Universiti Malaysia Terengganu Journal of Undergraduate Research eISSN: 2637-1138 Volume 3 Number 1, January 2021: 25-36 © Penerbit UMT

ADAPTIVE TIME-STEPPING FOR RUNGE-KUTTA METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS

CHUA KAH WAI, LOY KAK CHOON* AND RUWAIDIAH IDRIS

Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia.

*Corresponding author: [email protected] http://doi.org/10.46754/umtjur.2021.01.004

Abstract: Ordinary Differential Equations (ODEs) are usually used in numerous fields especially in solving the modelling problem. Numerical methods are one of the vital mathematical tools to solve the ODEs that appear in various modelling problems by determining the approximation solution close to the in exact solution if it exists. Runge- Kutta methods (RK) are the numerical methods used to integrate the ODEs by applying multistage methods at the midpoint of an interval which can efficiently produce a more accurate result or small magnitude of error. We proposed Runge-Kutta methods (RK) to solve the 1st_ order nonlinear stiff ODEs. The RK methods used in this research are known as the RK-2, RK-4, and RK-5 methods. We proved the existence and uniqueness of the ODEs before we solved it numerically. We also proved the absolute-stability of the RK methods to determine the overall stability of these methods. We found two suitable test cases which are the standard test problem and manufactured solution. We proved that by combining the adaptive step size with RK methods can result in more efficient computation. We implemented the 2nd_, 4th_ and 5th_ order of RK methods with step size adaptively algorithm to solve the test problem and manufactured solution via Octave programming language. The resulting numerical error and the stability of each method can be studied. We compared our results using several error plots versus the Central Processing Unit (CPU) time required to compute a given nonlinear 1st_ order stiff ODE problem. In a conclusion, RK methods which combine with the adaptive step size can result in more efficient computation and accuracy compare with the fixed step size RK methods.

Keywords: Ordinary ; Runge-Kutta; Multistage method; Adaptive step size; non-linear stiff equation. Introduction methods such as the exact method, separation Ordinary Differential Equations of variables, integrating factors, variation of parameters, and method of undetermined A differential equation is an equation which coefficients. In general, nonlinear ODEs are contains the function and its derivatives. A hard to be solved analytically because there differential equation is said to be linear if are many functions that we can construct it can be written as a linear combination of using a finite combination of finite number the derivatives. In general, the differential of elementary functions. ODEs can be equation can be divided into two categories grouped into two types which are linear which are the ordinary differential equations and nonlinear differential equations. For the (ODEs) and partial differential equations first-order linear ODEs, the function f(t,y) (PDEs). can be expressed in a linear form Ordinary differential equations (ODEs) f(t,y) = a(x)y+b(x), (1.1) is a function that involves one or higher order derivatives with only one independent where a(x) and b(x) are arbitrary variable. Normally, solving the first order functions of x only. The first-order nonlinear linear ODEs is easy by using various type of ODEs can be simply written as

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(1.2) control requires that the stepping algorithm signal information about its performance In our daily life, ODEs are used in a and the important condition is an estimation wide variety of disciplines, from biology, of its truncation error. physics, chemistry, economics, and engineering. Some examples of real-world application by using ODEs are such as in Methodology medicine for modelling cancer growth or Runge-Kutta Methods For Stiff ODEs the spread of a disease, in engineering for In this section, we will use Runge-Kutta describing the movement of electricity, in methods to solve the 1st_ order nonlinear physics to describe the motion of waves, stiff ordinary differential equation (ODE). pendulum or chaotic systems and in Runge-Kutta methods have been chosen chemistry for modelling chemical reactions in this project is because this method is and half-life for decaying radioactive a multistage method which has better isotopes. stability even though they are categorized as a fully explicit method. RK method does Runge-Kutta Methods not require initialization and it is efficient since high order RK methods are already There are many known methods to solve available. Hence, we will introduce the ODE problems. Runge-Kutta methods are general formula of 2nd_, 4th_ and 5th_ order the most common ODE solvers. Runge- Runge-Kutta methods to handle stiff Kutta methods are the generalization of the ordinary differential equations (ODEs). concept used in Modified Euler’s method. Runge-Kutta methods are numerical The general equation of Runge-Kutta methods that can integrate the ordinary method can be expressed below: differential equation by using multistep trial at the midpoint of an interval. Runge-Kutta has the advantage of requiring only explicit (2.1) evaluation of the derivatives of f(t,y).The step-size can be modified any time. In this is called an -stage RK method. project, I can use Runge-Kutta methods to where solve the first-order nonlinear stiff ODEs with adaptive step size.

Adaptive Step Size

During the progress, good ODE solvers may where the Butcher tableau with coefficients have some adaptive control that making are written as repeated changes in its step size which can make the computation more efficient. Normally the objective of adaptive step size control is to obtain some predetermined accuracy in the solution with minimum computational effort. Sometimes numerical accuracy is not directly demanded in the solution itself, but in some related conserved quantity that can be monitored. Implementation of adaptive step size

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A general explicit is given by where g(t) is a function of t alone. Furthermore, we assume a stiff solution to (2.6) be 20t with Butcher tableau y(t) = sin (t) e , (2.11) where the initial condition y (0) = 0 0 0 (by direct substitution) , Now, we need to findg (t): 1 yˈ = 20sin(t) e20t,+ cos(t) e20t while the implicit Euler method is given by (by product rule) (2.12) Then by putting y and y̕ in the ODE (2.7) equation (2.10), we have yˈ-4y2+y+3t = 20sin(t)e20t+cos(t)e20t with its Butcher tableau -4sin2 (t)e40t+sin(t)e20t+3t (2.13)

1 1 Finally, we have manufactured an ODE . yˈ- 4y2+y=20sin(t)e20t+cos(t)e20t 1 -4sin2(t) e40t+sin(t)e20t (2.14)

where y(0) = 0 with an exact solution y(t) Test Problem and Manufactured Solution = sin(t)e20t. Test Problem of 1st-Order Nonlinear Stiff ODE The Existence and Uniqueness of In this section, we provide a 1st- order nonlinear ODEs nonlinear stiff ODE with an exact Before we solve the 1st-order nonlinear solution. By using this test problem is to ODE, we need to find their existence and demonstrate the numerical illustration and uniqueness by using the concept of Lipchitz we will use the exact solution to validate continuity. The stiffness on ODEs can be the programming code. The test case for regarded as a nonlinear ODE problem with st 1 -order order nonlinear ODEs has the a large Lipschitz constant. The theorem following condition: of the existence and uniqueness states yˈ = -15y, y = 1 whem t = 0, (2.8) that if f is uniformly Lipschitz continuous over some time period which is 0 ≤ t ≤ T, where the exact solution is given then the unique solution will be obtained through any value α from initial value α. y(t) = e -15t (2.9) For the explicit methods, it is important to take a reasonably small step size to obtain st_ Manufacture a 1 Order Nonlinear Stiff accurate numerical results. ODE The theory for the existence of a In this section, we will discuss about the solution to the initial value problem (IVP) manufacturing of an ODE equation. Hence, of the 1st-order ODE is in the form: we let a 1st_ order nonlinear stiff ODE to have the following expression: yˈ(t) = f(t,y) y(0) = α (2.15)

2 y’ - 4y + y + 3t = g(t), with y(0) = y0 (2.10)

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The function (yˈ) is said to be Lipschitz called as the zero-stability, a notion used to continuous in y over some range of t and differentiate this from other type of stability y, such that there exists some constant L>0 that produces the same significance. such that LeVeque (2004) mentioned that an | f(t,y) - f(t,y*)|≤ L| y-yˈ| (2.16) r-step (LMM) is said to be zero-stable if the roots of the P(θ) satisfy the following for all y and yˈ in this range. Lipschitz conditions: continuity slightly stronger than continuity which requires that |f(t,y)-f(t,y*)|→0 as |θj|≤ 1 for j = 1,2,3,... y→y* whereas Lipschitz continuity only If θj is a repeated root, then |θj|1 . requires: In this section, we can conclude that RK-2, | f(t,y) - f(t,y*)|≤ 0as y→y* (2.17) RK-4 and RK-5 are all zero-stable methods. For example: Using Taylor expansion, we have Absolute stability Basically, the local approximations for (2.18) many problems can be nullified in the long term are due to the unstable behaviour. or A way to exploring what is needed for a method to solve these problems in stable (2.19) manner is to check the condition of simple by Mean Value Theorem. linear equation yˈ λy = to be solved over a Taking bound on the both side single time step. λ is the constant that can be a complex number, however we just (2.20) focus our attention on this quantity scaled in (2.21) terms of the step size, which is the complex number z = hλ. (2.22) We defined the stability function as Therefore, we had proved that the Lipschitz the rational function R(z) = 1 + zbT (I-zA)- 1 constant, L=| fy(t,y*)|. e, where e∈R has every component equal to 1 for a given a Runge-Kutta method. Zero stability of Runge-Kutta methods The stability region is a set on the complex plane such that |R(z)|≤1. When solving the Linear Multistep Method (LMM) is known problem yˈ=λy, we have to choose step size, to be zero stable and we will proof in this h small enough to ensure that z=hλ is in the section. Usually, a method is said to have stability region, and otherwise the sequence a special type of if the of approximations produced in many steps global errors can be bounded in term of sum will be unbounded. Then we will interpret of all the one-step error and has the same it as an unstable behaviour. The stability asymptotic behaviour as the local truncation region for explicit method with s=p≤6 is error approaches 0. In the ODE theory, it is shown in figure below.

Universiti Malaysia Terengganu Journal of Undergraduate Research Volume 3 Number 1, January 2021: 25-36 ADAPTIVE TIME-STEPPING FOR RUNGE-KUTTA METHODS FOR ORDINARY 29 DIFFERENTIAL EQUATIONS

Figure 3.1: Stability domain for explicit runge-kutta methods

In the case of a stiff problem, even Results and Discussion though it may not have a linear form but Analysis of Error there will be growth factors corresponding to values of z which may lie anywhere in In this section, we will analyze the error the left half-plane. In this particular case, between three different orders of Runge- we are also interested to have methods Kutta methods and exact solution for which are “A-stable”, in which all complex manufactured solution. numbers with negative real part lie in the stability region. Analysis of Error of the Test Problem

Figure 3.1: Numerical error versus step size h for test problem

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In Figure 3.1, the graph was plotted for the manufactured solution is getting using the logarithm value of base 10 for the smaller as the step size h decreases. When numerical error versus the similar logarithm we set up our h = 0.0125, the 2nd-, 4th- value of base 10 for step size h. The graph , and 5th-order of Runge-Kutta methods consists of three different methods used produce the numerical error of 6.47x10- which is 2nd-, 4th-, and 5th-order of Runge- 4, 1.51x10-6, 3.06x10-8, respectively. At h Kutta methods. The red line represents 2nd- = 0.0015625, the numerical error for 2nd-, order of Runge-Kutta method, the blue line 4th-, and 5th-order of Runge-Kutta methods represents 4th-order of Runge-Kutta method which are 1.13x10-5, 3.213x10-10, 8.18x10- and the green line represents 5th-order of 13, respectively. From these results, we can Runge-Kutta method. From Figure 3.1, we conclude that 5th-order of Runge-Kutta analyze the global error of the numerical methods produce smallest numerical error approximations with Runge-Kutta methods with same value of step size compared to the along t[a,b]. By observing the graph, we 2nd- and 4th-order of Runge-Kutta methods. can clearly see that all the numerical error

Analysis of Error of the Manufactured Solution

Figure 3.2: Numerical error versus step size h for manufactured solution

In Figure 3.2, the graph was plotted analyze the global error of the numerical using the logarithm value of base 10 for the approximations with Runge-Kutta methods numerical error versus the similar logarithm along t[a,b]. By observing the graph, we value of base 10 for step size h. The graph can clearly see that all the numerical error consists of three different methods used for the manufactured solution is getting which is 2nd-, 4th-, and 5th-order of Runge- smaller as the step size h decreases. When Kutta methods. The red line stands for 2nd- we set up our, h = 0.0125, the 2nd-, 4th, and order of Runge-Kutta method, the blue line 5th-order of Runge-Kutta methods produce stands for 4th-order of Runge-Kutta method the numerical error of 3.27x10-4, 5.54x10- and the green line stands for 5th-order of 7,5.65x10-9, respectively. At h=0.025, the Runge-Kutta method. From Figure 3.2, we numerical error is 1.30x10-3, 8.84x10-6

Universiti Malaysia Terengganu Journal of Undergraduate Research Volume 3 Number 1, January 2021: 25-36 ADAPTIVE TIME-STEPPING FOR RUNGE-KUTTA METHODS FOR ORDINARY 31 DIFFERENTIAL EQUATIONS and 2.14x10-7 for 2nd-, 4th-, and 5th-order of Efficiency Analysis Runge-Kutta methods, respectively. From In this section, we are going to analyze the th these results, we can conclude that 5 -order efficiency of nd2 -, 4th- and 5th-order Runge- of Runge-Kutta methods produces smallest Kutta methods using the plot of numerical numerical error with same value of step size error versus the CPU time. We obtained nd th compared to the 2 - and 4 -order of Runge- the CPU time by recording the total time of Kutta methods. calculation in second starting from (n =1) until the end of stopping algorithm (n =N).

Efficiency Analysis of Test Problem

Figure 3.3: Numerical error versus CPU Time for test problem In Figure 3.3, the graph was plotted pattern when the numerical error is getting using the logarithm value of base 10 decreases. CPU time is recorded before for numerical error versus the (central (n=1) and after (n=N) the loop of time- processor unit) CPU time required to stepping algorithm. Based on the efficiency complete the computation in seconds result, at step size h=0.05, the required CPU (s). The graph consists of three different time for 2nd-, 4th-, and 5th-order of Runge- methods used which is 2nd-, 4th-, and 5th- Kutta methods are 1.28x10-3s,4.27x10-3s, order of Runge-Kutta methods. The red line and 6.17x10-3s, respectively. At h=0.0125, is for 2nd-order of Runge-Kutta method, the the required CPU time for 2nd-, 4th-, and 5th- blue line is for 4th-order of Runge-Kutta order of Runge-Kutta methods are 7.53x10- method and the green line is for 5th-order 3s, 9.82x10-3s, and 2.28x10-2s, respectively. of Runge-Kutta method. From the graph As a conclusion, the CPU time increases above, we clearly can see that the CPU when the step size h decreases for all three time for each method increase in nonlinear Runge-Kutta methods.

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Efficiency Analysis of Manufactured Solution

Figure 3.4: Numerical error versus CPU Time for manufactured solution

In Figure 3.4, the graph was plotted 7.43x10-2s, 1.06x10-1s, and 1.631x10-1s , using the logarithm value of base 10 for respectively. As a conclusion, the CPU time numerical error versus the (central processor is increasing when the step size h is getting unit) CPU time required to complete the smaller for all three Runge-Kutta methods. computation in seconds (s). The graph consists of three different methods used Convergence Analysis which is 2nd-, 4th-, and 5th-order of Runge- Kutta methods. The red line stands for 2nd- The setting of the step sizes is also important order of Runge-Kutta method, the blue line to verify the order of the convergence for the stands for 4th-order of Runge-Kutta method numerical solution. We consider the y the and the green line stands for 5th-order of exact value and we denote y the numerical Runge-Kutta method. From the graph approximation where it depends the step above, we can clearly see that the CPU time size h. From here, we can compute the order for each method increases in a nonlinear of convergence k of any numerical methods fashion when the numerical error decreases. for ODE is mathematically expressed as: CPU time is recorded before(n=1) and after (n=N) within the loop of time-stepping algorithm. Based on the efficiency results, at step size h=0.05, the required CPU time nd th th for 2 -, 4 -, and 5 -order of Runge-Kutta (3.31) methods are 4.41x10-3s, 7.08x10-3s, and, 1.04x10-2s respectively. As we setting h where k is the order of convergence, y is nd th = 0.0015625, the CPU time for 2 -, 4 -, the exact solution and yh is the numerical and 5th-order of Runge-Kutta methods are approximation which depends on the step

Universiti Malaysia Terengganu Journal of Undergraduate Research Volume 3 Number 1, January 2021: 25-36 ADAPTIVE TIME-STEPPING FOR RUNGE-KUTTA METHODS FOR ORDINARY 33 DIFFERENTIAL EQUATIONS size h, is the norm of relative error Comparison On Efficency Error Between Fixed and Adaptive Step Size which depends on the value of h, the step In this section, we will compare the h efficiency error between the fixed and size and the half of, 2 the chosen step size. adaptive step size of 5th-order Runge-Kutta method for test problem and manufactured solution. Efficiency Error Between Fixed and Adaptive Step Size for Test Problem

Figure 3.6: Efficiency error between the fixed and adaptive step size ofth 5 -order of Runge-Kutta method. (Test Problem)

In Figure 3.6, the graph was plotted with adaptive step size is more efficient using the logarithm value of base 10 than 5th-order of Runge-Kutta method for numerical error versus the (central with fixed step size. We also clearly see processor unit) CPU time required to that the numerical error for adaptive step complete the computation in seconds (s). size of 5th-order of Runge-Kutta method is The graph consists of two different methods smaller than the fixed step size. Based on used which is 5th-order of Runge-Kutta the result, at CPU time s=0.0086773, the method with adaptive step size and 5th-order computed error for fixed and adaptive step of Runge-Kutta method with fixed step size. size of 5th-order of Runge-Kutta methods The red line is for 5th-order of Runge-Kutta are 3.68x10-2, and 8.66x10-5, respectively. method with adaptive step size, and the As a conclusion, the adaptive step size will blue line is for 5th-order of Runge-Kutta result in more efficient and produce smaller method with fixed step size. From the graph numerical error than fixed step size for 5th- above, we clearly can see that the CPU time order Runge-Kutta method. for the 5th-order of Runge-Kutta method

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Efficiency Error Between Fixed and Adaptive Step Size for Manufactured Solution

Figure 3.7: Efficiency error between the fixed and adaptive step size ofth 5 -order of Runge-Kutta method. (Manufactured solution) In Figure 3.7, the graph was plotted using Conclusion the logarithm value of base 10 for numerical In this research, the purpose is to study error versus the (central processor unit) CPU the RK methods when solving 1st-order time required to complete the computation nonlinear stiff ODEs. We have used two in seconds (s). The graph consists of two test cases which are the test problem and th different methods used which is 5 -order manufactured solution to determine the of Runge-Kutta method with adaptive step accuracy and efficiency of the numerical th size and 5 -order of Runge-Kutta method method. We provided the general formula th with fixed step size. The red line is for 5 - of 2nd-, 4th-, and 5th-order RK methods order of Runge-Kutta method with adaptive with the Butcher tableau. Then, we th step size, and the blue line is for 5 -order proposed a test problem and constructed of Runge-Kutta method with fixed step the manufactured solution with certain size. From the graph above, we clearly can assumptions for the purpose of numerical th see that the CPU time for the 5 -order of illustration. Before that, we verified the Runge-Kutta method with adaptive step existence and uniqueness of the test case by th size is more efficient than 5 -order of using Lipschitz continuity. Runge-Kutta method with fixed step size. We can observe that the numerical error for By using RK methods, the numerical nd th th adaptive step size of 5th-order of Runge- schemes of 2 -, 4 -, and 5 -order are Kutta method is smaller than the fixed constructed for both two test cases. step size. Based on the result, at CPU time Then, we verified the RK methods in s=0.11613, the computed error for fixed and term of numerical error and the order adaptive step size of 5th-order of Runge- of convergence for all accuracy by Kutta methods are 9.60x10-6, and 2.79x10-7, implementing the algorithm using Octave respectively. As a conclusion, the adaptive programming language. We also compared step size will result in more efficient and the efficiency of the RK methods in term of produces smaller numerical error than fixed the plot of numerical error versus the CPU step size for 5th-order Runge-Kutta method. time for both two test cases. Then, we also

Universiti Malaysia Terengganu Journal of Undergraduate Research Volume 3 Number 1, January 2021: 25-36 ADAPTIVE TIME-STEPPING FOR RUNGE-KUTTA METHODS FOR ORDINARY 35 DIFFERENTIAL EQUATIONS compared the fixed and adaptive step size Finally, I would also like to thank my friends for 5th-order RK method in term of the plot who are under the same supervision with my on efficiency error versus CPU time graph both supervisors for their encouragement for both two test cases. and willingness to share their valuable ideas Based on the obtained results, we and knowledge which are very helpful to can conclude that 5th-order Runge-Kutta me for completing this research. method is the most efficient for solving the first-order nonlinear stiff ODEs problem. References We also can conclude that the adaptive step Aliyu, B. K., Osheku C. A.,Funmilayo, A. size is more efficient and produce smaller A., & Musa, J. I. (2014). Identifying computed error than the fixed step size of Stiff Ordinary Differential Equations 5th-order Runge-Kutta method. and Problem Solving Environmetns (PSEs). Journal of Scientific Research Acknowledgements & Reports, 3(11), 1420-1448. At the end of my thesis, I would like to thank Balac, S., & Mahe, F. (2013). Embedded all people who made this thesis possible and Runge-Kutta scheme for step- conducting this final year project is such an size control in the interaction enjoyable experience. picture method.Computer Physics First and foremost, I wish to express Communications, 184(4), 1211-1219. my appreciation to Dr Loy Kak Choon, Butcher, J. C. (1996). A history of Runge- supervisor and Dr Ruwaidiah Binti Idris, Kutta methods. Applied Numerical co-supervisor of this thesis for giving me Mathematics, 20(3), 247-260. many useful advices and guidance with Butcher, J. C. (2005). Numerical Methods patients. Thank you for inspiring me to for Ordinary Differential Equations have a better understanding my research Second Edition. England: John Wiley and giving me many useful information, & Sons Ltd. which helped me to complete this research. I really appreciate it. Butcher, J. C. (2007). Runge-Kutta methods. Scholarpedia, 2(9), 3147.http://www. Next, my sincere gratitude goes to scholarpedia.org/article/Runge-Kutta_ Dr. Mohamed Saifullah Bin Hussin and methods [21 October 2011] Dr. Zabidin Bin Salleh as my coordinators for Final Year Project. All the information LeVeque, R. J. (2004). Finite Difference of writing a formal thesis was given to us Methods for Differential Equation. through workshops. These workshops had Washington: University of Washington. helped me to prepare my project in a proper Oleg G. (2007). Adaptive Stepsize way. Numerical Methods for Solving Most importantly, I would like to Ordinary Differential Equations. express my deepest gratitude for a constant Richard, L. B., & Douglas, F. J. (2011). support, emotional understanding and love , Ninth Edition. that I received from my family especially Error Control and the Runge-Kutta- my mother, Liew Siew Foong. Thank Fehlberg Method, pp. 294-302. you for their motivational help, support Salau, O., & Ajide, O. (2017). Adaptive and encouragement for the success of Time Steps Runge-Kutta Methods: my research. Without their support, this Comparative Analysis of Simulation research cannot be completed with a great Time in Nonlinear and Harmonically success.

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Excited Pendulum and Duffing Thohura, S., & Rahman, A. (2010). Oscillators.British Journal of Applied Comparison of Numerical Methods Science & Technology, 19(5), 1-13. for Solving Initial Value Problems for Serdar I. (2012). A Runge-Kutta Model Stiff Differential Equations. Journal of Based Adaptive Predictive Controller Bangladesh Mathematical Society, 30, for Nonlinear Systems. IFAC 122-132. Proceedings Volumes, 45(17), 151- Wang Z., Wang Q., & Klinke D.Z. (20160. 156. Simulation Study on Effects of Order Stephane B. (2013). High order embedded and Step Size of Runge-Kutta Methods Runge-Kutta scheme for adaptive that Solve Contagious Disease and step-size control in Interaction Picture Tumor Models. Journal of Computer method. Journal of the Korean Science & System Biology, 9(5), 163- Society for Industrial and Applied 172. Mathematics, 17(4), 238-266. Wen S. (2015). An Introduction to Syed Hyder, A. M, & Arnold Willem, Numerical Computation. Penn State H. (2013). Adaptive time stepping University, USA, pp. 173-216. algorithm for Lagrangian transport Zhao, F., & Utkin, V. (1996). Adaptive models: Theory and idealised test Simulation and control of variable- cases, 68, 9-21. structure control systems in sliding regimes. Automatica, 32(7), 1037- 1042.

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