Determination of Inner and Outer Bounds of Reachable Sets
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1 Determination of Inner and Outer Bounds of Reachable Sets Francisco Rego Abstract - To verify the safety of a dynamic system, one important procedure is to compute its reachable set of states. There are different methods of computing reachable sets, namely interval integration, capture basin, methods involving the minimum time to reach function and level set methods. This work deals with interval integration to compute subpavings to over or under approximate reachable sets. An algorithm to over and under estimate sets through subpavings, which potentially reduces the computational load when the test function or the contractor is computationally heavy, is implemented and tested. This algorithm is used to compute inner and outer approximations of reachable sets. The test function and the contractors used in this work to obtain the subpavings involve guaranteed integration, provided either by the Euler method or by VSPODE. The devised methods were applied to compute inner and outer approximations of reachable sets for the double integrator example. From the results it was observed that using contractors instead of test functions yields much tighter results. It was also confirmed that for a given minimum box size there is an optimum time step such that with a greater or smaller time step we obtain worse results. Unexpectedly, when using VSPODE for the guaranteed integration we obtain worse results than with the Euler method, suggesting a possible flaw in the application of the program or in the program itself. Introduction 1.1 Background The concept of reachable sets consists of determining which states can be reached by a particular system with known dynamics from a set of initial conditions (forwards reachable set), or from which initial conditions is it possible to reach a specified target set of states (backwards reachable set) as illustrated in figure 1. One method of estimating reachable sets is simulating a single trajectory of the system at a time. However, this is a computationally expensive method if we are dealing with systems with many different state values and input signals. Also, using only simulations, it is possible to miss important but isolated trajectories that lead to unsafe sets. To avoid those problems, there are methods of estimating reachable sets capturing the behavior of entire groups of trajectories at once. This concept of capturing the behavior of groups of trajectories is present in works about Region of Attraction (RoA) analysis and reachability for systems with limited energy inputs, solving a Sum-of-Squares (SOS) relaxation (Packard, Topcu, Seiler, & Balas, 2009). Another work (Djeridane & Lygeros, 2006) discusses the use of neural networks to approximate the solution of the Hamilton- Jacobi-Isaacs (HJI) PDE to obtain the evolution of reachable sets. A MATLAB level set toolbox which approximates Hamilton- Jacobi-Isaacs PDEs through finite differences was developed by Mitchell (Mitchell, 2002). The method of Jaulin (Lhommeau, Jaulin, & Hardouin, 2011) aims to compute guaranteed bounds on the capture basin of a dynamical system using Interval Analysis. To obtain guaranteed bounds on reachable sets one solution is to compute reachable sets using wrappers, such as interval vectors, instead of points. Guaranteed integration using Interval Analysis is a widely developed field (Neher, Jackson, & Nedialkov, 2007) which gives guaranteed solutions of Initial Value Problems (IVP). Interval analysis was applied also on several other aerospace related topics such as finding trim-points for nonlinear aircraft models (van Kampen, Chu, Mulder, & van Emden, 2007), pilot model identification (van Kampen, Zaal, de Weerdt, Chu, & Mulder, 2010), integer ambiguity resolution for aircraft attitude determination (van Kampen, de Weerdt, Chu, & Mulder, 2009) (de Weerdt, van Kampen, Chu, & Mulder, 2008), spacecraft reentry optimization (Filipe, de Weerdt, van Kampen, Chu, & Mulder, 2009) and fuel optimization for constrained spacecraft formation rotations (de Weerdt, Chu, & Mulder, Global Fuel Optimization for Constrained Spacecraft Formation Rotations, 2009). The Problem - Reachable Sets 2.1 Definition In this work we will consider non-autonomous dynamical systems with a control and a disturbance input. Such systems can be described by the following formulation. 1
2 (2-1) where, is the state, is the control input, is the disturbance input, is an initial set of states, and is a target set of states. The spaces of admissible control input and disturbance input trajectories are denoted as the spaces of piecewise continuous functions, and respectively. The information pattern is specified by a strategy for the control input, which is a map specifying the input signal as a function of the disturbance signal. In this work we will consider non-anticipative strategies: (2-2) Formally, we can define the forward and backwards reachable sets as follows. Definition I.1 (Forwards Reachable Set). The forward reachable set at time, of the system (2-1) from the initial set, is the set of all states such that there exists a strategy, for which is reachable from some, for all disturbance inputs,along a trajectory satisfying (2-1). Definition I.2 (Backwards Reachable Set). The backward reachable set at time, of the system (2-1) from the target set, is the set of all states such that there exists a strategy, for which some is reachable from, for all disturbance inputs, along a trajectory satisfying (2-1). trajectory. One possible way to obtain guaranteed bounds is using interval analysis, which consists of working with intervals instead of real numbers and extends some of the usual operations of real analysis into operations between intervals (Moore, Kearfott, & Cloud, 2009). Two important characteristics of an interval are its width and midpoint defined by equations (3-1) and (3-2) respectively. (3-1) (3-2) For interval vectors (also referred to as boxes) the width, midpoint and volume are defined as follows: (3-3) (3-4) (3-5) Using interval analysis we can search for inclusion functions, from intervals to intervals, such that if is contained in, is contained in. One property of an inclusion function is that it can be inclusion isotonic (if then ). Inclusion functions can have many forms such as the interval extension, the centered form the mean value form, the slope form and the Taylor expansion (Moore, Kearfott, & Cloud, 2009). The simplest form to have an inclusion function is to substitute the operations in the expression of with their equivalent on interval arithmetic. In this case we have an interval extension. The interval arithmetic operations are an extension to intervals of the arithmetic operations between real numbers, namely the sum, subtraction, multiplication and division, and are computed as expressed in equations (3-6) to (3-9). (3-6) (3-7) Fig. 1 The Approach - Interval Analysis 3.1 Interval Analysis Grid based methods for computing reachable sets are not able to give guaranteed bounds, since it is possible to miss some important (3-8) (3-9) The rational interval functions are then a subset of the interval valued functions defined according to definition II. 2
3 Definition II. A rational interval function is an interval-valued function whose values are defined by a specific sequence of interval arithmetic operations. The next theorem holds for rational inclusion functions but it could be extended for various other functions well defined on interval analysis such as or. Theorem I If is a rational expression in the interval variables and a finite set of constant intervals with interval arithmetic operations, then implies for every set of interval numbers for which the interval arithmetic operations in are defined. From this theorem we can conclude that for smaller intervals where the inclusion function is evaluated, the estimation on provided by the inclusion function is tighter. For practical implementations of interval analysis there are several libraries, on many different programming languages, that create a new interval variable type and define the basic functions on interval variables. Two of the most popular libraries that implement interval analysis are PROFIL\BIAS (Knüppel, 1999) for C++ and INTLAB for MATLAB (Rump, 1999). Finally, when dealing with finite precision, another problem arises. There is a basic requirement that the result of a computation with finite precision encloses the real result. To satisfy this requirement, a library must make use of outward rounding, which consists of performing rounding towards for the lower bound of an interval and rounding towards for the upper bound. 3.2 Guaranteed Integration It is also possible to integrate interval functions in the same way we integrate real valued functions (Moore, Kearfott, & Cloud, 2009). For an interval valued function and a uniform subdivision of we can define finite sum. Then we can define the integral of follows. (3-10) as (3-11) In practice, one possible way to compute an enclosure of the integral of an interval function is to use the above definition stopping at a finite. Using the above method or other to perform integrations, it is possible to get an inclusion of the solution of an initial value problem if we perform Picard iterations. The homogeneous initial value problem, and, is formulated as the following integral equation. where is defined as follows. (3-12) (3-13) is an interval majorant of the operator if for. The interval operator is inclusion isotonic if implies. If is an inclusion isotonic interval enclosure of the following interval operator is inclusion isotonic. (3-14) If is an inclusion function of the solution of an initial value problem then is also an inclusion function of the solution of an IVP and. Therefore a finite number of the Picard iterates on (3-13) are desirable to obtain a solution to an initial value problem. This method will be useful to compute the enclosures of the trajectories on section Subpavings A subpaving of a box is a union of non-overlapping subboxes of with non-zero width. Non-overlapping means that the intersection of two such subboxes must be empty or have zero volume, that is, they may only intersect on their boundaries. Subpavings are useful to provide over- ( ) and underapproximations ( ) of a set such that. The theory associated with subpavings is thoroughly discussed in the book of Jaulin (Jaulin, Kiefer, Didrit, & Walter, 2001). A subpaving of can be obtained through successive bisections and eliminations, on that case it is called a regular subpaving. A bisection is a process of dividing a box into two smaller boxes of equal width. and are obtained as follows. Let s consider the index 3
4 of the first component of of maximum width, that is, is defined from (3-15). (3-15) The boxes and are then defined as follows. (3-16) (3-17) To under- and over-approximate a set by two subpavings and,, an algorithm was developed by Jaulin (Jaulin, Kiefer, Didrit, & Walter, 2001) which is named SIVIA (Set Inverter Via Interval Analysis). It utilizes an interval Boolean function that if it gives as result, then, if it gives as result, then, and it gives as result if no conclusion could be made. The SIVIA algorithm in pseudo code is as follows: Algorithm SIVIA(in:,, ; inout: ) 1. ; 2. if return; 3. if { ; return;}; 4. SIVIA(,,, ); SIVIA(,,, ); The Proposed Solution 4.1 Formulation for Reachable sets A similar procedure from that of the paper of Jaulin (Lhommeau, Jaulin, & Hardouin, 2011) could be used to compute enclosures for reachable sets and is based on Assumption I, illustrated in figure 2. Assumption I If and are such that, if is a box of states, if and if then (4-1) (4-2) Algorithm SIVIA(in:,, ; inout:, ) 1. if return; 2. if { ; ; return;}; 3. if { ; return;}; 4. SIVIA(,,,, ); SIVIA(,,,, ); In SIVIA a natural choice for initializing and would be. The choice of controls the precision of the enclosure, since the smallest boxes on the subpaving are the first boxes, obtained by successive bisections, for which. A smaller yields a tighter enclosure but increases the computational time. A possible way of improving the overapproximation is the use of contractors instead of test functions. Contractors are functions from interval vectors to interval vectors that satisfy the following conditions ( is a set associated with the contractor): (contractance), (correctness), The SIVIA algorithm can be adapted for use with contractors, yielding a non-regular subpaving. The algorithm is as follows. Fig. 2 In Assumption I we consider a trajectory of the type which is the solution of the initial value problem for the initial vector, To get tighter bounds and to improve convergence, it is better to use contractors instead of test functions. For that purpose we can use Assumption II, illustrated on figure 3. In Assumption II we consider an inverse trajectory that, if is the solution of, is the solution of, where and. is the complementary set of. Assumption II If and are such that, if is a box of states, if and if then: The interval function (4-3) Is a contractor for the constraint 4
5 The interval function a box contains the sides of. where set (4-4) Is a contractor for the constraint Fig. 4 Fig. 3 From Assumptions I and II, by starting with a box and having a description of and, using a branch and prune algorithm we can obtain an inner or an outer bound for the reachable set. To obtain a tighter enclosure, given the target set we can compute inner and outer bounds for using a number of time steps and using and as intermediate results, where is an integer from 0 to. Using intermediate time steps can help reduce the effects associated with the overestimation of, because it increases non-linearly with. On the other hand, with more time steps the accumulated error increases since on each time step we are overestimating the reachable set at most by the size of the smallest box on the subpaving plus the highest overestimation of and the error associated with assumption I or II. Computational Application 5.1 Branch and Prune (B&P) algorithm with flag To implement the method described in the previous section two algorithms of determining subpavings that enclose a given set were implemented and tested. They were named Branch and Prune algorithm and Branch and Prune algorithm with flag. For the test function of assumption I both algorithms yield the same result, however the Branch and Prune algorithm with flag, that is described in figure 14, is more efficient. Figure 4 illustrates six iterations of the algorithm used to enclose a circle starting with 5.2 Branch and Contract (B&C) On figure 15 we describe an algorithm, based on the B&P algorithm with flag described before, of computing subpavings using contractors. This algorithm will be named Branch and Contract (B&C). For the B&C algorithm we need the function that computes a subpaving of. The function is defined as follows: On the Branch and Contract algorithm, we use a contractor, which is the term of a sequence, under some stopping criteria. The stopping criteria considered were defining some and the series stops at or the series stops when, on the last case is a measure of the required precision. Figure 5 illustrates six iterations of the B&C algorithm of the same problem of figure 4. Fig. 5 5
6 Figure 6 contains the eliminated boxes after the sixth iteration of figure 5. To reduce the overestimation it is preferable to compute the enclosure of the integration in time steps. For that purpose, naming the Euler method as, the application of the Euler method to compute on the rest of this work will be performed as follows. Algorithm (inout:,,, ) 1. ; ; ; if return; 4. ; goto 2 ; Fig Applying the Euler Method To compute the guaranteed integration of the trajectories one possible method is the Euler method (Lhommeau, Jaulin, & Hardouin, 2011). The Euler method computes an inclusion function of the trajectory that determines the states reached in a time for initial conditions in the box. The flow is evaluated in a box, that includes, which is an expansion of as is shown in figure 7, where. Then we compute. If the result is valid and, otherwise it is not possible to reach a conclusion and we consider. The box is computed with the function, where the function is defined as follows: (5-1) 5.4 Applying Interval Taylor Models Besides boxes, a more flexible type of wrappers are Interval Taylor Models which consist of a polynomial, symbolic, part and an interval part. To compute enclosures of solutions of initial value problems using Interval Taylor Models the possibility considered in this work is using the VSPODE solver (Lin & Stadtherr, 2006). Application Example 6.1 Branch and Prune The devised methods were tested for the double integrator system (6-1) where the control input is contained in (6-1) To obtain a general picture, the results of and, for for 10 time steps, a minimum box size of and with are displayed on figure 9. The area of computed with the sum of the area ( ) of all boxes of the subpaving describing is and the area of computed with the same method is. Fig. 7 6
7 Fig Branch and Contract This section contains the results obtained when using the B&C algorithm. The results displayed on this section are for and, with a target set and with a minimum box size of. The initial box, root of the subpaving is. The line in blue in the following figures is the analytical solution. The computed set obtained with one time step is displayed on figure 10. The area of the computed set is. Fig VSPODE To observe the effects of using VSPODE to perform the guaranteed integration, the results of and using VSPODE for state propagation, for 10 time steps with a minimum box size of are displayed in figure 13. The target set is and the initial box is. Fig. 10 The sets and computed with 10 time steps are in figure 11. The area of computed with 10 time steps is and the area of is. Fig. 13 From the results of figure 13 the computed area of is and the area of is. Conclusions and recommendations Fig. 11 The results and for 20 time steps are displayed in figure 12. The area of is and of is. 7.1 Conclusions We have seen that using the field of interval analysis, namely the construction of subpavings through set inversion and guaranteed integration, we can compute inner and outer bounds of reachable sets defined by the union of non-overlapping boxes. An algorithm of constructing subpavings (B&P) was implemented where only boxes in contact with already eliminated boxes or in contact with 7
8 a pre-defined set of boxes can be eliminated. The algorithm was tested and it was proven to work as expected, however suffering from the curse of dimensionality, the time to run grows exponentially with the dimension of the problem. From the results of the devised method applied to compute bounds on the reachable set of the double integrator system we can draw some conclusions. We observe that using a branch and contract algorithm yield much tighter results than using a branch and prune algorithm. Increasing the number of time steps above a certain number may have a detrimental effect on the outer bound since at each time step we have a wrapping effect of the intermediate solution, but has a positive effect on the inner bound since we are considering more samples of the control input. The application of VSPODE showed that it yields worse results than Euler integration indicating that there is still work to be done either on how the program is applied or on VSPODE itself. 7.2 Recommendations As a recommendation of future work, to devise new guaranteed integration methods and apply them on the methods discussed on this work or to apply other available guaranteed integration programs may improve the effectiveness of the discussed methods. Also, the application of VSPODE on the created programs may be improved or possibly another program that implements interval Taylor models may be created since it looks that they can have a great advantage comparing to use boxes only. It was observed that programming on C++ yielded much more computationally efficient programs than those implemented on MATLAB the applicability of the devised methods may improve significantly if the methods were programmed on C++. Finally, another possible future work is, instead of using samples of the control input or disturbance input, we can estimate the optimal control input or disturbance input knowing that the optimal control input is the input that maximizes the system function on the direction towards the reachable set at a particular moment and state, and the optimal disturbance input is the input that minimizes the state function towards the reachable set at that moment. References de Weerdt, E., Chu, Q., & Mulder, J. (2009). Global Fuel Optimization for Constrained Spacecraft Formation Rotations. AIAA Guidance, Navigation, and Control Conference. Chicago. de Weerdt, E., van Kampen, E., Chu, Q., & Mulder, J. (2008). Integer Ambiguity Resolution Using Interval Analysis. ION Journal of Navigation, Djeridane, B., & Lygeros, J. (2006). Neural approximation of PDE solutions: An application to reachability computations. 45th IEE Conference on Decision & Control, (pp ). Filipe, N., de Weerdt, E., van Kampen, E., Chu, Q., & Mulder, J. (2009). Terminal Area Energy Management Trajectory Optimzation Using Interval Analysis. AIAA Guidance, Navigation, and Control Conference. Chicago. Jaulin, L., Kiefer, M., Didrit, O., & Walter, E. (2001). Applied Interval Analysis. Springer. Knüppel, O. (1999). PROFIL/BIAS V 2.0 report TUHH. Lhommeau, M., Jaulin, L., & Hardouin, L. (2011). Capture basin approximation using interval analysis. International Journal of Adaptative Control and Signal Processing, 25, Lin, Y., & Stadtherr, M. A. (2006). Validated solution of ODEs with parametric uncertainties. Computer Aided Chemical Engineering, 21, Mitchell, I. (2002, August). Application of level set methods to control and reachability problems in continuous and hybrid systems. PhD dissertation, Stanford University. Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to Interval Analysis. SIAM. Neher, M., Jackson, K. R., & Nedialkov, N. S. (2007). On Taylor Model Based Integration of ODEs. SIAM Journal on Numerical Analysis, 45, Packard, A., Topcu, U., Seiler, P., & Balas, G. (2009, June 9). Quantitative Local Analysis of Nonlinear Systems Using Sum-of-Squares Decomposition, Main presentation. St.Louis, Missouri, USA: 2009 American Control Conference. 8
9 Rump, S. (1999). INTLAB - INTerval LABoratory. In T. Csendes, Developments in Reliable Computing (pp ). Dordrecht: Kluwer Academic Publishers. van Kampen, E., Chu, Q., Mulder, J., & van Emden, M. (2007). Nonlinear Aircraft Trim Using Interval Analysis. AIAA Guidance, Navigation, and Control Conference and Exhibit. Determination Using GPS and an Interval Integer Ambiguity Resolution Algorithm. AIAA Guidance, Navigation, and Control Conference. Chicago. van Kampen, E., Zaal, P., de Weerdt, E., Chu, Q., & Mulder, J. (2010). Optimization of Human Perception Modeling Using Interval Analysis. Journal of Guidance, Control, and Dynamics, 33, van Kampen, E., de Weerdt, E., Chu, Q., & Mulder, J. (2009). Aircraft Attitude Fig. 14 9
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