Singular Event Detection

Transcription

1 Singular Event Detetion Rafael S. Garía Eletrial Engineering University of Puerto Rio at Mayagüez Faulty Mentor: S. Shankar Sastry Researh Supervisor: Jonathan Sprinkle Graduate Mentor: Aaron Ames Summer Undergraduate Program in Engineering at Berkeley (SUPERB) 2004 Department of Eletial Engineering & Computer Sienes College of Engineering University of California, Berkeley

2 Singular Event Detetion Rafael S. Garía Abstrat The main objetive in the study of event loation is to find when the solution to an Ordinary Differential Equation (ODE) rosses a swithing surfae. Due to the fat that it is almost always impossible to find the exat solution to an arbitrary ODE, the event must be loated with a numerial approximation to the exat solution. Sine approximate solutions are by nature inexat, the question is whether the real solution displays the same qualitative behavior as its numerial approximation. This raises the need for a number that gauges the validity of this approximation, i.e., a ondition number. The purpose of this researh is to show how numerial approximations of the solutions of simple systems linear systems with simple swithing surfaes hyperplanes an lead to inorret event detetion. Motivated by these examples, a andidate ondition number will be proposed in an attempt to quantify this failure. I. INTRODUCTION Consider an Initial Value Problem (IVP), for a set of ODEs suh that: { f ẋ = + (x) if g(x) 0 f (x) if g(x) < 0 where tɛ[a, b], x(a) = x 0, and x R 2. An event is said to our at time t if g(ˆx + (t )) = 0; the point at whih the vetor field hanges from f + to f. Due to the fat that the majority of the Ordinary Differential Equations (ODEs) do not have an exat solution, it is imperative to utilize numerial methods to approximate the solution of the IVP stated above. As a result, a level of unertainty is introdued in the event loation, whih leads to the need for a guarantee of its reliability. Thus reating a neessity of a ondition number that ould be apable of quantifying the level of orretness of the event loation, i.e, the orretness the approximated solution with respet to the atual solution. In vetor algebra, the ondition number, κ, of a square matrix, A, is defined as: κ(a) = A A 1 where, is a valid vetor norm. Alternatively, if the square matrix A is singular deomposed as A = UΣV T, the ondition number of a matrix an also be stated as: κ(a) = σ max σ min where, σ max and σ min are, respetively, the maximum and minimum salar values of the diagonal matrix Σ. The ondition number of a matrix provides a measure of how sensitive is a matrix to numerial operations and errors in the data. Additionally, it indiates the auray of matrix inversion and the auray of the solution of the lineal equation. If the ondition number of a matrix is approximately 1, the matrix is said to be well-onditioned. In ontrast, if the ondition number of the matrix is muh greater than one, it is said to be ill-onditioned. Ill-onditioned matries do not yield preise numerial solutions [2].This study seeks to propose a ondition number for event loation. This number will serve as a mean to numerially judge its reliability. In fat, the proposed ondition number is: ond#(f) = f + (ˆx(t )) f (ˆx(t )) +g(ˆx(t )) g(ˆx(t )) t p P sign(g(ˆx(t p ))) g(ˆx(t p )) h ˆx(t p ) ˆx(t ) t p P

3 For a given subset, P, suh that: P = {t p R n d : +g(ˆx(t p )) = 0 dt +g(ˆx(t p)) > 0} Further details of this proposed number will be desribed later on. II. NUMERICAL ODE SOLVERS Already knowing that is impossible to find an exat solution to all of the existing ODEs, it is imperative to find numerial ways to approximate the solutions to these equations. In this study, it was desirable to find a numerial solution to an IVP in a given interval a t b. The studied ODEs were of the form: ẋ = ẋ 1 ẋ 2. ẋ n = f(t, x) with given initial values x(a). These equations are of speial interest, for they are utilized to desribe the dynamis and behavior of swithing systems. From [7], it is known that for a given tolerane, τ, the numerial solver an approximate a solution, ˆx, suh that is aurate to the order of r(τ). Furthermore, it an be assumed that the solution is bounded by x(t) ˆx(t) r(h), where h is step size of the solver and r(h) 0 as h 0. Following is a desription of several ODEs that were used to solve the ODEs of the event detetion algorithms performed in setion 3 and 5. All of the methods utilized were either one step methods or multi step methods. The one step methods inluded: Forward Euler s, 2nd order Taylor s, and 4th order Runge-Kotta. Conversely, the multistep methods inluded 3rd order Preditor Correted Adams-Moulton (AM3) and Bakward Differene Formula (BDF). The equation ẋ = f(t, x) was onsidered 2 dimensional, and solved with these methods. The final system of equations is omposed of equations (1) and (2) as follows: A. Forward Euler s Method ẋ 1 = F(t, x 1, x 2 ) (1) ẋ 2 = G(t, x 1, x 2 ) (2) The forward Euler method is attrative beause it is easy to implement and visualize. Although, it has some drawbaks as is the ase of major error results with inreasing step size numbers. This method is based on the one-term Taylor series. From [6] (pg. 142), it an be found that the numerial solution of a system of two variables an be approximated using: x 1 (j + 1) = x 1 (j) + hf(t(j), x 1 (j), x 2 (j)) x 2 (j + 1) = x 2 (j) + hg(t(j), x 1 (j), x 2 (j)) where h is the time interval, h = t j+1 t j, and j=0,1,2,...n. N represents the number of steps in the interval studied. In this method the numerial solution is found by reusing the previously omputed value to approximate the subsequent value of the solution. This proess ontinues iteratively until the desired interval is overed.

4 B. Seond-Order Taylor Method The seond-order Taylor method offers a better loal error than Euler s method; although, it has also showed very unpleasing results with inreasing step size numbers. In [6](pg. 143), it is shown that the numerial solution to an ODE using a seond-order Taylor expansion is given by: x 1 (j + 1) = x 1 (j) + hf (t(j), x 1 (j), x 2 (j)) + h2 2 [ F t (t(j), x 1(j), x 2 (j)) + F t (t(j), x 1(j), x 2 (j))f (t(j), x 1 (j), x 2 (j))] x 2 (j + 1) = x 2 (j) + hg(t(j), x 1 (j), x 2 (j)) + h2 2 [ G t (t(j), x 1(j), x 2 (j)) + G t (t(j), x 1(j), x 2 (j))g(t(j), x 1 (j), x 2 (j))] where h is the time interval, h = t j+1 t j. As in the Forward Euler s method, the numerial solution is found by reusing the previously omputed value to approximate the subsequent value of the solution. C. Runge-Kotta Method The Runge-Kotta method offers another alternative to approximate the solution of an ODE. This method is widely used due to its higher auray in omparison to other one-step methods. Additionally, it offers a better loal error than Euler s and Taylor s, showing good approximations with inreasing step size numbers. From [1], it is found a 4th order Runge-Kotta numerial solution for the system of equations (1) and (2) as follows: where, ˆx 1 (j + 1) = ˆx 1 (j) + h 6 [k 1 + 2k 2 + 2k 3 + k 4 ] ˆx 2 (j + 1) = ˆx 2 (j) + h 6 [m 1 + 2m 2 + 2m 3 + m 4 ] k 1 = F (t(j), ˆx 1 (j), ˆx 2 (j)) m 1 = G(t(j), ˆx 1 (j), ˆx 2 (j)) k 2 = F (t(j) + h 2, ˆx 1(j) + h 2 k 1, ˆx 2 (j) + h 2 m 1) m 2 = G(t(j) + h 2, ˆx 1(j) + h 2 k 1, ˆx 2 (j) + h 2 m 1) k 3 = F (t(j) + h 2, ˆx 1(j) + h 2 k 2, ˆx 2 (j) + h 2 m 2) m 3 = G(t(j) + h 2, ˆx 1(j) + h 2 k 2, ˆx 2 (j) + h 2 m 2) k 4 = F (t(j) + h, ˆx 1 (j) + hk 3, ˆx 2 (j) + hm 3 ) m 4 = G(t(j) + h, ˆx 1 (j) + hk 3, ˆx 2 (j) + hm 3 ) In this method, h is one again the time interval, h = t(j + 1) t(j). The numerial solution is found by reusing the previously omputed value to approximate the subsequent point of the solution. D. Adam-Moulton Method As opposite to the one-step methods developed by Euler, Taylor, and Runge-Kutta; the Adam-Moulton method has the attrative of utilizing previously alulated points to find the subsequent point in the iteration. Methods that follow this desription are often referred as memory methods. Although, it is not self starting and it has to be given previously alulated values to start its omputations depending on the order of its implementation. In order to ahieve a better auray, it is good to utilize a good self-starting one-step method. For this implementation, Runge-Kutta was hosen to perform this task for the three first points of the solution.

5 From [3], it is found that one those 3 initial points are reahed the algorithm used for the 3rd order Preditor Correted Adams-Moulton (AM3) is the following: x 1 (j + 1) = ˆx 1 (j) + h 12 [23F (t(j), ˆx 1(j), ˆx 2 (j)) 16F (t(j 1), ˆx 1 (j 1), ˆx 2 (j 1)) +5F (t(j 2), ˆx 1 (j 2), ˆx 2 (j 2))] x 2 (j + 1) = ˆx 2 (j) + h 12 [23G(t(j), ˆx 1(j), ˆx 2 (j)) 16G(t(j 1), ˆx 1 (j 1), ˆx 2 (j 1)) +5G(t(j 2), ˆx 1 (j 2), ˆx 2 (j 2))] ˆx 1 (j + 1) = h 12 [5F (t(j + 1), x 1(j + 1), x 2 (j + 1)) + 8F (t(j), ˆx 1 (j), ˆx 2 (j)) F (t(j 1), ˆx 1 (j 1), ˆx 2 (j 1))] ˆx 2 (j + 1) = h 12 [5G(t(j + 1), x 1(j + 1), x 2 (j + 1)) + 8G(t(j), ˆx 1 (j), ˆx 2 (j)) F (t(j 1), ˆx 1 (j 1), ˆx 2 (j 1))] As in the previous methods, h represents the time interval, h = t j+1 t j. In this equation the preditor is used to aquire a better approximation. Finally, the solution is found by reusing the previously omputed value to alulate the subsequent point of the solution. E. Bakward Differene Formula Method The BDF method was another memory method implemented beause of its attrativeness of utilizing previously alulated points to find the subsequent point in the iteration. As in the AM3 method, the memory harateristi yields more aurate results than one-step methods suh as similar to the AM3 method, the BDF method is not self starting, it has to be given previously alulated values to start its omputations. For auray issues, a fourth order Runge-Kutta was used to alulate the three first points of the solution. From [6] (pag. 183), it is found that one those 3 initial points are reahed, the algorithm to solve the system is the following: ˆx 1 (j + 1) = 6 11 [6F (t(j + 1), ˆx 1(j + 1), ˆx 2 (j + 1)) + 3ˆx 1 (j) 3 2 ˆx 1(j 1) ˆx 1(j 2)] ˆx 2 (j + 1) = 6 11 [6G(t(j + 1), ˆx 1(j + 1), ˆx 2 (j + 1)) + 3ˆx 2 (j) 3 2 ˆx 2(j 1) ˆx 2(j 2)] As in the previous methods, h also represents the time interval, h = t j+1 t j. In this equation the preditor is used to aquire a better approximation. Finally, the solution is found by reusing the previously omputed value to alulate the subsequent point of the solution. III. EVENT DETECTION TECHNIQUES The next task is to find and apply event detetion tehniques (EDTs), to the already stated IVP given by: { f ẋ = f(x) = + (x) if g(x) 0 f (3) (x) if g(x) < 0 for the given interval tɛ[a, b], and initial ondition x = x 0, in whih x R 2. In order to simplify further explanations, an event detetion as (3) will be referred as f = (f +, f, g) EDTs utilize different algorithms with the objetive of finding the most aurate way to loate events. Two different algorithms were simulated utilizing MATLAB. The first method will be referred as the zero-loation method and is similar to that of [4], while the seond method is referred as the minimum-neighborhood method, whih is similar to the interval bisetion algorithm found in [5], but with some slight hanges.

6 A. Zero-Loation Method The zero-loation method (ZLM) starts by numerially solving the ODE, ẋ = f + (x). The approximated solution is denoted by ˆx + (t) with initial ondition ˆx + (0) = x 0. Subsequently, at eah time step, t n of the interval, it is heked if g(x + (t n )) > 0 and g(x + (t n+1 )) < 0. If this ondition holds, an event has ourred. Consequently, a linear interpolation between the points ˆx + (t n ) and ˆx + (t n+1 ) is performed, and denoted by ˆx + (t). Then, it is numerially solved for t suh that g(ˆx + (t )) = 0 After the ourrene of an event, the same proess is started again, exept that ẋ = f (x) is solved with ˆx (0) = ˆx + (t ). Finally, this algorithm is repeated (with its respetive sign hanges) if multiple events are to be deteted. B. Minimum-Neighborhood Method The minimum-neighborhood method (MNM)follows the first ouple of steps as the zero-loation method. It starts by numerially solving ẋ = f + (x), where the approximated solution is denoted by ˆx + (t) with initial ondition ˆx + (0) = x 0. Subsequently, at eah time step t n of the interval, it is heked if g(x + (t n )) > 0 and g(x + (t n+1 )) < 0. If this ondition holds, an event has ourred and the next step is to subdivide the interval [t n, t n+1 ]. This is performed by seleting times τ 0 = t n < τ 1 < < τ k = t n+1 where k is input by the user. Subsequently, values for i are found suh that g(ˆx(τ i )) > 0 and g(ˆx(τ i+1 )) < 0 holds. One this ondition is true, the interval is deremented again for the points in whih it holds, until values are found suh that g(ˆx(τ i )) and g(ˆx(τ i+1 )) are approximately zero. Consequently, t is set to τ i. After loating the event, the same proess is started again, exept that ẋ = f (x) with initial ondition ˆx (0) = ˆx + (t ). C. Example The following IVP was implemented to illustrate the solutions of both algorithms: ( ) ( ) 1 1 x1 ẋ = 1 1 with an h = 0.01 and x + (a) = ( 2, 2) T. By looking at the graphs, it is notieable that both methods follow x vs. X 1 Euler s INT Runge Kotta INT Euler s MNM Runge Kotta MNM Event at X 1 = X 1 Fig. 1. Event detetion for both methods a very similar behavior. Although, they are slightly different in terms of omputational issues. For instane, the MNM offers greater preision but at the expense of more omputational time. In our ase, the omputational

7 9.148 vs. X 1 Euler s INT Runge Kotta INT Euler s MNM Runge Kotta MNM Event at X 1 = X 1 Fig. 2. Euler s proximity for both methods vs. X 1 Euler s INT Runge Kotta INT Euler s MNM Runge Kotta MNM Event at X 1 = X 1 Fig. 3. Runge-Kotta proximity for both methods time was an issue for the alulation of the ondition number. As a result,the ZLM was used to performed the event detetion tasks. IV. SOLUTION BEHAVIOR Up to this point, the importane of the existene of a ondition number has been pointed out but not properly defined in terms of good or bad behavior of an IVP as given in (3). In order to do this, it is important to define what onstitutes a good senario and what does not. A good senario would be one in whih { r x(t) ˆx(t) + (h) if t [a, t ] r (h) if t [t, b] when x + (t) ˆx + (t) r + (h) and x (t) ˆx (t) r (h) where ˆx ± (t) are the approximate solutions of x ±. This definition of a good senario an be simplified by taking r(h) = max{r + (h), r (h)} yielding the following definition:

8 Definition 4.1: The solution of the IVP f = (f +, f, g) is said to be well-posed if x + (t) ˆx + (t) r(h) and x (t) ˆx (t) r(h) for some r(h) implies that x(t) ˆx(t) r(h) for all t [a, b]. This definition of a well-posed event an be used to give the definition of an ill-posed event Definition 4.2: The solution of the IVP f = (f +, f, g) is said to be ill-posed if x + (t) ˆx + (t) r(h) and x (t) ˆx (t) r(h) for some r(h), but x(t) ˆx(t) > r(h) for some t [a, b]. Alternatively, an f is ill-posed if and r(h) > r(h) for all h. x(t) ˆx(t) r(h) An ill-posed solution reates a very unstable behavior in whih the aurate detetion of an event annot be guaranteed. The above definitions involve knowing the atual solution in order to detet wether an IVP f is ill-posed. Sine it annot be assumed that the atual solution is known, the auses some diffiulty. To irumvent this, we introdution the following lemma: Lemma 4.1: If two different approximate solutions ˆx(t) and, ŷ(t) of an IVP f = (f +, f, g), are ill-posed, i.e., if x ± (t) ˆx ± (t) r(h) and x ± (t) ŷ ± (t) r(h) but, ˆx(t) ŷ(t) r(h) then the atual solution of the IVP is ill-posed: x(t) ˆx(t) r(h) and x(t) ŷ(t) r(h). V. CONDITION NUMBER Following is a detail explanation of the set of speifiations utilized to propose a ondition number for event detetion tehniques. Consequently, the proposed onditioned number will be expressed and thoroughly explained. Finally, two examples are going to be presented with the objetive of providing an idea on the behavior of the ondition number in event detetion appliations.

9 A. Speifiations The following are a set of speifiations reated with the objetive of narrowing the behavior of any prospetive andidate for the ondition number: 1) For the perfet ase, i.e., a ase in whih the solution hits the surfae orthogonally, the ondition number should be one. 2) As the ondition number inreases, the auray of the approximate solution dereases; beoming progressively ill-posed. 3) The ondition number is defined to be infinite for a singular event, i.e., when the vetor field is tangent to the surfae at the event. In order to understand these speifiations, it is important to know their impliation. First of all, these onditions imply that the ondition number will depend on a funtion that is apable of desribing the way in whih the approximate solution intersets a swithing surfae. As a result, the Lie Derivative of g with respet to f, referred as g, and given by g(x) = g(x) x f(x) was targeted to perform this task. This funtion relates the hange of the funtion g(x) in the diretion of f(x). Therefore, in the ase in whih g = 0 at an event, it means that the swithing surfae G 0 = {x g(x) = 0} is tangent to the vetor field at this point, i.e., a singular event loation. Conversely if g = 1, it means that the vetor field is orthogonal to the surfae G 0 and the approximate solution is ompletely reliable, i.e, well-posed. To state ondition (2) of the speifiations in a more preise way, onsider two IVP f 1 = (f1, f 1 +, g 1) and f 2 = (f2, f 2 +, g 2). Beause these are ill-posed, if then x ± 1 (t) ˆx± 1 (t) r 1(h) and x ± 2 (t) ˆx± 2 (t) r 2(h) x 1 (t) ˆx 1 (t) r 1 (h) and x 2 (t) ˆx 2 (t) r 2 (h) where r 1 (h) > r 1 (h) and r 2 (h) > r 2 (h). Now a mesure of how ill-posed these events are an be obtained by onsidering r 1 (h) r 1 (h) and r 2 (h) r 2 (h). Therefore if ond#(f 1 ) denotes the ondition number of f 1 and ond#(f 2 ) denotes the ondition number of f 2, speifiation number (2) an be stated preisly as: if ond#(f 1 ) > ond#(f 2 ) r 1 (h) r 1 (h) > r 2 (h) r 2 (h). (4) This ondition is extremely important in that if a prospetive ondition number an be shown to satisfy this ondition (as well as speifiations number (1) and (3)), then it is a valid ondition number for the event detetion problem. B. Proposed Condition Number Consider the set P given by: P = {t p R n d : +g(ˆx(t p )) = 0 dt +g(ˆx(t p)) > 0} where +g and g are the Lie Derivatives of the event funtion at the right and left side of the event respetively and t p is the point at whih the approximate solution yields an g equal to zero, i.e., the time at whih the approximate solution is tangential to the solution. Subsequently, for a given funtion f = (f +, f, g), in whih g is a hyperplane of the form g(x) = x + α. Having said this, the proposed ondition number will have the form: ond#(f) = γ 1 (ˆx(t )) +g(ˆx(t )) g(ˆx(t )) t p P γ 2 (ˆx(t p )) g(ˆx(t p )) t p P ˆx(t p ) ˆx(t ) γ 3 (ˆx(t p )) (5) where the gamma funtions γ 1, γ 2 and γ 3 will be disussed in the remainder of this setion. This ondition number an be extended to the ase when g(x) is an arbitrary (smooth) funtion beause of a speifi

10 transformation. Namely, for the IVP f = (f, f +, g) setting T (x) = (x, z = g(x)) T we obtain an equivalent IVP f = ( f, f +, g) by setting ( ) ( f x f = ) ( ) ( (x) f z g(x) + x f = + ) (x) g(x, z) = z. z +g(x) Note that in this ase, g has the desired form. Using this transformation, for an IVP f = (f, f +, g) with an arbitrary smooth event funtion g, the ondition number an be defined by ond#(f) = ond#( f). So, from this point on, it an be assumed that g(x) = x + α. In order to disuss the gamma funtions in more detail, some speial ases will be onsidered. In the ase in whih, P =, i.e., there is not a tp that satisfies +g(ˆx(t p )) = 0, the ondition number beomes: ond#(f) = γ 1 (ˆx(t )) +g(ˆx(t )) g(ˆx(t )) Alternatively, when there are no events in the interval [a, b] of the integration, i.e., there does not exists a t suh that g(ˆx(t )) = 0, and P then the ondition number beomes: ond#(f) = γ 2 (ˆx(t p )) g(ˆx(t p )) t p P Finally, if P = and there are no events, the ondition number is not defined. In this ase the IVP is a normal differential equation with no swithing, so a ondition number is not needed. C. The gamma funtions γ 1 (x): The first gamma funtion was taken to be γ 1 (ˆx(t )) = f + (ˆx(t )) f (ˆx(t )) The objetive of this funtion is to sale the ondition number, so that it is insensitive the hanges in the magnitude of the vetor field as it rosses the swithing surfae and sensitive to hanges in the orientation of the vetor field at the swithing surfae. To justify this, onsider the ase when P = but an event ours, i.e., there is not a t p that satisfies +g(ˆx(t p )) = 0, (5) resulting in: ond#(f) = f + (ˆx(t )) f (ˆx(t )) +g(ˆx(t )) g(ˆx(t )). This is the ase for the IVP given by f = (f, f +, g) where ( ) f + (x) = f (x) =, g(x) = x 0 In this ase, without γ 1 (ˆx(t )), for two onstants > > 0, 1 + g(ˆx(t )) g(ˆx(t )) = 1 < 1 1 = + g(ˆx(t ))L f g(ˆx(t )) so the ondition number would sale with magnitude, while it should remain onstant under this saling. By adding the term γ 1 (ˆx(t )), the magnitude of the vetor field is saled down by its magnitude. As a result, the ondition number is no longer sensitive to the magnitude of the vetor field at the swithing surfae, i.e., we have ond#(f ) = f + (ˆx(t )) f (ˆx(t )) + g(ˆx(t )) g(ˆx(t )) = = 1 = = f + (ˆx(t )) f (ˆx(t )) + g(ˆx(t ))L f g(ˆx(t )) = ond#(f )

11 Alternatively, γ 1 (ˆx(t )) also aounts for the orientation of the vetor field at the swithing surfae. For example, onsider the IVP given by f = (f, f +, g) where ( ) f + (x) = f 1 (x) =, g(x) = x without the γ 1 (ˆx(t )) term, the ondition number would not be sensitive to hanges in orientation of vetor field at the swithing surfae beause for two onstants > > 0, 1 + g(ˆx(t )) g(ˆx(t )) = 1 = 1 + g(ˆx(t ))L f g(ˆx(t )) so as the vetor beause more and more tangent to the swithing surfae these values do not hange. Now, with the γ 1 (ˆx(t )) term, we have ond#(f ) = f + (ˆx(t )) f (ˆx(t )) + g(ˆx(t )) g(ˆx(t )) = > 1 + ( ) 2 = f + (ˆx(t )) f (ˆx(t )) + g(ˆx(t ))L f g(ˆx(t )) = ond#(f ) so as the vetor field beomes more tangent to the swithing surfae, the ondition number inreases as desired. γ 2 (x): Another ase for the ondition number is when there are no events our in the interval [a, b], but the set P is not null, i.e., t suh that g(ˆx(t )) = 0, and so P. Assuming for simpliity that P = {p} (5) beomes: ond#(f) = γ 2(ˆx(t p )) g(ˆx(t p )) In this ase we would like ond#(f) to measure how lose the solution gets to the swithing surfae G 0. With this in mind take γ 2 (ˆx(t p )) = sign(g(ˆx(t p ))). Wherein it follows that γ 2 (ˆx(t p )) g(ˆx(t p )) = 1 d(x(t p ), G 0 ) where d(x(t p ), G 0 ) is the distane between x(t p ) and a subset of points G 0 = {x g(x) = 0}. This is due to the fat that the event funtion was hosen to be of the form g(x) = x + α. To see this, note that but beause g(x) = x + α, Now, substituting into (6), as desired. d(x(t p ), G 0 ) = min y G 0 d(x(t p ), y) = min y G 0 x(t p ) y min x(t p ) y = g(x(t p )) = sign(g(x(t p )))g(x(t p )) y G 0 γ 2 (ˆx(t p )) g(ˆx(t p )) = 1 sign(g(x(t p )))g(x(t p )) = 1 d(x(t p ), G 0 ) γ 3 (x): The last ase to analyze is when there is an event in the interval [a, b], but P. In this ase the ondition number will be: ond#(f) = γ 1 (ˆx(t )) +g(ˆx(t )) g(ˆx(t )) t p P γ 2 (ˆx(t p )) g(ˆx(t p )) ˆx(t p ) ˆx(t ) γ 3 (ˆx(t p )) t p P (6)

12 This last equation ontains all of the previously indiated onstants exept from γ 3 (ˆx(t p )). In this ase, γ 3 (ˆx(t p )) has the objetive of aounting for the step size seletion in the ondition number equation. As a result, it was hosen to be equal to h. Additionally, the equation ontains the term ˆx(t p ) ˆx(t ). This term has the objetive of taking into aount how far the point where the event ours, x(t ), and the point tangential to the surfae, x(t p ). This ompensating for big strethes and ompressions of the solution near the surfae. Meaning that the further apart these points are, the more strethed and ill-posed the solution beomes. Conversely, the losest these points are, the more ompressed and well-posed should beome. Therefore, we an summarize by saying that the proposed ondition number is: D. Examples ond#(f) = f + (ˆx(t )) f (ˆx(t )) +g(ˆx(t )) g(ˆx(t )) t p P sign(g(ˆx(t p ))) g(ˆx(t p )) t p P h ˆx(t p ) ˆx(t ) (7) Following are two main examples that were simulated to alulate their respetive ondition numbers. The first example has a linear ODE funtion in f +, while the seond example has a non-linear/polynomial ODE. Reall that the ondition number was simulated with funtions of the form g(x) = bx + α. Also, 4th order Runge-Kotta was utilized to solve the funtion with a time step of ) Linear Funtion: The following IVP was implemented to find its ondition number: ( ) ( ) 1 1 x1 if g(x) x 2 ẋ = ( 0 1 ) if g(x) < 0 with g(x) = x 2 + α were α varies from α = 9 to 10 at 0.01 steps and x + (a) = ( 2, 2) T. Fig. 5, shows the 150 vs. X 1 Euler Runge Kotta x 2 = alpha X 1 Fig. 4. Zoom-in of the ODE solution plot of ondition number as alpha hanges. From this plot, it an be seen that the ondition number inreases and the solution to the IVP beomes progressively ill-posed, i.e, as α Looking at Fig. 4, it an be seen that the two approximate solutions have larger errors than what it should be expeted. This number is lose to where it is supposed to happen exatly when the approximate solution is tangent to the surfae, whih agrees with the large ondition number aquired in Fig. 5.

13 120 Condition Number vs. alpha Condition Number alpha Fig. 5. Condition Number for α = 9 : 0.01 : 10 2) Non-Linear Funtion: The following IVP was also analyzed to find its ondition number: ( ) ẋ = 1 10( 5x 4 1+β 4 ) β 5 ( 0 1 ) if g(x) 0 if g(x) < 0 with g(x) = x 2 + α were α varies from α = 4 to 8 at 0.01 steps and x + (a) = (2.5β, f + 2 (2.5β))T. vs. X Euler Runge Kotta = alpha X 1 Fig. 6. ODE solution In this example, two different things were tested. First of all, the value of alpha was varied to see variations of the ondition number throughout the interval. Seond of all, values of β in f + were varied to see if the ondition number behaved as expeted with hanges in x(t ) and x(t p ). Fig. 6 shows that the event beomes ill-pose at approximately x 2 = Also, Fig. 7, shows the plot for the ondition number as alpha varies and β remains onstant at 1. In this ase, the plot shows that the IVP beomes progressively ill-posed as α Fig.8 shows the effet that β produes on the ondition number. The graph shows that as β inreases, the value of the ondition number inreases as well. This is due to the fat that x(t ) and x(t p ) will be further apart.

14 Condition Number vs. alpha Condition Number alpha Fig. 7. Condition Number for α = 4 : 0.01 : 8 Condition Number vs. alpha Condition Number Fig. 8. Condition number for multiple β This an also be seen in Fig. 9. This figure was saled with the objetive of making all of the magnitudes equal and fousing on the behavior of the graph. As it an be seen, the slope of the bottom part of the surfae inrements as β inrements. As a result, the value of β is taken into aount as expeted. REFERENCES [1] Ordinary Differential equations.. Available Online, June [2] Matrix Condition Number. Available Online, June [3] Ordinary Differential Equations. Computational Siene Eduation Program, [4] W. H. Enrht, K. R. Jakson, S. P. Nørset and P. G. Tompsen. Effetive Soulution of disontinuous IVPs a Rounge-Kotta formula pair with interpolants. Appl. Math. Comp., vol. 27 (1998), pg [5] C. Moler. Are we there yet? Matlab News and Notes. Simulink 2 Speial Edition (1997), pg [6] L. F. Shampine. Numerial Solutions for Ordinary Differential Equations. Chapman and Hall, [7] L. F. Shampine, S. Thompson. Event Loation for Ordinary Differential Equations. Computers with Mathematis and Appliations, vol. 39 (2000), pg

15 Fig. 9. Relation between β,γ, and ondition number