysis and Nonlinear Control of Wastewater Plants with Parameter Uncertainty. IFAC Proc. Vol.
**2005**, 38, 55–60. [Google Scholar] [CrossRef][Green Version] - Rauh, A.; Kersten, J.; Aschemann, H. Techniques for Verified Reachability Analysis of Quasi-Linear Continuous-Time Systems. In Proceedings of the 24th International Conference on Methods and Models in Automation and Robotics 2019, Miedzyzdroje, Poland, 26–29 August 2019. [Google Scholar]
- Kersten, J.; Rauh, A.; Aschemann, H. State-Space Transformations of Uncertain Systems With Purely Real and Conjugate-Complex Eigenvalues Into a Cooperative Form. In Proceedings of the 23rd International Conference on Methods and Models in Automation and Robotics 2018, Miedzyzdroje, Poland, 27–30 August 2018. [Google Scholar]
- Jaulin, L.; Kieffer, M.; Didrit, O.; Walter, É. Applied Interval Analysis; Springer–Verlag: London, UK, 2001. [Google Scholar]
- Mayer, G. Interval Analysis and Automatic Result Verification; De Gruyter Studies in Mathematics, De Gruyter: Berlin, Germany; Boston, MA, USA, 2017. [Google Scholar]
- Kühn, W. Rigorous Error Bounds for the Initial Value Problem Based on Defect Estimation; Technical Report. 1999. Available online: http://www.decatur.de/personal/papers/defect.zip (accessed on 13 January 2021).
- Kersten, J.; Rauh, A.; Aschemann, H. Interval Methods for Robust Gain Scheduling Controllers: An LMI-Based Approach. Granul. Comput.
**2020**, 5, 203–216. [Google Scholar] [CrossRef] - Bünger, F. A Taylor Model Toolbox for Solving ODEs Implemented in MATLAB/INTLAB. J. Comput. Appl. Math.
**2020**, 368, 112511. [Google Scholar] [CrossRef] - Müller, M. Über die Eindeutigkeit der Integrale eines Systems gewöhnlicher Differenzialgleichungen und die Konvergenz einer Gattung von Verfahren zur Approximation dieser Integrale; Sitzungsbericht Heidelberger Akademie der Wissenschaften; Walter de Gruyter GmbH & Co KG: Berlin, Germany, 1927. [Google Scholar]
- Rauh, A.; Kersten, J.; Aschemann, H. Interval Methods and Contractor-Based Branch-and-Bound Procedures for Verified Parameter Identification of Quasi-Linear Cooperative System Models. J. Comput. Appl. Math.
**2020**, 367, 112484. [Google Scholar] [CrossRef] - Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V. Linear Matrix Inequalities in System and Control Theory; SIAM: Philadelphia, PA, USA, 1994. [Google Scholar]
- Sturm, J. Using SeDuMi 1.02, A MATLAB Toolbox for Optimization over Symmetric Cones. Optim. Methods Softw.
**1999**, 11–12, 625–653. [Google Scholar] [CrossRef] - Löfberg, J. YALMIP: A Toolbox for Modeling and Optimization in MATLAB. In Proceedings of the IEEE International Symposium on Computer Aided Control Systems Design, Taipei, Taiwan, 2–4 September 2004; pp. 284–289. [Google Scholar]
- Rump, S. IntLab—INTerval LABoratory. In Developments in Reliable Computing; Csendes, T., Ed.; Kluver Academic Publishers: Dordrecht, The Netherlands, 1999; pp. 77–104. [Google Scholar]
- Freihold, M.; Hofer, E. Derivation of Physically Motivated Constraints for Efficient Interval Simulations Applied to the Analysis of Uncertain Dynamical Systems. Appl. Math. Comput. Sci.
**2009**, 19, 485–499. [Google Scholar] [CrossRef][Green Version] - Freihold, M.; Rauh, A.; Hofer, E.P. Physically Motivated Constraints for Efficient Interval Simulations Applied to the Analysis of Uncertain Models of Blood Cell Dynamics. In Progress in Industrial Mathematics at ECMI 2008; Fitt, A.D., Norbury, J., Ockendon, H., Wilson, E., Eds.; Springer: Berlin/Heidelberg, Germany, 2010; pp. 563–569. [Google Scholar]
- Bünger, F. DEMOTAYLORMODEL Short Demonstration of the Taylor Model Toolbox. Available online: www.ti3.tuhh.de/intlab/demos/html/dtaylormodel.html (accessed on 16 February 2021).
- Bünger, F. Preconditioning of Taylor Models, Implementation and Test Cases. Nonlinear Theory Its Appl. IEICE
**2021**, 12, 2–40. [Google Scholar] [CrossRef] - Bünger, F. Shrink Wrapping for Taylor Models Revisited. Numer. Algorithms
**2018**, 78, 1001–1017. [Google Scholar] [CrossRef] - Rauh, A.; Kersten, J. Toward the Development of Iteration Procedures for the Interval-Based Simulation of Fractional-Order Systems. Acta Cybern.
**2020**. [Google Scholar] [CrossRef] - Rauh, A.; Kersten, J. Verification and Reachability Analysis of Fractional-Order Differential Equations Using Interval Analysis. In Proceedings 6th International Workshop on Symbolic-Numeric Methods for Reasoning about CPS and IoT, Online, 31 August 2020; Electronic Proceedings in Theoretical Computer Science; Dang, T., Ratschan, S., Eds.; Open Publishing Association: Den Haag, The Netherlands, 2021; Volume 331, pp. 18–32. [Google Scholar] [CrossRef]
- Rauh, A.; Jaulin, L. Novel Techniques for a Verified Simulation of Fractional-Order Differential Equations. Fractal Fract.
**2021**, 5, 17. [Google Scholar] [CrossRef]

**Figure 1.**State enclosures by means of exploiting cooperativity for the simple benchmark system in (20).

**Figure 3.**State enclosures in terms of their lower and upper bounds for the step-down converter after backward transformation into the original state-space using the procedure in Section 2.4.

**Figure 4.**State enclosures for the step-down converter using the novel procedure according to Section 3 in combination with the real-valued transformation approach of Section 2.3.

**Figure 5.**Partitioning of the parameter domain according to Section 3 for two different parameterizations of the multi-sectioning strategy.

**Figure 6.**Comparison of the proposed approach with $L=10$ with the results of

`verifyode`for the

**case 1**.

**Figure 7.**Comparison of the proposed approach with $L=10$ with the results of

`verifyode`for the

**case 2**.

**Figure 8.**Comparison of the proposed approach with $L=10$ with the results of

`verifyode`for the

**case 3**.

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**MDPI and ACS Style**

Rauh, A.; Kersten, J. Transformation of Uncertain Linear Systems with Real Eigenvalues into Cooperative Form: The Case of Constant and Time-Varying Bounded Parameters. *Algorithms* **2021**, *14*, 85.
https://doi.org/10.3390/a14030085

**AMA Style**

Rauh A, Kersten J. Transformation of Uncertain Linear Systems with Real Eigenvalues into Cooperative Form: The Case of Constant and Time-Varying Bounded Parameters. *Algorithms*. 2021; 14(3):85.
https://doi.org/10.3390/a14030085

**Chicago/Turabian Style**

Rauh, Andreas, and Julia Kersten. 2021. "Transformation of Uncertain Linear Systems with Real Eigenvalues into Cooperative Form: The Case of Constant and Time-Varying Bounded Parameters" *Algorithms* 14, no. 3: 85.
https://doi.org/10.3390/a14030085