Control Problems in DAE. with Application to Path Planning Problems for. Volker H. Schulz. published as: Preprint 96-12,

Transcription

1 Reduced SQP Methods for Large-Scale Optimal Control Problems in DAE with Application to Path Planning Problems for Satellite Mounted Robots Volker H Schulz PhD thesis at the University of Heidelberg published as: Preprint 96-12, Interdisziplinares Zentrum fur Wissenschaftliches Rechnen, Universitat Heidelberg, 1996 Gutachter: Prof Dr Hans Georg Bock Prof Dr Gabriel Wittum Tag der Einreichung: 13 November 1995 Tag der mundlichen Prufung: 6 Februar 1996

2 Acknowledgements This thesis has been prepared during my work at the Interdisciplinary Center for Scientic Computing (IWR) of the Ruprecht-Karls-Universitat Heidelberg I would like to thank Prof HG Bock for providing a very creative and inspiring atmosphere within his research group, of which I am lucky enough to be a member I am indebted to him and especially to Dr JP Schloder for much invaluable advice and many stimulating discussions I would also like to thank Prof RW Longman for the collaboration on satellite mounted robotics, the outcome of which is part of this thesis I'd like to give thanks to my colleagues and friends Dr Marc Steinbach, Reinhold von Schwerin and Michael Winckler for fruitful joint work and many useful suggestions and critical comments I would like to thank Dr M Zedd for providing realistic data of the shuttle and its remote manipulator system and all other people who helped me during my work on this thesis Finally I wish to thank the graduate program \Modellierung und Wissenschaftliches Rechnen in Mathematik und Naturwissenschaften" of the IWR both for travel support and additional hardware This book is dedicated to my wife Petra She not only proofread the nal text, but helped very much to get things done with her continued and loving encouragement

3 Contents 1 Introduction 3 11 The mathematical problem formulation : : : : : : : : : : : : : : : : : : : : 7 12 Notational conventions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2 The Collocation Discretization Collocation for two point BVP in ODE : : : : : : : : : : : : : : : : : : : : Choice of collocation points : : : : : : : : : : : : : : : : : : : : : : The polynomial representation : : : : : : : : : : : : : : : : : : : : : A tempting combination : : : : : : : : : : : : : : : : : : : : : : : : Collocation for BVP in DAE with invariants : : : : : : : : : : : : : : : : : DAE models from mechanics : : : : : : : : : : : : : : : : : : : : : : Collocation discretization of two point BVP in DAE : : : : : : : : : Exploiting invariants : : : : : : : : : : : : : : : : : : : : : : : : : : 21 3 Partially reduced SQP methods From feasible path methods to reduced SQP : : : : : : : : : : : : : : : : : The reduced gradient seen as a directional derivative : : : : : : : : : : : : RSQP methods and their convergence properties : : : : : : : : : : : : : : : Position of function evaluations : : : : : : : : : : : : : : : : : : : : Information used for the update of the reduced Hessian : : : : : : : The concept of partially reduced SQP methods : : : : : : : : : : : : : : : Local convergence of PRSQP methods : : : : : : : : : : : : : : : : : : : : Basic assumptions : : : : : : : : : : : : : : : : : : : : : : : : : : : Linear convergence : : : : : : : : : : : : : : : : : : : : : : : : : : : Superlinear convergence : : : : : : : : : : : : : : : : : : : : : : : : Local convergence of PRSQP methods with structure preserving updates : Globalization strategies : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 61 4 Direct PRSQP methods for optimal control problems in DAE The indirect approach : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Direct methods and the boundary value problem approach : : : : : : : : : The discretized problem : : : : : : : : : : : : : : : : : : : : : : : : A new family of direct methods for optimal control problems : : : : : : : : 73 1

4 2 Contents 44 Aspects of the numerical realization : : : : : : : : : : : : : : : : : : : : : : A PRSQP algorithm : : : : : : : : : : : : : : : : : : : : : : : : : : Recursions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : PRSQP with block updates : : : : : : : : : : : : : : : : : : : : : : Mesh selection for the states : : : : : : : : : : : : : : : : : : : : : : : : : : Mesh selection for the controls : : : : : : : : : : : : : : : : : : : : : : : : : 87 5 Path Planning for Satellite Mounted Robots Modeling a Satellite Mounted Robot : : : : : : : : : : : : : : : : : : : : : Theoretical Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Minimum attitude disturbance paths : : : : : : : : : : : : : : : : : Minimum time paths : : : : : : : : : : : : : : : : : : : : : : : : : : Minimum maximal acceleration paths : : : : : : : : : : : : : : : : : Principal plane paths : : : : : : : : : : : : : : : : : : : : : : : : : : Circles as shortest paths in principal planes : : : : : : : : : : : : : Numerical results Technical preliminaries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Optimal control of ground based robots : : : : : : : : : : : : : : : : : : : Optimal paths for the shuttle RMS : : : : : : : : : : : : : : : : : : : : : : D paths : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : D paths : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 120 Summary 131 A Quaternions 133 B Dimensions of the shuttle and its RMS 135 Bibliography 139

5 Chapter 1 Introduction Er-Kennen Er-Rechnung einer Heinz von Foerster 1973 [44] Large scale nonlinear optimization is a vivid research area of growing importance A main source of large scale optimization problems is the optimization of systems whose behavior can be described by dierential equations A typical eld of application lies in engineering Due to recent developments in numerical simulation techniques and increasing capacities of computers there is a natural demand for optimization, which some think of as the ultimate aim of all simulation Mathematical optimization may therefore be considered a new key technology [26] General purpose optimization codes from software libraries mostly work best for small scale problems which are not too nonlinear There is a lack of ecient general purpose solvers for medium and large scale problems The reason is that such problems possess certain structures which have to be considered for eciency reasons, but are generalizable only to a certain extent Therefore special optimization methods have to be developed for special problem classes Such structures arise, eg, from discretizations of dierential equations Thus the successful solution of optimization problems in such applications requires a blend of several methodologies and techniques: discretization and solution techniques for discretized problems, optimization theory and techniques, modeling and theoretical investigations of the application This exhibits the interdisciplinary character of work in this eld, which thus is a distinctive part of scientic computing 3

6 discretization optimization modeling 4 1 Introduction In the present thesis new ecient optimization methods are developed for optimal control problems in dierential algebraic equations (DAE), and they are applied to path planning problems for satellite mounted robots Each constituent of the solving of large optimization problems from applications mentioned above is taken into consideration For each item and their combinations new aspects and for some of them completely new points of view are presented The main topic of this thesis is the introduction of the new class of partially reduced sequential quadratic programming (PRSQP) methods for large scale optimization problems and their application to optimal control problems in DAE A second central point of this thesis is the investigation of path planning problems for satellite mounted robots using the new optimization methods and the provision of a basic understanding of the nature of the solutions to these path planning problems The whole thesis is motivated by the above application problem, which is reected on in this introduction The mathematical formulation, however, is much easier, if the theoretical ground is laid down rst, to be built on up until the solution of the motivating application problem So the main results of the thesis are now briey described in a top down manner, where gure 11 may help to keep track of the actual position within the thesis Figure 11: Dependency structure of the thesis' chapters The topic of path planning for robots, which receives considerable attention for ground based robots [77, 76, 113, 114], is signicantly more complicated in space The reason is that the process of manipulating a load at one end of the robot can cause

7 the base of the robot to change its attitude the robot-satellite system is nonholonomic Therefore one aims at the construction of maneuvers of the robot so that the nal attitude of the satellite matches its initial attitude The rst solutions for this problem presented by Longman in [85] are feasible, but complex and undesirable Together with Bock and Longman the author of this thesis developed in [108] the optimal control problem approach, to overcome this decit Its basic idea is to formulate an optimal control problem with the conguration constraints for the satellite attitude as boundary conditions and an objective functional describing the properties of desired paths Though this approach seems to be quite natural from a mathematical point of view, it is unusual from an engineering standpoint, mainly because of a supposed lack of ecient solution methods for optimal control problems In chapter 6 numerical results for optimal paths are presented using realistic data of the space shuttle Numerical results, however, are hardly intelligible, if there is no basic theoretical understanding of the nature of optimal paths Therefore in chapter 5 we discuss several models and present novel theoretical results about the properties of optimal paths for satellite mounted robots, which are necessary for a thorough understanding of the numerical results presented in chapter 6 Chapter 5 and chapter 4 serve as the groundwork for the numerical results in chapter 6, which on the other hand presents also performance results of the numerical methods developed in chapter 4 There are two distinct approaches to the treatment of optimal control problems in DAE, which are both investigated in chapter 4 The indirect approach formulates necessary conditions for the solution of optimal control problems in terms of a boundary value problem Based on classical conditions for ordinary dierential equations (ODE) [31, 72, 23], necessary conditions for DAE of index 1 are derived From a practical point of view the direct approach is much more preferable Its fundamental principle is to discretize the control function and to employ nonlinear programming techniques for the solution of a resulting nite dimensional optimization problem The rst direct methods were typically gradient-type algorithms based on single shooting, see, eg, Breakwell, Speyer, and Bryson [30] or Miele [88] A major breakthrough in solving real-life problems by direct methods has been achieved by Bock and coworkers with the development of the boundary value problem approach [20, 21, 24, 102] This approach is characterized by a discretization of the system states using, eg, multiple shooting or collocation The discretization equations are treated as side conditions in the optimization problem and thus simultaneously solved together with the optimization problem The code MUSCOD [94, 24, 108, 27] is based on multiple shooting in connection with a structured SQP (Sequential Quadratic Programming) method and high-rank block updates of the Hessian of the Lagrangian It has been successfully applied, eg, to trajectory optimization problems in robotics [77, 76, 114, 108] Another multiple shooting code, based on this algorithm and its theory, has later been developed in [73] Some direct collocation methods are described by Betts [16] and von Stryk [118] Both use a collocation discretization critically discussed in section 213 Direct reduced SQP methods for 5

8 6 1 Introduction discretized optimal control problems are described, eg, in [80, 17, 70] These methods do not consider inequality constraints or allow only for box constraints on the controls Thus MUSCOD is among the codes dening the state of the art in this area In chapter 4 direct methods are presented, which employ reduced SQP methods based on the partial reduction concept introduced in chapter 3 These methods are able to take into account inequality constraints for the controls as well as for the states In fact a whole new family of such methods is introduced, where one implementation (OCPRSQP) is described in detail It is compared with MUSCOD in section 62 Furthermore, the practically important issue of adaptivity is addressed in chapter 4 An analysis of the inuence of the discretization error of the states of the dynamical system is performed and new mesh selection strategies for the control discretization are presented Chapter 4 is based on discretization considerations in chapter 2 and optimization considerations in chapter 3 Chapter 3 contains the description of PRSQP methods as the main new concept at the mathematical core of this thesis It is concerned with the numerical solution of large structured nonlinear optimization problems, as they result, eg, from discretizations of optimal control problems as described in chapter 4 In contrast to SQP methods, reduced SQP methods deal only with the projected Hessian of the Lagrangian, which is a matrix of the size of the \true" degrees of freedom of the problem, and which is known to be positive semidenite at a solution point Depending on the structure of the optimization problem, reduced SQP methods possess a great eciency potential compared to SQP methods The disadvantage is that problems with inequalities or without a global chart of the \true" degrees of freedom are scarcely tractable by these methods After a thorough review of existing reduced SQP methods, PRSQP methods are introduced in chapter 3, which avoid these diculties by combining the advantage of reduced SQP methods small quadratic subproblems with the advantage of (full) SQP methods convenient treatment of inequality constraints The basic idea of these methods is to formulate the reduced SQP method only wrt those constraints, which allow a straight forward parameterization and to treat the remaining constraints in the same way as usual SQP methods do, but reduced onto the kernel of the rst mentioned constraints Indeed SQP and reduced SQP methods can be considered as special cases of PRSQP methods Convergence proofs are given The application of structure preserving update formula for the reduced Hessian is considered as well Based on the theory developed in this chapter, a large scale shape optimization problem [106, 111] has already been successfully solved, which is briey referred to in section 32 In chapter 2 an overview about the collocation discretization of boundary value problems in ODE and DAE is given, mainly in a formulation established by Ascher and coworkers (eg, [6, 8, 10]) Furthermore, the notion of a structure preserving collocation implementation is informally introduced Typical implementations [16, 118] for optimal control problems are shown to be not structure preserving and thus potentially increase the degree of nonlinearity of the optimization problem Finally, the techniques for the ecient exploitation of invariants, which frequently appear if higher index DAE are reduced to

9 11 The mathematical problem formulation 7 lower index, developed in [110], are discussed and compared with related techniques The remainder of this introductory chapter is devoted to the denition of the mathematical problem class to be treated in this thesis and to the explanation of some notational conventions 11 The mathematical problem formulation There are various formulations of optimal control problems for dynamical systems The basic problem treated in this thesis is of the form (t 2 [0; T ]): subject to min (y(t ); T ) (1:1) _y 1 (t) = f(y 1 (t); y 2 (t); u(t)); (12) 0 = g(y 1 (t); y 2 (t); u(t)); (13) r(y 1 (0); y 2 (0); y 1 (T ); y 2 (T )) = 0; (1:4) u min i u i (t) u max i ; i = 1; : : : ; n u ; (1:5) s(y 1 (t); y 2 (t); u(t)) 0: (1:6) Here denotes the real valued objective criterion to be minimized The semi-explicit dierential algebraic equation (DAE) (12, 13) describes the system dynamics for the time dependent dierential (y 1 (t) 2 R ny 1 ) and algebraic (y2 (t) 2 R ny 2 ) state variables and the time dependent control variables (u(t) 2 R n u ) It is of index l 2 N, if l is the minimal number of dierentiations of (13) wrt time, t, so that the resulting system can be algebraically transformed to an ordinary dierential equation We only consider DAE of index 1, ie DAE 2 is always nonsingular In (14) n r boundary conditions are formulated, which have to be satised by the solution of the optimal control problem Equation (16) species n s path constraints All problem functions ; f; g; r; s are assumed to be at least twice continuously differentiable, and to possess higher order derivatives when required: f 2 C 2 (R ny 1 R ny 2 R nu ; R ny 1 ); g 2 C 2 (R ny 1 R ny 2 R nu ; R ny 2 ); r 2 C 2 (R ny 1 R ny 2 R ny 1 R ny 2 ; R n r ); s 2 C 2 (R ny 1 R ny 2 R nu ; R ns ) 2 C 2 (R ny R;R): This formulation of the optimal control problem is not the most general one In the following we comment on possible generalizations and reformulations: Non-autonomous problems can be included in the context above by dening an additional dierential state variable y 1;ny1 +1 satisfying the initial value problem _y 1;ny1 +1(t) = 1, y 1;ny1 +1(0) = 0 In a similar way parameters p 2 R np can be included by using a trivial ordinary dierential equation _p = 0

10 8 1 Introduction In theoretical considerations the nal time T can be xed or free For numerical computations it is preferable to reformulate a problem with free nal time to one with xed nal time, eg, in the following way: The nal time T is treated as a new system parameter and the system time is substituted by t := T, where 2 [0; 1] Thus also the right hand side in (12) has to be multiplied with T in order to describe the dynamics in the new time variable The Mayer cost function in (11) is one of the most typical cost functions Another type of similar importance is the Lagrange cost function min Z T 0 L(y 1 (t); y 2 (t); u(t))dt for xed nal time T, which models a continuous dependency of the cost on the whole trajectory Sometimes a combination of both appears in the literature, the Bolza cost function min (y 1 (T ); y 2 (T )) + Z T 0 L(y 1 (t); y 2 (t); u(t))dt: Problems formulated with these alternative cost functions can be equivalently formulated as optimal control problems with a Mayer cost functional Many dynamical systems are formulated as DAE of index > 1 In that case we assume that index reduction is applied employing successive dierentiations of the algebraic constraint (13) wrt time t Resulting invariants can be exploited in the numerical solution of the optimal control problem as described in section 22 and in [110] The index of a DAE as dened above is the dierential index There are other denitions like the perturbation index [65], which are equivalent to the dierential one in the case of semi-explicit DAE Typical DAE from chemical engineering [27] are quasi-linear and semi-explicit B(y 1 (t); y 2 (t); u(t)) _y 1 (t) = f(y 1 (t); y 2 (t); u(t)); B nonsingular 0 = g(y 1 (t); y 2 (t); u(t)): This form can be conveniently implemented in a collocation discretization, but it makes the formulas very clumsy The box constraints on the controls (15) can be considered as special cases of the conditions (16) Since they are usually treated dierently, they are formulated explicitly

11 12 Notational conventions 12 Notational conventions 9 Here we summarize some notational conventions, which are used throughout the thesis The vector spaces used are of the type R n with a norm k:k Linear mappings A : If not otherwise indicated the identity R n! R m are considered matrices A 2 R mn matrix is denoted by I 2 R nn The kernel and the range of a matrix A are denoted by N (A) and R(A) respectively The norms for matrices are assumed to be induced by the corresponding vector space norms The derivative of a dierentiable function, g : R n! R m : x 7! g(x), in many cases is denoted by an index or by the corresponding capital letter (here G), so that g x =: G The derivative of a scalar function, f : R n! R : x 7! f(x), is also called a gradient and then considered a column vector, so that r x f := f > x = (@f=@x)>

12 10 1 Introduction

13 Chapter 2 The Collocation Discretization Collocation is one of the two main discretization methods for boundary value problems (BVP) in ordinary dierential equations (ODE) and dierential algebraic equations (DAE) In this chapter we recall basic denitions and results for collocation, as established by Ascher and coworkers Furthermore, critical aspects of special well known implementations are discussed and the numerical exploitation of invariants is demonstrated 21 Collocation for two point BVP in ODE In this section the basic properties of the collocation method for ODE are recalled following the lines of [6] We consider the following parameter dependent two point BVP in the interval [0; T ]: where y(t) 2 R ny ; p 2 R np _y(t) = f(y(t); p); (21) r(y(0); y(t ); p) = 0; (22) and f 2 C 2 (R ny R np ; R ny ), r 2 C 2 (R ny R ny ; R nr ), with n r = n y +n p The collocation discretization is based on the approximation of the solution of the ODE (21) by a piecewise polynomial of degree k on a mesh : 0 = 1 < : : : < m?1 < m = T : (2:3) On each of these subintervals the solution is approximated by a polynomial yj (t; j) = k+1 P s=1 j;l l t?j j+1? j ; t 2 [j ; j+1 ] (24) j;s 2 R ny ; s = 1; : : : ; k + 1; j = 1; : : : ; m? 1; where f s g k+1 s=1 is an appropriately chosen basis of the space P k+1 [0; 1] of polynomials of order k + 1 (b= degree k) on the intervall [0; 1] To determine the coecients f j;s g, the 11

14 12 2 The Collocation Discretization approximate solution is required to satify the ODE (21) on a subdivision of this mesh, the collocation points t jl := j + l ( j+1? j ); l 2 [0; 1]: _y j (t jl ; j ) = f(y j (t jl ; j ); p); l = 1; : : : ; k; j = 1; : : : ; m? 1; (2:5) and to be continuous at the mesh points y j ( j+1 ; j )? y j+1( j+1 ; j+1 ) = 0; j = 1; : : : ; m? 1: (2:6) Here the continuity condition in the case j = m? 1 means that we dene variables y m approximating y(t ) and require y m?1( m ; m?1 )? y m = 0 Additionally, the boundary conditions have to be satised as well r(y ( 1 1; 1 ); ym ) = 0: (2:7) In each collocation interval yj ( j+1 ; j ) can be interpreted as the result of one step of an implicit Runge-Kutta-method starting from yj ( j ; j ) The corresponding IRK-scheme is 1 a 11 : : : a 1k R a il = i R L l (s)ds; b i = 1 L i (s)ds; 1 i; l k; k a k1 : : : a kk b 1 : : : b k 0 L l (s) = k Q i=1 i6=l 0 (s? i )= k Q i=1 i6=l ( l? i ) These special Runge-Kutta-schemes are called collocation schemes The convergence properties of the collocation discretization are determined by the convergence properties of the corresponding IRK-method, which results from a related quadrature formula observing the fact that in the interval [ j ; j+1 ] with length h j := j+1? j y( j+1 ) = y( j ) + h j Z 1 0 = y( j ) + h j kx l=1 _y( j + sh j )ds f( j + l h j ; y( j + l h j ); p) Z 1 L l (s)ds 0 {z } b l q +h j O(hj): Here q is the maximal order of polynomials that are integrated exactly by the quadrature formula dened by fb l g and the collocation points f l g If the collocation points are well chosen, the order of the error, q, is greater than k + 1, which is referred to as the superconvergence property, since k + 1 is the order which can be expected for any quadrature formula dened by collocation points with i 6= j ; 8i 6= j Due to Radau [99] the highest order achievable is q = 2k This is the case for Gaussian points, which are dened as the roots of the polynomials fp k g that are dened by the recursion: (k + 1)P k+1 (t) = (2k + 1)(2t? 1)P k (t)? kp k?1 (t); P 0 (t) 1; P 1 (t) = 2t? 1: (2:8) The roots of P k are symmetric wrt the midpoint of the intervall [0,1] For k = 1 ( 1 = 1=2) this results in the implicit midpoint rule For k = 3 the IRK-scheme is:

15 21 Collocation for two point BVP in ODE 13 1? p p p p Gauss 2? p p ? p ? p The order q = 2k? 1 is achieved for the roots of the polynomials fp k + P k?1 ; 2 Rg, which gives room to x one of the collocation points to a preassigned value For 1 = 0 (collocation points = roots of P k +P k?1 ) or k = 1 (collocation points = roots of P k?p k?1 ) this results in the Radau-collocation-schemes, whose simplest members (k = 1) are the Euler-method and the backward-euler-method For k = 3 the IRK-schemes are 0 6? p p ?1? p p p p 6 36 Radau IA?1+ p ?43 p ?7 p ? p ? p p ?7 p p ? p ? p ?169 p p p p 6 36 Radau IIA?2+3 p 6 225?2?3 p For the next lower order q = 2k? 2 it is possible to choose values for two collocation points If one assigns 1 = 0 and k = 1, this results in the Lobatto points, which are the roots of P k? P k?2 The corresponding IRK-scheme for k = 3 is ? Lobatto IIIA For a more thorough discussion of quadrature formulas see [29] Under mild regularity assumptions concerning unique solvability and conditioning of the BVP, the following convergence properties can be proved (for details see [3]): convergence for all t 2 [ j ; j+1 ]:! t? y (i) (t)? y (i) (t) = h k?i+1 j y (k+1) ( j ) (i) j + O(h k?i+2 j ) + O(h q ); (29) h j 0 i k; () = 1 Z ky (s? ) (s? l )ds; k! 0 l=

16 14 2 The Collocation Discretization superconvergence at mesh points j and for the parameters: y( j )? y ( j ) = O(h q ); p? p = O(h q ); (2:10) where h = max h j and the superscripts in parentheses mean derivatives wrt time j 211 Choice of collocation points The choice of the collocation points inuences the stability and convergence properties of the resulting collocation method Algebraic stability is one of the relevant stability criteria For collocation schemes algebraic stability is equivalent to AN-stability All three alternatives investigated here (Gauss, Radau, Lobatto) are A-stable But only Gauss- and Radau-schemes are algebraically stable (see [5]) On the other hand, collocation at Gaussian points is reported in [6] to loose its superconvergence properties if applied to singularly perturbed problems Furthermore, the schemes possess dierent decoupling properties of increasing and decreasing modes of a system of ODE Radau IIA yields a stable integration for rapidly increasing modes as dened in [6], while Radau IA does it for the reverse Since in nonlinear ODE explicit decoupling of increasing and decreasing modes is rarely applicable, symmetric schemes are preferable, since they read the same in both directions As Ascher [6] points out, collocation at Gaussian points oers better decoupling properties than collocation at Lobatto points Since there are no \optimal" collocation points for all cases, a practical collocation method should include a way to oer the user a selection of several collocation methods 212 The polynomial representation There are various possibilities to represent the piecewise polynomial collocation approximation The most popular ones are B-splines, Hermite-splines and the Runge-Kutta-basis [13] They have similar properties wrt storage requirements and conditioning of the resulting linear system We choose the so called Runge-Kutta-representation because of its notational ease The collocation solution in the interval [ j ; j+1 ] is represented in the following way: y (t) = y j + h j k P l=1 z jl l t?j h j 8t 2 [j ; j+1 ]; where (211) y j := y ( j ) 2 R ny ; z jl := _y ( j + l h j ) 2 R ny ; h j := j+1? j ; l 2 P k+1 [0; 1] : l (0) = 0; _ l( i ) = ( 1; if l = i; 0; else: The name Runge-Kutta-representation is motivated by the fact that the variables fz jl g k l=1 are the stages of the corresponding IRK-scheme, which can be written as:

17 21 Collocation for two point BVP in ODE ( 1 ) : : : k ( 1 ) k 1 ( k ) : : : k ( k ) 213 A tempting combination 1(1) : : : k (1) In particular in the context of optimization problems for DAE systems, a special combination of collocation points and local polynomial representation is widely used (see, eg, [14, 118, 16]) It is the combination of Hermite-splines of order 4 with 3 Lobatto points in a special formulation of the resulting discretization equations (H 1 L 3, see below) The tempting aspect of this combination lies in the fact that it yields a relatively small number of variables to be determined numerically Hermite-splines of order 4 in the intervall [ j ; j+1 ] are dened as the interpolating polynomial of the states y j := y( j ), y j+ := y( j+1 ) and the derivatives z j := _y( j ), z j+ := _y( j+1 ) at the mesh points The equations for collocation at Lobatto points are then: collocation conditions: () 0 = y j? y j+ + h j 6 continuity conditions: " z j = f(y j ; p); (212)! _y ( j + h j 2 ) = f y ( j + h j 2 ); p (213) z j + z j+ + 4f y j + y j+ 2 + h j 8 (z j? z j+ ); p!# z j+ = f(y j+ ; p); (214) y j+ = y j+1 : (215) In the following the system ( ) will be referred to as the H 4 L 3 -system Obviously equation (215) can be solved immediately by substituting y j+ with y j+1 in the equations (213) and (214) From the equations (213, 214) now follows: z j+ = z j+1 Using this in (213) leads to the system: " z j = f(y j ; p) (212) 0 = y j? y j+1 + h j y j + y j+1 z j + z j+1 + 4f + h j (z j? z j+1 ); p : (213 0 ) The system (212, ) will be referred to as the H 2 L 3 -system Again H 2 L 3 can be reduced by substituting z j from (212) in (213 0 ), leading to the equation: 0 = y j? y j+1 + h j 6 " z j + z j+1 + 4f y j + y j+1 2 ;!# + h j 8 (f(y j; p)? f(y j+1 ; p)) ; p!# ( )

18 16 2 The Collocation Discretization as the only equation to be solved in the intervall [ j ; j+1 ] This will be referred to as H 1 L 3 It is widely used because of its small number of necessary variables { only fy j g instead of fy j ; y j+ ; z j ; z j+ g But there are severe drawbacks in nonlinear systems with parameters where Newtontype iterations have to be applied, as is shown in the following example: Example 21: Consider the ODE-BVP _y = y 2 p; y(0) = 1; y(1) = 5 Its solution parameter is p = 0:8 Here we want to investigate the eect of the dierent formulations above on the nonlinearity of the resulting discretization equations Therefore the BVP is solved on an equidistant mesh with 20 meshpoints by using the dierent discretization formulations Then the iterations are performed again, but started for the y- and z-values at the discrete solution and for the p-value at the values p 0 in the following table The necessary numbers of Newton-iterations to solve the corresponding nonlinear equations are listed The iteration process is stopped, when knewton-incrementk 2 =(#variables) 10?4 # Newton-iterations p H 4 L H 2 L H 1 L Obviously H 4 L 3 and H 2 L 3 need only 1 Newton-iteration, while H 1 L 3 behaves completely dierent This indicates that in H 1 L 3 the smaller amount of variables is bought by an increase of the nonlinearity of the problem This eect is explained by the following theorem Theorem 21 Consider a parameter dependent ODE-BVP, which is linear wrt the parameter p, of the type _y = G(y)p + g(y); R(y(t 0 ); y(t f ))p + r(y(t 0 ); y(t f )) = 0; (2:16) where G(y) 2 R nnp ; r(y(t 0 ); y(t f ); p) 2 R n+np and R(y(t 0 ); y(t f )) 2 R (n+np)np The BVP is assumed to be well dened If the collocation equations dening the solution of the discretized BVP are linear wrt p as well, then the following is valid: A Newton iteration started on the collocation solution of the BVP, but with an arbitrary parameter value p 0, converges after one iteration step Proof: Dene x 2 R nx as the vector of all collocation variables The whole nonlinear system of discretization equations is assumed to be of the type: F (x; p) = A(x)p + a(x)! = 0; (2:17)

19 22 Collocation for BVP in DAE with invariants 17 where A(x) 2 R (nx+np)np Since the start values x 0 for x are assumed to be the solution values for the system, but p is assumed to be started at p 0 = ^p+p, where ^p is the solution parameter value, one gets the Taylor expansion F (x 0 ; p 0 ) = F (x 0 ; F (x0 ; ^p)p = A(x 0 )p: (2:18) {z } 0 The Newton step is determined by the linearization of F (x0 ; p 0 )x + A(x 0 )p =?F (x 0 ; p 0 ) (2:18) =?A(x 0 )p: (2:19) According to the regularity assumptions this linear system of equations has a unique solution, which is obviously (x; p) = (0;?p) Remark: This \one-step-convergence" theorem can be generalized to parameter identication problems in DAE [104] The theorem explains the dierences between the systems H 4 L 3, H 2 L 3 and H 1 L 3 in example 21 H 1 L 3 does not preserve the structure in (216) Additionally it can easily be seen that H 1 L 3 hardly preserves any structure possibly inherent in the right hand side of the ODE 22 Collocation for BVP in DAE with invariants Dierential-algebraic equations (DAE) are widely used to model dynamical systems Often the algebraic part of these equations models linking conditions or is used as a shortcut for otherwise rather complicated and expensive models The following sections give examples for DAE models in mechanics and summarize the discretization of DAE-BVP using collocation with the aim to use it in an optimization context 221 DAE models from mechanics Beginning with the Euler-Lagrange-equations for mechanical multibody systems, we examine some typical types of higher index DAE In mechanical systems, the dierential variables representing the states of the system are usually denoted by p, and the algebraic variables by Dening T = T (p; _p) as the kinetic energy of a mechanical system, the Euler-Lagrange-equations are of the form d (p; (p; = Q(p; _p) + G(p) > ; (220) g(p) = 0; (221)

20 18 2 The Collocation Discretization where p 2 R np are the generalized coordinates, _p the velocities, Q(p; _p) are exterior and interior forces and are the Lagrange multipliers associated with the constraint equations g, where G The constraints g model links in the model, which are not yet satised by the coordinates p Performing the dierentiations in (220) the system (220, 221) results in M(p)p = f(p; _p) + G(p) > ; (222) g(p) = 0; (223) where M(p) 2 T=@ _p 2 and f := Q? (@ 2 T=(@p@ _p)) _p This is a DAE of index 3, which is called the descriptor form for multibody systems Details about the proper numerical treatment of such systems for simulation purposes can be found in [1, 2, 9, 42, 46, 28] and in the context of optimization in [110, 112] If we introduce additional variables q _p in (220, 221), we get the system (p; Q(p; _p)? G(p) > = 0; (p; _p) _p = 0; (225) g(p) = 0; (226) which is due to the \Lagrangian approach" in [100] The algebraic variables are here and _p This formulation saves tedious calculations if the dynamical equations are to be derived by hand As it is pointed out in [100], it is also a very sparse formulation and therefore computationally less expensive Since the resulting equations are less involved, they may also give more theoretical insight, which is used, eg, in chapter 5 If there are no constraint equations (226) (and therefore no term G(p) > in 224), the DAE (224, 225) is of index Collocation discretization of two point BVP in DAE We consider the two point DAE boundary value problem in the interval [0; T ]: _y 1 = f(y 1 ; y 2 ); (227) 0 = g(y 1 ; y 2 ); (228) r(y 1 (0); y 2 (0); y 1 (T ); y 2 (T )) = 0; (2:29) where the time (t) dependent variables y 1 (t) 2 R ny 1 are the dierential variables, y 2 (t) 2 R ny 2 the algebraic variables and f 2 C 2 (R ny 1R ny 2 ; R n y1), g 2 C 2 (R ny 1R ny 2 ; R n y2), r 2 C 2 (R ny 1 R ny 2 R ny 1 R ny 2 ; R n r ) with n r = n y1 Following the lines of [10], again we dene a collocation grid : 0 = 1 < : : : < m?1 < m = T (2:30)

21 22 Collocation for BVP in DAE with invariants 19 and collocation points f 1 ; : : : k g [0; 1] The dierential variables are discretized in the interval [ j ; j+1 ] by y 1 (t) = y 1;j + h j k P l=1 z jl l t?j h j 8t 2 [j ; j+1 ]; where (231) y 1;j := y 1 ( j) 2 R ny 1 ; zjl := _y 1 ( j + l h j ) 2 R ny 1 ; hj := j+1? j ; l 2 P k+1 [0; 1] : l (0) = 0; _ l( i ) = The algebraic variables are discretized by the vectors ( 1; if l = i; 0; else: x jl 2 R ny 2 ; l = 1; : : : ; k; j = 1; : : : ; m? 1 representing the solution values y 2 (t jl ); t jl := j + l h j, at the collocation points A polynomial interpolation of fx j1 ; : : : ; x jk g yields an approximation y 2 (t) in the whole interval [ j ; j+1 ] In the following considerations and in chapter 4 the symbols \z" and \x" will also be used with only one index dened as: z j := (z > j1 ; : : : ; z> jk )> ; x j := (x > j1 ; : : : ; x> jk )> : The collocation discretization of the boundary value problem ( ) consists of the following system of equations: collocation conditions: z jl = f(y 1;j + h j kx s=1 0 = g(y 1;j + h j kx continuity conditions for y 1 : y 1;j + h j and boundary conditions: s=1 z js l ( s ); x jl ); l = 1; : : : ; k; j = 1; : : : ; m? 1 (232) z js l ( s ); x jl ); (233) kx z js l (1)? y 1;j+1 = 0; j = 1; : : : ; m? 1; (2:34) s=1 r(y 1;1 ; y 2 (0); y 1;m ; y 2 (T )) = 0: (2:35) As in (26), the continuity condition in the case j = m? 1 means that we dene variables y 1;m approximating y 1 (T ) and formulate the condition with these variables These equations are well dened for index 1 systems In the case of DAE of index 2 the boundary conditions have to be consistent with the derivative (wrt time t) of the algebraic condition Furthermore, for a straight forward implementation one has to require l 6= 0, which excludes Lobatto schemes In [66] a rather complicated workaround for this special case is suggested According to [4, 66], the following orders for the global error of the dierential (y 1 ) and algebraic (y 2 ) variables at the mesh points for a collocation discretization with k collocation points are achieved

22 20 2 The Collocation Discretization Gaussian points Radau points Lobatto points y 1 y 2 y 1 y 2 y 1 y 2 index 1 O(h 2k ) O(h k+1 ) O(h 2k?1 ) O(h 2k?1 ) O(h 2k?2 ) O(h 2k?2 ) index 2 (k odd) (k even) O(h k+1 ) O(h k ) O(h k?1 ) O(h k?2 ) O(h 2k?1 ) O(h k ) O(h 2k?2 ) O(h k?1 ) O(h k ) In the case of DAE of index 2 the superconvergence property for Gaussian collocation is lost The reason is that in this discretization the mesh points are no collocation points, where the algebraic conditions are required In order to retain the superconvergence property also for collocation schemes with l 6= 1, Ascher and Petzold [7, 8] introduce projected collocation methods These methods are dened in [8] only in the context of quasilinearization of the DAE In the concept of quasilinearization, however, the order of discretization and linearization is reverse, if compared with the boundary value approach (see below) Using quasilinearization, the DAE is rst linearized and then discretized Let y 1 ; y 2 be the solution of the linearized DAE of index 2 _y 1 (t) = F 1 (t)y 1 (t) + F 2 (t)y 2 (t) + f(t); (236) 0 = G(t)y 1 (t) + g(t): (237) Then the dierentiable part of the solution of the projected collocation method, ^y 1;j, at the meshpoints j is dened by the equations ^y 1;j = y 1;j + F 2 ( j ) j ; (238) 0 = G( j )^y 1;j + g( j ); (239) where y 1;j are the solution values of the unprojected collocation discretization of the linearized DAE and j are additional variables in order to dene the projection Each Newton-iteration of a projected collocation method therefore contains two steps: (1) compute the solution y of the linearized BVP, (2) project the values at the mesh points according to (238, 239) These iterations are shown to be locally quadratically convergent By using this special kind of projection which is not orthogonal to the manifold g(y) 0 the usual superconvergence properties (eg O(h 2k ) for Gaussian collocation) are shown to hold Collocation methods with k = 1 are equivalent to their projected versions dened above In the context of optimization the two step iteration above is not advisable, since one needs the DAE dicretization in the form of a system of equations and not of an algorithm However, it is not hard to see that for the DAE (227, 228) a nonlinear condition of the type ^y 1;j = y 1;j f(^y 1;j ; y 2 ( j )) 2 0 = g(^y 1;j )

23 22 Collocation for BVP in DAE with invariants 21 could serve to construct a system of equations which is equivalent to the original formulation of projected collocation But it includes additional variables (^y 1;j, j ) and involves second derivatives of f in the linearization process of these equations Another way of insuring that known equations for the states are satised at mesh points is explained in the following section for invariants But it can be applied to the enforcement of the algebraic equations at the collocation mesh points as well 223 Exploiting invariants Up to now we have only explained how to discretize DAE of index up to 2 However, in section 221 we mentioned that in practice there are models involving DAE of index at least 3 Direct discretization of higher index DAE does not lead to stable schemes Therefore in these cases index reduction is suggested This means that the algebraic part of the DAE is dierentiated wrt time as often as necessary in order to get a DAE of index 1 or 2 For multibody descriptor models repeated dierentiations of the position constraints (223) lead to the constraints on velocity (240) and acceleration level (241) 0 = G(p) p _ ; (240) 0 = G(p)p (G(p) _p) _p: Equations (222, 241) form a DAE of index 1 The original onstraint (223) and its derivative (240) serve as invariants There may, of course, be other sources of invariants like, eg, energy conservation, geometric invariants or the conservation of the Hamiltonian in optimal control problems Here we dene invariants of the solution of the DAE-BVP ( ) of index 1 as additional equations h(y 1 (t)) = 0; t 2 I [0; T ] (2:42) known to be satised in some subinterval I by the exact solution of the boundary value problem For the sake of clarity of the presentation, in (242) the invariant is formulated depending only on the dierentiable variables, which is of course true for invariants resulting from index reduction Using only the index 1 formulation (222, 241) for the numerical solution of multibody descriptor form DAE, one may observe the well known drift phenomenon (see, eg, [1, 2, 28, 42]) Due to an accumulation of discretization errors the numerical solution does no longer satisfy the invariants, which leads to inacceptable errors in the solution Therefore there have been undertaken many eorts to devise techniques for the numerical conservation of these invariants These range from Baumgarte techniques [15, 110, 9] over symplectic integration schemes for Hamiltonian systems [82] and additionally projecting Runge-Kutta-schemes [9] up to general projecting integration schemes [1, 42, 2] for initial value problems and boundary value problems [110] The adequate exploitation of invariants results in a stabilizing eect on the numerical solution of the DAE Besides, there is an additional payo with regard to conditioning in the case of multistage projection BVP methods This is motivated by the following lemma

24 22 2 The Collocation Discretization Lemma 22 Consider the linear system of equations for x 1 ; x 2 2 R n W x 1 = x 2 ; W 2 R nn nonsingular; (2:43) where x 1 is to be expressed in terms of x 2 Assume that x 1 and x 2 satisfy the relation Hx 1 = Hx 2 (2:44) for some matrix H 2 R n hn of full rank with n h < n Then there is an orthogonal matrix Q 2 R nn with Q > Q = I, so that Q > W Q = " # I 0 Z > ; (2:45) W Z where the columns of Z 2 R n(n?n h) form an orthogonal basis of N (H) Proof: Consider a QR-factorization of the matrix H > h i" # " H > R R = Y Z =: Q 0 0 Then there is a unique representation of the vectors x 1 and x 2 as From (244) follows y 1 = y 2 and therefore yielding # : x i = Y y i + Zz i ; i = 1; 2: (2:46) y 1 = Y > x 1 = Y > x 2 = Y > W x 1 = Y > W (Y y 1 + Zz 1 ); Y > W h Y Z i = h I 0 i : The linear system (243) can be interpreted as a linearized continuity condition for a discretized ODE, while (244) represents linear invariants By using the decomposition (246), the linear system (243) with the property (244) can be solved in the following way: Z > W Zz 1 = z 2? Z > W Y R?1 Hx 2 ; where y 1 = R?1 Hx 2 The condition number of Z > W Z is not worse than that of W itself, because only orthogonal transformations are involved On the other hand Z > W Z often is better conditioned than W, which is conrmed by numerical experience [110] Thus lemma 22 shows that the knowledge of an invariant can also be exploited for the conditioning of linear systems Similar systems as in lemma 22 arise in the multistage least squares approach proposed in [110] Since a DAE discretization consisting of collocation conditions, continuity conditions, boundary conditions and invariants is formally overdetermined, we introduce a

25 22 Collocation for BVP in DAE with invariants 23 ranking of the conditions leading to a weaker formulation of the continuity conditions in the form of least squares conditions By using the notation introduced in ( ), the original continuity condition (234) for collocation (26) is replaced by the condition min y 1;j+1 y 1;j + h j kp 2 z js l (1)? y 1;j+1 s=1 2 (247) st h(y 1;j+1 ) = 0; (248) if the invariant h (242) is known to hold at j+1, ie j+1 2 I This formulation is equivalent to the orthogonal projection of the continuity conditions onto the invariant manifold M = fy j h(y) = 0g Therefore its solution coincides with the solution of the invariant conservation technique proposed in [8], which involves an additional projection step as in (238, 239), instead of F 2 in (238) and h instead of g in (239) The least squares formulation above has two advantages It is generalizable to optimization boundary value problems in a straight forward way It exploits possible gains in the conditioning of the overall linear system due to the presence of invariants By employing a generalized Gauss-Newton-method [22, 102], the local least squares problems (247, 248) are linearized \under the norm" resulting in the linear least squares problems where c j = y 1;j + h j k P s=1 min y j+1 ky 1;j + jz j? y 1;j+1 + c j k 2 2 (249) st H j+1 y 1;j+1 + h j+1 = 0; (250) z js l (1)? y 1;j+1, j = h j h H j+1 := (@h=@y 1 )(y 1;j+1 ) QR-decomposition of leads to the equivalent linear system H > j+1 = h Y j+1 Z j+1 i " R j+1 0 1(1)I : : : k (1)I i, h j+1 := h(y j+1 ) and Z > j+1 y 1;j + Z > j+1 jz j? Z > j+1 y 1;j+1 + Z > j+1 c j = 0; (251) # H j+1 y 1;j+1 + h j+1 = 0: (252) This system replaces the usual linearized continuity conditions at each node If the invariant holds at j as well, we transform the variables y 1;l ; l = j; j + 1 so that y 1;l = Y l d Y l + Z l d Z l ; l = j; j + 1;

26 24 2 The Collocation Discretization where Y j ; Z j are dened by a QR-factorization of H j > the projected continuity conditions := (@h=@y 1 )(y 1;j ) > Thus we obtain Z > j+1z j d Z j + Z > j+1 jz j? d Z j+1 + Z > j+1c j+1? Z > j+1y j R?> j+1 h j+1 = 0; (2:53) where d Y j+1 is determined by R > j+1d Y j+1+ h j+1 = 0 Thus the projected continuity conditions have the same form as the original ones (234) Remark: The linearization \under the norm" in (249, 250) describes the tangential space of the nonlinear least-squares conditions (247, 248) up to an error of the order of the residual in (247), which is the discretization error Therefore, in the context of optimization the solution of the multistage optimization problem is disturbed by an additional error of the order of the discretization error

27 Chapter 3 Partially reduced SQP methods In this chapter the algorithmic concept of partially reduced SQP (PRSQP) methods is introduced First we motivate and review the algorithmic features of reduced SQP methods Then the partial reduction concept is formulated and the convergence properties of resulting optimization algorithms are investigated Basic concepts of nonlinear optimization, like necessary and sucient optimality conditions, are not covered They can be found in any textbook about nonlinear optimization (eg, [59, 43]) 31 From feasible path methods to reduced SQP The aim of this section is to give a brief introduction into the basic terminology of reduced SQP methods (RSQP) We consider the constrained optimization problem minf(x; p) x;p (31) st c(x; p) = 0 2 R nx ; (32) where p 2 R np and x 2 R nx, the problem functions f and c are at least twice continuously dierentiable and C x is nonsingular A common way to solve optimization problem (31, 32) is to consider x as a function of p and thus the optimization problem as an unconstrained one in only the variables p That means that one uses the implicit function theorem in order to state the existence of a function : R np! R nx, so that x = (p) Then the constrained optimization problem (31, 32) is transformed to an unconstrained optimization problem: min p ~ f(p); with ~ f(p) := f((p); p): (3:3) Methods based on this simple variable reduction process will be called feasible path methods in the sequel The advantages of this approach are the reduction of the number of optimization variables and the fact that the Hessian of ~ f(p) is positive semi-denite at the solution, if there exists a solution at all The drawback is that at each step of an 25