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Models

We also provide several frequently used models for tests.

Van der Pol oscillator​

d2Xdt2−μ(1−X2)dXdt+X=0\frac{d^2X}{dt^2}-\mu (1-X^2) \frac{dX}{dt} +X =0

We provide a two-dimensional form as

x˙=yy˙=μ(1−x2)y−xμ=1\begin{align*} \dot{x} &=y \\ \dot{y} &=\mu (1-x^2)y-x \\ \mu &=1 \end{align*}

References​

[1]: Wikipedia contributors. (2022, May 3). Van der Pol oscillator. In Wikipedia, The Free Encyclopedia. Retrieved 09: 47, June 8, 2022, from https://en.wikipedia.org/w/index.php?title=Van_der_Pol_oscillator&oldid=1085958541

Coupled Van der Pol oscillator​

x˙0=x1x˙1=(1−x02)x1−x0+(x2−x0)x˙2=x3x˙3=(1−x22)x3−x2+(x0−x2)\begin{align*} \dot{x}_{0} &= x_{1} \\ \dot{x}_{1} &= (1-x_{0}^{2})x_{1} - x_{0} + (x_{2}-x_{0}) \\ \dot{x}_{2} &= x_{3} \\ \dot{x}_{3} &= (1-x_{2}^{2})x_{3} - x_{2} + (x_{0}-x_{2}) \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

2D LTV System​

x˙=−x+ty+t+u1+v1y˙=t2x+y−t+u2+v2\begin{align*} \dot{x} &= -x +ty+t+u_{1} +v_{1} \\ \dot{y} &= t^2x+y-t+u_{2} +v_{2} \end{align*}

References​

[1]: 2-dimensional Linear Time-Varying(LTV) system with Time-Varying(TV) and Time-Invariant(TI) uncertainties, from https://flowstar.org/benchmarks/2-dimensional-ltv-system/

Laub-Loomis​

x˙0=1.4x2−0.9x0x˙1=2.5x4−1.5x1x˙2=0.6x6−0.8x1x2x˙3=2−1.3x2x3x˙4=0.7x0−x3x4x˙5=0.3x1−3.1x5x˙6=1.8x5−1.5x1x6\begin{align*} \dot{x}_{0} &= 1.4 x_{2} - 0.9 x_{0} \\ \dot{x}_{1} &= 2.5 x_{4} - 1.5 x_{1} \\ \dot{x}_{2} &= 0.6x_{6} - 0.8 x_{1} x_{2} \\ \dot{x}_{3} &= 2 - 1.3 x_{2} x_{3} \\ \dot{x}_{4} &= 0.7 x_{0} - x_{3}x_{4} \\ \dot{x}_{5} &= 0.3 x_{1} - 3.1 x_{5} \\ \dot{x}_{6} &= 1.8 x_{5} - 1.5 x_{1} x_{6} \end{align*}

References​

[1]: Laub, M. T., & Loomis, W. F. (1998). A molecular network that produces spontaneous oscillations in excitable cells of Dictyostelium. Molecular biology of the cell, 9(12), 3521-3532.

Cellular p53​

x˙0=(0.5−9.963×10−6x0x4−1.925×10−5x0)×3600x˙1=(1.5×10−3+1.5×10−2(x02/(547600+x02))−8×10−4x1)×3600x˙2=(8×10−4x1−1.444×10−4x2)×3600x˙3=(1.66×10−2x2−9×10−4x3)×3600x˙4=(9×10−4x3−1.66×10−7x32−9.963×10−6x4x5)×3600x˙5=(0.5−3.209×10−5x5−9.963×10−6x4x5)×3600\begin{align*} \dot{x}_{0} &= (0.5 - 9.963 \times 10^{-6} x_{0} x_{4} - 1.925 \times 10^{-5} x_{0}) \times 3600 \\ \dot{x}_{1} &= (1.5 \times 10^{-3} + 1.5 \times 10^{-2} (x_{0}^{2} / (547600 + x_{0}^{2})) - 8 \times 10^{-4} x_{1}) \times 3600 \\ \dot{x}_{2} &= (8 \times 10^{-4} x_{1} - 1.444 \times 10^{-4} x_{2}) \times 3600 \\ \dot{x}_{3} &= (1.66 \times 10^{-2} x_{2} - 9 \times 10^{-4} x_{3}) \times 3600 \\ \dot{x}_{4} &= (9 \times 10^{-4} x_{3} - 1.66 \times 10^{-7} x_{3} 2 - 9.963 \times 10^{-6} x_{4} x_{5}) \times 3600 \\ \dot{x}_{5} &= (0.5 - 3.209 \times 10^{-5} x_{5} - 9.963 \times 10^{-6} x_{4} x_{5}) \times 3600 \end{align*}

References​

[1]: Leenders, G. B., & Tuszynski, J. A. (2013). Stochastic and deterministic models of cellular p53 regulation. Frontiers in oncology, 3, 64.

9D Genetic oscillator​

x˙0=50x2−0.1x0x5x˙1=100x3−x0x1x˙2=0.1x0x5−50x2x˙3=x1x5−100x3x˙4=5x2+0.5x0−10x4x˙5=50x4+50x2+100x3−x5(0.1x0+x1+2x7+1)x˙6=50x3+0.01x1−0.5x6x˙7=0.5x6−2x5x7+x8−0.2x7x˙8=2x5x7−x8\begin{align*} \dot{x}_{0} &= 50 x_{2} - 0.1 x_{0} x_{5} \\ \dot{x}_{1} &= 100 x_{3} - x_{0} x_{1} \\ \dot{x}_{2} &= 0.1 x_{0} x_{5} - 50 x_{2} \\ \dot{x}_{3} &= x_{1} x_{5} - 100 x_{3} \\ \dot{x}_{4} &= 5 x_{2} + 0.5 x_{0} - 10 x_{4} \\ \dot{x}_{5} &= 50 x_{4} + 50 x_{2} + 100 x_{3} - x_{5} (0.1 x_{0} + x_{1} + 2 x_{7} + 1) \\ \dot{x}_{6} &= 50 x_{3} + 0.01 x_{1} - 0.5 x_{6} \\ \dot{x}_{7} &= 0.5 x_{6} - 2x_{5} x_{7} + x_{8} - 0.2 x_{7} \\ \dot{x}_{8} &= 2 x_{5} x_{7} - x_{8} \end{align*}
References​

[1]: Vilar, J. M., Kueh, H. Y., Barkai, N., & Leibler, S. (2002). Mechanisms of noise-resistance in genetic oscillators. Proceedings of the National Academy of Sciences, 99(9), 5988-5992.

Water Tank 6Eq​

x˙0=u0+0.1+k1(4−x5)−k02gx0x˙1=k02g(x0−x1)x˙2=k02g(x1−x2)x˙3=k02g(x2−x3)x˙4=k02g(x3−x4)x˙5=k02g(x4−x5)\begin{align*} \dot{x}_{0} &= u_{0} + 0.1 + k_{1} (4 -x_{5}) - k_{0} \sqrt{2gx_{0}} \\ \dot{x}_{1} &= k_{0} \sqrt{2g (x_{0}-x_{1})} \\ \dot{x}_{2} &= k_{0} \sqrt{2g (x_{1}-x_{2})} \\ \dot{x}_{3} &= k_{0} \sqrt{2g (x_{2}-x_{3})} \\ \dot{x}_{4} &= k_{0} \sqrt{2g (x_{3}-x_{4})} \\ \dot{x}_{5} &= k_{0} \sqrt{2g (x_{4}-x_{5})} \end{align*}

where k0=0.015,k1=0.01,g=9.81k_{0} = 0.015, k_{1}= 0.01, g=9.81

References​

[1]: Althoff, M., Stursberg, O., & Buss, M. (2008, December). Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. In 2008 47th IEEE Conference on Decision and Control (pp. 4042-4048). IEEE.

2D ODE​

x˙0=−0.5x0−0.5x1+0.5x0x1x˙1=−0.5x1+1+u\begin{align*} \dot{x}_{0} &= -0.5x_{0}-0.5x_{1}+0.5x_{0} x_{1} \\ \dot{x}_{1} &= -0.5x_{1} + 1 + u \end{align*}

References​

[1]: Xue, B., Li, R., Zhan, N., & Fränzle, M. (2021, May). Reach-avoid analysis for stochastic discrete-time systems. In 2021 American Control Conference (ACC) (pp. 4879-4885). IEEE.

Lotka-Volterra model of 2 variables​

x˙0=1.5x0−x0x1x˙1=−3x1+x0x1\begin{align*} \dot{x}_{0} &= 1.5 x_{0} - x_{0} x_{1} \\ \dot{x}_{1} &= -3 x_{1} + x_{0} x_{1} \\ \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

Lotka-Volterra model of 5 variables​

x˙0=x0(1−(x0+0.85x1+0.5x4))x˙1=x1(1−(x1+0.85x2+0.5x0))x˙2=x2(1−(x2+0.85x3+0.5x1))x˙3=x3(1−(x3+0.85x4+0.5x2))x˙4=x4(1−(x4+0.85x0+0.5x3))\begin{align*} \dot{x}_{0} &= x_{0} (1-(x_{0}+0.85 x_{1} + 0.5 x_{4})) \\ \dot{x}_{1} &= x_{1} (1-(x_{1}+0.85 x_{2} + 0.5 x_{0})) \\ \dot{x}_{2} &= x_{2} (1-(x_{2}+0.85 x_{3} + 0.5 x_{1})) \\ \dot{x}_{3} &= x_{3} (1-(x_{3}+0.85 x_{4} + 0.5 x_{2})) \\ \dot{x}_{4} &= x_{4} (1-(x_{4}+0.85 x_{0} + 0.5 x_{3})) \\ \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

2-state with polynomial vector field​

x˙0=−0.42x0−1.05x1−2.3x02−0.5x0x1−x03x˙1=1.98x0+x0x1\begin{align*} \dot{x}_{0} &= -0.42x_{0} - 1.05 x_{1} - 2.3 x_{0}^{2} -0.5 x_{0} x_{1} - x_{0}^{3} \\ \dot{x}_{1} &= 1.98 x_{0} + x_{0} x_{1} \end{align*}

References​

[1]: Tan, W., & Packard, A. (2008). Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of-squares programming. IEEE Transactions on Automatic Control, 53(2), 565-571.

Synchronous Machine​

x˙0=x1x˙1=0.2−0.7sin⁡x0−0.05x1\begin{align*} \dot{x}_{0} &= x_1 \\ \dot{x}_{1} &= 0.2 - 0.7 \sin{x_{0}} -0.05x_{1} \end{align*}

References​

[1]: Xue, B., She, Z., & Easwaran, A. (2016). Under-approximating backward reachable sets by polytopes. In Computer Aided Verification: 28th International Conference, CAV 2016, Toronto, ON, Canada, July 17-23, 2016, Proceedings, Part I 28 (pp. 457-476). Springer International Publishing.

Biological model of 7 variables​

x˙0=−0.4x0+5x2x3x˙1=0.4x0−x1x˙2=x1−5x2x3x˙3=5x4x5−5x2x3x˙4=−5x4x5+5x2x3x˙5=0.5x6−5x4x5x˙6=−0.5x6+5x4x5\begin{align*} \dot{x}_{0} &= -0.4 x_{0} + 5 x_{2} x_{3} \\ \dot{x}_{1} &= 0.4 x_{0} - x_{1} \\ \dot{x}_{2} &= x_{1} - 5 x_{2} x_{3} \\ \dot{x}_{3} &= 5 x_{4} x_{5} - 5 x_{2} x_{3} \\ \dot{x}_{4} &= -5 x_{4} x_{5} + 5 x_{2} x_{3} \\ \dot{x}_{5} &= 0.5 x_{6} - 5 x_{4} x_{5} \\ \dot{x}_{6} &= -0.5 x_{6} + 5 x_{4} x_{5} \\ \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

Biolgoical model of 9 variables​

x˙0=3x2−x0x5x˙1=x3−x1x5x˙2=x0x5−3x2x˙3=x1x5−x3x˙4=3x2+5x0−x4x˙5=5x4+3x2+x3−x5(x0+x1+2x7+1)x˙6=5x3+x1−0.5x6x˙7=5x6−2x5x7+x8−0.2x7x˙8=2x5x7−x8\begin{align*} \dot{x}_{0} &= 3 x_{2} - x_{0} x_{5} \\ \dot{x}_{1} &= x_{3} - x_{1} x_{5} \\ \dot{x}_{2} &= x_{0} x_{5} - 3 x_{2} \\ \dot{x}_{3} &= x_{1} x_{5} - x_{3} \\ \dot{x}_{4} &= 3 x_{2} + 5 x_{0} - x_{4} \\ \dot{x}_{5} &= 5 x_{4} + 3 x_{2} + x_{3} - x_{5} (x_{0}+x_{1}+2x_{7}+1) \\ \dot{x}_{6} &= 5 x_{3} + x_{1} - 0.5 x_{6} \\ \dot{x}_{7} &= 5 x_{6} - 2 x_{5} x_{7} + x_{8} - 0.2 x_{7} \\ \dot{x}_{8} &= 2 x_{5} x_{7} - x_{8} \\ \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

Jet engine​

x˙0=−x1−1.5x02−0.5x03−0.5x˙1=3x0−x1\begin{align*} \dot{x}_{0} &= -x_{1} - 1.5 x_{0}^2 -0.5 x_{0}^3 -0.5 \\ \dot{x}_{1} &= 3 x_{0} - x_{1} \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

Jet engine with time-varying disturbances​

x˙0=−x1−1.5x02−0.5x03−0.5+u0x˙1=3x0−x1+u1\begin{align*} \dot{x}_{0} &= -x_{1} - 1.5 x_{0}^2 -0.5 x{0}^3 -0.5 + u_{0} \\ \dot{x}_{1} &= 3 x_{0} - x_{1} + u_{1} \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

Spring-pendulum​

x˙0=x2x˙1=x3x˙2=x0x32+gcos⁡θ−k(x0−L)x˙3=−2x2x3+gsin⁡θx0k=2L=1g=9.8\begin{align*} \dot{x}_{0} &= x_{2} \\ \dot{x}_{1} &= x_{3} \\ \dot{x}_{2} &= x_{0} x_{3}^2 + g \cos{\theta} - k (x_{0}-L) \\ \dot{x}_{3} &= - \frac{2 x_{2} x_{3} + g \sin{\theta}}{x_{0}} \\ k &= 2 \\ L &= 1 \\ g &= 9.8 \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

Brusselator​

x˙0=A+x02x1−Bx0−x0x˙1=Bx0−x02x1A=1B=1.5\begin{align*} \dot{x}_{0} &= A + x_{0}^2 x_{1} - B x_{0} - x_{0} \\ \dot{x}_{1} &= B x_{0} - x_{0}^2 x_{1} \\ A &= 1 \\ B &= 1.5 \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

Lorentz system​

x˙0=σ(x1−x0)x˙1=x0(ρ−x2)−x1x˙2=x0x1−βx2σ=10ρ=83β=28\begin{align*} \dot{x}_{0} &= \sigma (x_{1} - x_{0}) \\ \dot{x}_{1} &= x_{0} (\rho - x_{2}) - x_{1} \\ \dot{x}_{2} &= x_{0} x_{1} - \beta x_{2} \\ \sigma &= 10 \\ \rho &= \frac{8}{3} \\ \beta &= 28 \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

Rossler attractor​

x˙0=−x1−x2x˙1=x0+ax1x˙2=b+x2(x0−c)a=0.2b=0.2c=5.7\begin{align*} \dot{x}_{0} &= -x_{1} - x_{2} \\ \dot{x}_{1} &= x_{0} + a x_{1} \\ \dot{x}_{2} &= b + x_{2} (x_{0} - c) \\ a &= 0.2 \\ b &= 0.2 \\ c &= 5.7 \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

PI controller with disturbance​

x˙0=−0.101(x0−20)+1.3203(x1−0.1616)−0.01x02x˙1=−(−0.101(x0−20)+1.32.3(x1−0.1616)−0.01x02)+3(20−x0)+u\begin{align*} \dot{x}_{0} &= -0.101 (x_{0}-20) + 1.3203 (x_{1} - 0.1616) - 0.01 x_{0}^2 \\ \dot{x}_{1} &= -(-0.101 (x_{0}-20) + 1.32.3 (x_{1} - 0.1616) - 0.01 x_{0}^2) + 3 (20-x_{0}) + u \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).

lac operon model​

x˙0=−2k3x02k8Rx12+τk3x02+μ+2k−3F+(k5I−(k9+k−5)x0)k−2Xk4μ(k3x02+μ)k7(k2(k8Rx12+τ)+k−2(k3x02+μ))x˙1=−2k8Rx12+2k−8(k8Rx12+τ)k3x02+k9x0k−2Xk4η(k3x02+μ)k7(k2(k8Rx12+τ)+k−2(k3x02+μ))k2=4×105    k−2=0.03    k3=0.2k−3=60    k4=1    k5=0.6k−5=0.006    k6=3×10−6    k7=3×10−6k8=0.03    k−8=1×10−5    k9=5000R=0.01    X=0.002002    μ=0.005F=0.0001    I=91100    τ=0.008\begin{align*} \dot{x}_{0} &= -2 k_{3} x_{0}^2 \frac{k_8 R x_{1}^2 + \tau}{k_3 x_{0}^2 + \mu} + 2 k_{-3} F + \frac{(k_5 I - (k_9 + k_{-5}) x_{0}) k_{-2} \mathcal{X} k_4 \mu (k_3 x_{0}^2 + \mu)} {k_7(k_2(k_8 R x_{1}^2 + \tau) + k_{-2} (k_3 x_{0}^2 + \mu))} \\ \dot{x}_{1} &= -2 k_8 R x_{1}^2 + \frac{2 k_{-8} (k_8 R x_{1}^2 + \tau)}{k_3 x_{0}^2} + \frac{k_9 x_{0} k_{-2} \mathcal{X} k_{4} \eta (k_3 x_{0}^2 + \mu)}{k_7(k_2(k_8 R x_{1}^2 + \tau) + k_{-2} (k_3 x_{0}^2 + \mu))} \\ k_2 &= 4 \times {10}^5 \ \ \ \ k_{-2} = 0.03 \ \ \ \ k_{3} = 0.2 \\ k_{-3} &= 60 \ \ \ \ k_{4} = 1 \ \ \ \ k_{5} = 0.6 \\ k_{-5} &= 0.006 \ \ \ \ k_{6} = 3 \times {10}^{-6} \ \ \ \ k_{7} = 3 \times {10}^{-6} \\ k_{8} &= 0.03 \ \ \ \ k_{-8} = 1 \times {10}^{-5} \ \ \ \ k_{9} = 5000 \\ R &= 0.01 \ \ \ \ \mathcal{X} = 0.002002 \ \ \ \ \mu = 0.005 \\ F &= 0.0001 \ \ \ \ I = 91100 \ \ \ \ \tau = 0.008 \end{align*}

References​

[1]: Chen, X. (2015). Reachability analysis of non-linear hybrid systems using taylor models (Doctoral dissertation, Fachgruppe Informatik, RWTH Aachen University).