2013, 3(3): 519-548. doi: 10.3934/naco.2013.3.519

Approximation of reachable sets using optimal control algorithms

1. 

Applied Mathematics, Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany

2. 

Institute of Mathematics and Applied Computing (LRT), University of the Federal Armed Forces at Munich, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany

Received  October 2011 Revised  February 2013 Published  July 2013

We investigate and analyze a computational method for the approximation of reachable sets for nonlinear dynamic systems. The method uses grids to cover the region of interest and the distance function to the reachable set evaluated at grid points. A convergence analysis is provided and shows the convergence of three different types of discrete set approximations to the reachable set. The distance functions can be computed numerically by suitable optimal control problems in combination with direct discretization techniques which allows adaptive calculations of reachable sets. Several numerical examples with nonconvex reachable sets are presented.
Citation: Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519
References:
[1]

H. Attouch and R. J.-B. Wets, Isometries for the Legendre-Fenchel transform,, Trans. Amer. Math. Soc., 296 (1986), 33. doi: 10.1090/S0002-9947-1986-0837797-X.

[2]

J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, "Viability Theory. New Directions,", Second edition, (2011). doi: 10.1007/978-3-642-16684-6.

[3]

J.-P. Aubin, T. Bernado and P. Saint-Pierre, A viability approach to global climate change issues,, in, (2005), 113.

[4]

R. Baier, "Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen,", Bayreuth. Math. Schr., 50 (1995).

[5]

R. Baier, Selection strategies for set-valued Runge-Kutta methods,, in, (2005), 149.

[6]

R. Baier, Ch. Büskens, I. A. Chahma and M. Gerdts, Approximation of reachable sets by direct solution methods of optimal control problems,, Optim. Methods Softw., 22 (2007), 433. doi: 10.1080/10556780600604999.

[7]

R. Baier, I. A. Chahma and F. Lempio, Stability and convergence of Euler's method for state-constrained differential inclusions,, SIAM J. Optim., 18 (2007), 1004. doi: 10.1137/060661867.

[8]

R. Baier and M. Gerdts, A computational method for non-convex reachable sets using optimal control,, in, (2009), 23.

[9]

W.-J. Beyn and J. Rieger, Numerical fixed grid methods for differential inclusions,, Computing, 81 (2007), 91. doi: 10.1007/s00607-007-0240-4.

[10]

W.-J. Beyn and J. Rieger, The implicit Euler scheme for one-sided Lipschitz differential inclusions,, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 14 (2010), 409. doi: 10.3934/dcdsb.2010.14.409.

[11]

H. G. Bock, "Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen,", Bonner Mathematische Schriften, 183 (1987).

[12]

O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption,, SIAM J. Control Optim., 48 (2010), 4292. doi: 10.1137/090762075.

[13]

O. Bokanowski, A. Désilles, and H. Zidani, HJB approach for motion planning and reachabilty analysis,, in, (2011), 28.

[14]

N. Bonneuil, Computing the viability kernel in large state dimension,, J. Math. Anal. Appl., 323 (2006), 1444. doi: 10.1016/j.jmaa.2005.11.076.

[15]

N. Bonneuil, Maximum under continuous-discrete-time dynamic with target and viability constraints,, Optimization, 61 (2012), 901. doi: 10.1080/02331934.2011.605127.

[16]

Y. Cao, S. Li, L. R. Petzold and R. Serban, Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution,, SIAM J. Sci. Comput., 24 (2003), 1076. doi: 10.1137/S1064827501380630.

[17]

M. Caracotsios and W. E. Stewart, Sensitivity analysis of initial-boundary-value problems with mixed PDEs and algebraic equations,, Computers chem. Engng., 19 (1985), 1019.

[18]

I. A. Chahma, Set-valued discrete approximation of state-constrained differential inclusions,, Bayreuth. Math. Schr., 67 (2003), 3.

[19]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998).

[20]

D. Cohen-Or, D. Levin and A. Solomovici, Three-dimensional distance field metamorphosis,, ACM Trans. Graph., 17 (1998), 116.

[21]

E. Crück, A. Désilles and H. Zidani, Collision analysis for an UAV,, in, (2012).

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M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization,'', Second edition, (2011).

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M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in, 2 (2002), 221.

[24]

M. Dellnitz, O. Junge, M. Post and B. Thiere, On target for Venus-set oriented computation of energy efficient low thrust trajectories,, Celestial Mech. Dynam. Astronom., 95 (2006), 357.

[25]

A. L. Dontchev and E. M. Farkhi, Error estimates for discretized differential inclusions,, Computing, 41 (1989), 349.

[26]

A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order Runge-Kutta approximations in control constrained optimal control,, SIAM J. Numer. Anal., 38 (2000), 202. doi: 10.1137/S0036142999351765.

[27]

W. F. Feehery, J. E. Tolsma and P. I. Barton, Efficient sensitivity analysis of large-scale differential-algebraic systems,, Appl. Numer. Math., 25 (1997), 41. doi: 10.1016/S0168-9274(97)00050-0.

[28]

T. F. Filippova and E. V. Berezina, On state estimation approaches for uncertain dynamical systems with quadratic nonlinearity: theory and computer simulations,, in, (2008), 326.

[29]

H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wa.zewski's theorems on closed domains,, J. Differ. Equ., 161 (2000), 449. doi: 10.1006/jdeq.2000.3711.

[30]

J. E. Gayek, Approximating reachable sets for a class of linear control systems,, Internat. J. Control, 43 (1986), 441. doi: 10.1080/00207178608933477.

[31]

M. Gerdts, "User manual for OCPID-DAE1,", User manual, (2010).

[32]

M. Gerdts, "Optimal Control of ODEs and DAEs,", DeGruyter, (2011).

[33]

A. Girard and C. Le Guernic, Zonotope/hyperplane intersection for hybrid systems reachability analysis,, in, (2008), 22.

[34]

A. Griewank, "Evaluating Derivatives. Principles and Techniques of Algorithmic Differentiation,", volume 19 of, (2000).

[35]

L. Grüne, "Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,'', volume 1783 of, (1783).

[36]

G. Häckl, "Reachable Sets, Control Sets and Their Computation. With a Preface by F. Colonius,", volume 7 of, (1995).

[37]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system,, Numer. Math., 87 (2000), 247. doi: 10.1007/s002110000178.

[38]

O. Hájek, "Control Theory in the Plane,'', Second edition, (2008).

[39]

H. Hermes and J. P. Lasalle, Functional Analysis and Time Optimal Control,, in, (1969).

[40]

P. Kenderov, Dense strong continuity of pointwise continuous mappings,, Pacific J. Math., 89 (1980), 111. doi: 10.2140/pjm.1980.89.111.

[41]

N. Kirov and M. Krastanov, Volterra series and numerical approximations of ODEs,, in, 3401 (2005), 337.

[42]

E. K. Kostousova, State estimation for dynamic systems via parallelotopes: optimization and parallel computations,, Optim. Methods Softw., 9 (1998), 269. doi: 10.1080/10556789808805696.

[43]

E. K. Kostousova, State estimation for control systems with a multiplicative uncertainty through polyhedral techniques,, in, (2012), 12.

[44]

M. I. Krastanov and V. M. Veliov, High-order approximations to nonholonomic affine control systems,, in, (2010), 4.

[45]

A. B. Kurzhanski, I. M. Mitchell and P. Varaiya, Optimization techniques for state-constrained control and obstacle problems,, J. Optim. Theory Appl., 128 (2006), 499. doi: 10.1007/s10957-006-9029-4.

[46]

A. B. Kurzhanski and P. Varaiya, Ellipsoidal techniques for reachability analysis: internal approximation,, Systems Control Lett., 41 (2000), 201. doi: 10.1016/S0167-6911(00)00059-1.

[47]

A. B. Kurzhanski and P. Varaiya, Dynamic optimization for reachability problems,, J. Optim. Theory Appl., 108 (2001), 227. doi: 10.1023/A:1026497115405.

[48]

A. B. Kurzhanski and P. Varaiya, On ellipsoidal techniques for reachability analysis. Part I: external approximations,, Optim. Methods Softw., 17 (2002), 177. doi: 10.1080/1055678021000012426.

[49]

A. B. Kurzhanski and P. Varaiya, Ellipsoidal techniques for reachability under state constraints,, SIAM J. Control Optim., 45 (2006), 1369. doi: 10.1137/S0363012903437605.

[50]

D. Levin, Multidimensional reconstruction by set-valued approximations,, IMA J. Numer. Anal., 6 (1986), 173. doi: 10.1093/imanum/6.2.173.

[51]

T. Lorenz, Epi-Lipschitzian reachable sets of differential inclusions,, Syst. Control Lett., 57 (2008), 703. doi: 10.1016/j.sysconle.2008.01.007.

[52]

K. Malanowski, Ch. Büskens, and H. Maurer, Convergence of approximations to nonlinear optimal control problems,, in, (1997), 253.

[53]

T. Maly and L. R. Petzold, Numerical methods and software for sensitivity analysis of differential-algebraic systems,, Appl. Numer. Math., 20 (1996), 57. doi: 10.1016/0168-9274(95)00117-4.

[54]

I. M. Mitchell, Comparing forward and backward reachability as tools for safety analysis,, in, (2007), 428.

[55]

I. M. Mitchell and C. J. Tomlin, Overapproximating reachable sets by Hamilton-Jacobi projections,, J. Sci. Comput., 19 (2003), 323. doi: 10.1023/A:1025364227563.

[56]

J. Nocedal and S. J. Wright, "Numerical Optimization,'', Springer Series in Operations Research, (1999). doi: 10.1007/b98874.

[57]

A. Pietrus and V. M. Veliov, On the discretization of switched linear systems,, System Control Lett., 58 (2009), 395. doi: 10.1016/j.sysconle.2009.01.005.

[58]

A. Puri, V. Borkar, and P. Varaiya, $\epsilon$-Approximations of differential inclusions,, in, (1996), 20.

[59]

M. Quincampoix and V. M. Veliov, Optimal control of uncertain systems with incomplete information for the disturbances,, SIAM J. Control Optim., 43 (): 1373. doi: 10.1137/S0363012903420863.

[60]

J. Rieger, Shadowing and the viability kernel algorithm,, Appl. Math. Optim., 60 (2009), 429. doi: 10.1007/s00245-009-9083-z.

[61]

R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis,'', volume 317 of, (1998).

[62]

P. Saint-Pierre, Approximation of the viability kernel,, Appl. Math. Optim., 29 (1994), 187. doi: 10.1007/BF01204182.

[63]

M. Sandberg, Convergence of the forward Euler method for nonconvex differential inclusions,, SIAM J. Numer. Anal., 47 (): 308. doi: 10.1137/070686093.

[64]

P. Varaiya, Reach set computation using optimal control,, in, (2000), 323.

[65]

V. M. Veliov, Second order discrete approximations to strongly convex differential inclusions,, Systems Control Lett., 13 (1989), 263. doi: 10.1016/0167-6911(89)90073-X.

[66]

V. M. Veliov, Second order discrete approximation to linear differential inclusions,, SIAM J. Numer. Anal., 29 (1992), 439. doi: 10.1137/0729026.

[67]

P. R. Wolenski, The exponential formula for the reachable set of a Lipschitz differential inclusion,, SIAM J. Control Optim., 28 (1990), 1148. doi: 10.1137/0328062.

show all references

References:
[1]

H. Attouch and R. J.-B. Wets, Isometries for the Legendre-Fenchel transform,, Trans. Amer. Math. Soc., 296 (1986), 33. doi: 10.1090/S0002-9947-1986-0837797-X.

[2]

J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, "Viability Theory. New Directions,", Second edition, (2011). doi: 10.1007/978-3-642-16684-6.

[3]

J.-P. Aubin, T. Bernado and P. Saint-Pierre, A viability approach to global climate change issues,, in, (2005), 113.

[4]

R. Baier, "Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen,", Bayreuth. Math. Schr., 50 (1995).

[5]

R. Baier, Selection strategies for set-valued Runge-Kutta methods,, in, (2005), 149.

[6]

R. Baier, Ch. Büskens, I. A. Chahma and M. Gerdts, Approximation of reachable sets by direct solution methods of optimal control problems,, Optim. Methods Softw., 22 (2007), 433. doi: 10.1080/10556780600604999.

[7]

R. Baier, I. A. Chahma and F. Lempio, Stability and convergence of Euler's method for state-constrained differential inclusions,, SIAM J. Optim., 18 (2007), 1004. doi: 10.1137/060661867.

[8]

R. Baier and M. Gerdts, A computational method for non-convex reachable sets using optimal control,, in, (2009), 23.

[9]

W.-J. Beyn and J. Rieger, Numerical fixed grid methods for differential inclusions,, Computing, 81 (2007), 91. doi: 10.1007/s00607-007-0240-4.

[10]

W.-J. Beyn and J. Rieger, The implicit Euler scheme for one-sided Lipschitz differential inclusions,, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 14 (2010), 409. doi: 10.3934/dcdsb.2010.14.409.

[11]

H. G. Bock, "Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen,", Bonner Mathematische Schriften, 183 (1987).

[12]

O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption,, SIAM J. Control Optim., 48 (2010), 4292. doi: 10.1137/090762075.

[13]

O. Bokanowski, A. Désilles, and H. Zidani, HJB approach for motion planning and reachabilty analysis,, in, (2011), 28.

[14]

N. Bonneuil, Computing the viability kernel in large state dimension,, J. Math. Anal. Appl., 323 (2006), 1444. doi: 10.1016/j.jmaa.2005.11.076.

[15]

N. Bonneuil, Maximum under continuous-discrete-time dynamic with target and viability constraints,, Optimization, 61 (2012), 901. doi: 10.1080/02331934.2011.605127.

[16]

Y. Cao, S. Li, L. R. Petzold and R. Serban, Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution,, SIAM J. Sci. Comput., 24 (2003), 1076. doi: 10.1137/S1064827501380630.

[17]

M. Caracotsios and W. E. Stewart, Sensitivity analysis of initial-boundary-value problems with mixed PDEs and algebraic equations,, Computers chem. Engng., 19 (1985), 1019.

[18]

I. A. Chahma, Set-valued discrete approximation of state-constrained differential inclusions,, Bayreuth. Math. Schr., 67 (2003), 3.

[19]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998).

[20]

D. Cohen-Or, D. Levin and A. Solomovici, Three-dimensional distance field metamorphosis,, ACM Trans. Graph., 17 (1998), 116.

[21]

E. Crück, A. Désilles and H. Zidani, Collision analysis for an UAV,, in, (2012).

[22]

M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization,'', Second edition, (2011).

[23]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in, 2 (2002), 221.

[24]

M. Dellnitz, O. Junge, M. Post and B. Thiere, On target for Venus-set oriented computation of energy efficient low thrust trajectories,, Celestial Mech. Dynam. Astronom., 95 (2006), 357.

[25]

A. L. Dontchev and E. M. Farkhi, Error estimates for discretized differential inclusions,, Computing, 41 (1989), 349.

[26]

A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order Runge-Kutta approximations in control constrained optimal control,, SIAM J. Numer. Anal., 38 (2000), 202. doi: 10.1137/S0036142999351765.

[27]

W. F. Feehery, J. E. Tolsma and P. I. Barton, Efficient sensitivity analysis of large-scale differential-algebraic systems,, Appl. Numer. Math., 25 (1997), 41. doi: 10.1016/S0168-9274(97)00050-0.

[28]

T. F. Filippova and E. V. Berezina, On state estimation approaches for uncertain dynamical systems with quadratic nonlinearity: theory and computer simulations,, in, (2008), 326.

[29]

H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wa.zewski's theorems on closed domains,, J. Differ. Equ., 161 (2000), 449. doi: 10.1006/jdeq.2000.3711.

[30]

J. E. Gayek, Approximating reachable sets for a class of linear control systems,, Internat. J. Control, 43 (1986), 441. doi: 10.1080/00207178608933477.

[31]

M. Gerdts, "User manual for OCPID-DAE1,", User manual, (2010).

[32]

M. Gerdts, "Optimal Control of ODEs and DAEs,", DeGruyter, (2011).

[33]

A. Girard and C. Le Guernic, Zonotope/hyperplane intersection for hybrid systems reachability analysis,, in, (2008), 22.

[34]

A. Griewank, "Evaluating Derivatives. Principles and Techniques of Algorithmic Differentiation,", volume 19 of, (2000).

[35]

L. Grüne, "Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,'', volume 1783 of, (1783).

[36]

G. Häckl, "Reachable Sets, Control Sets and Their Computation. With a Preface by F. Colonius,", volume 7 of, (1995).

[37]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system,, Numer. Math., 87 (2000), 247. doi: 10.1007/s002110000178.

[38]

O. Hájek, "Control Theory in the Plane,'', Second edition, (2008).

[39]

H. Hermes and J. P. Lasalle, Functional Analysis and Time Optimal Control,, in, (1969).

[40]

P. Kenderov, Dense strong continuity of pointwise continuous mappings,, Pacific J. Math., 89 (1980), 111. doi: 10.2140/pjm.1980.89.111.

[41]

N. Kirov and M. Krastanov, Volterra series and numerical approximations of ODEs,, in, 3401 (2005), 337.

[42]

E. K. Kostousova, State estimation for dynamic systems via parallelotopes: optimization and parallel computations,, Optim. Methods Softw., 9 (1998), 269. doi: 10.1080/10556789808805696.

[43]

E. K. Kostousova, State estimation for control systems with a multiplicative uncertainty through polyhedral techniques,, in, (2012), 12.

[44]

M. I. Krastanov and V. M. Veliov, High-order approximations to nonholonomic affine control systems,, in, (2010), 4.

[45]

A. B. Kurzhanski, I. M. Mitchell and P. Varaiya, Optimization techniques for state-constrained control and obstacle problems,, J. Optim. Theory Appl., 128 (2006), 499. doi: 10.1007/s10957-006-9029-4.

[46]

A. B. Kurzhanski and P. Varaiya, Ellipsoidal techniques for reachability analysis: internal approximation,, Systems Control Lett., 41 (2000), 201. doi: 10.1016/S0167-6911(00)00059-1.

[47]

A. B. Kurzhanski and P. Varaiya, Dynamic optimization for reachability problems,, J. Optim. Theory Appl., 108 (2001), 227. doi: 10.1023/A:1026497115405.

[48]

A. B. Kurzhanski and P. Varaiya, On ellipsoidal techniques for reachability analysis. Part I: external approximations,, Optim. Methods Softw., 17 (2002), 177. doi: 10.1080/1055678021000012426.

[49]

A. B. Kurzhanski and P. Varaiya, Ellipsoidal techniques for reachability under state constraints,, SIAM J. Control Optim., 45 (2006), 1369. doi: 10.1137/S0363012903437605.

[50]

D. Levin, Multidimensional reconstruction by set-valued approximations,, IMA J. Numer. Anal., 6 (1986), 173. doi: 10.1093/imanum/6.2.173.

[51]

T. Lorenz, Epi-Lipschitzian reachable sets of differential inclusions,, Syst. Control Lett., 57 (2008), 703. doi: 10.1016/j.sysconle.2008.01.007.

[52]

K. Malanowski, Ch. Büskens, and H. Maurer, Convergence of approximations to nonlinear optimal control problems,, in, (1997), 253.

[53]

T. Maly and L. R. Petzold, Numerical methods and software for sensitivity analysis of differential-algebraic systems,, Appl. Numer. Math., 20 (1996), 57. doi: 10.1016/0168-9274(95)00117-4.

[54]

I. M. Mitchell, Comparing forward and backward reachability as tools for safety analysis,, in, (2007), 428.

[55]

I. M. Mitchell and C. J. Tomlin, Overapproximating reachable sets by Hamilton-Jacobi projections,, J. Sci. Comput., 19 (2003), 323. doi: 10.1023/A:1025364227563.

[56]

J. Nocedal and S. J. Wright, "Numerical Optimization,'', Springer Series in Operations Research, (1999). doi: 10.1007/b98874.

[57]

A. Pietrus and V. M. Veliov, On the discretization of switched linear systems,, System Control Lett., 58 (2009), 395. doi: 10.1016/j.sysconle.2009.01.005.

[58]

A. Puri, V. Borkar, and P. Varaiya, $\epsilon$-Approximations of differential inclusions,, in, (1996), 20.

[59]

M. Quincampoix and V. M. Veliov, Optimal control of uncertain systems with incomplete information for the disturbances,, SIAM J. Control Optim., 43 (): 1373. doi: 10.1137/S0363012903420863.

[60]

J. Rieger, Shadowing and the viability kernel algorithm,, Appl. Math. Optim., 60 (2009), 429. doi: 10.1007/s00245-009-9083-z.

[61]

R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis,'', volume 317 of, (1998).

[62]

P. Saint-Pierre, Approximation of the viability kernel,, Appl. Math. Optim., 29 (1994), 187. doi: 10.1007/BF01204182.

[63]

M. Sandberg, Convergence of the forward Euler method for nonconvex differential inclusions,, SIAM J. Numer. Anal., 47 (): 308. doi: 10.1137/070686093.

[64]

P. Varaiya, Reach set computation using optimal control,, in, (2000), 323.

[65]

V. M. Veliov, Second order discrete approximations to strongly convex differential inclusions,, Systems Control Lett., 13 (1989), 263. doi: 10.1016/0167-6911(89)90073-X.

[66]

V. M. Veliov, Second order discrete approximation to linear differential inclusions,, SIAM J. Numer. Anal., 29 (1992), 439. doi: 10.1137/0729026.

[67]

P. R. Wolenski, The exponential formula for the reachable set of a Lipschitz differential inclusion,, SIAM J. Control Optim., 28 (1990), 1148. doi: 10.1137/0328062.

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