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Mathematical and Computer Modelling of Dynamical Systems 1387-3954/01/0702-133$16.00 2001, Vol. 7, No. 2, pp. 133±143 Swets & Zeitlinger Aspects of Modeling Dynamical Systems by Differential-Algebraic Equations P.C. MULLER ABSTRACT In recent years the analysis and synthesis of control systems in descriptor form has been established. The general description of dynamical systems by differential-algebraic equations (DAE) is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. In this contribution the pros and cons of system modelling by differential-algebraic equations are discussed and an actual state of the art of descriptor systems is presented. Firstly, the advantages of modelling are touched in general and illustrated in detail by Lagranges equations of ®rst kind, by subsystem modeling and by the statement of the tracking control problem. Secondly, the development of tools for numerical integration is discussed resulting in the comment that today stable and ef®cient DAE solvers exist and that the simulation of descriptor systems is not a problem any longer. Thirdly, the methods of analyzing and designing descriptor systems are considered. Here, linear and nonlinear systems have to be distinguished. For linear descriptor systems more or less the required methods to solve usual control tasks are available in principal. But actually a related program package for fast and reliable application of these methods is still missed. However, in the near future such a toolbox is expected. Main dif®culties arise for nonlinear problems. A few results on stability and optimal control are known only and still a lot of research work has to be effected. In spite of these de®ciencies, all over the descriptor system approach is very attractive for modeling and simulation, and will become attractive more and more for analysis and design. Keywords: differential-algebraic equations, descriptor systems, singular control systems. 1 INTRODUCTION The investigation of dynamical systems in mechanical, electrical or chemical engineering usually requires a mathematical modeling of the system behavior. Safety Control Engineering, University of Wuppertal, D-42097 Wuppertal, Germany. E-mail: Mueller@srm.uni-wuppertal.de È 134 P.C. MULLER The increasing complexity of these processes lead on the one side to the development of computer programs automatically generating the governing system equations, cf. [37] for multibody systems, or on the other side to an increase of modular subsystem modeling of which the complete model is composed. Usually, this interconnection-oriented modeling describes the dynamic behavior of the single components by differential equations and the coupling of the subsystems by algebraic equations. Allover, the mathematical model is represented by a combined set of differential and algebraic, i.e., by differential-algebraic equations (DAE). In control engineering we speak about singular control systems or descriptor systems [18]. Today the analysis and synthesis of control systems in descriptor form has been established more or less. Therefore, in the following a survey of the state-of-art of descriptor systems is presented with respect to the aspects of modeling, simulation, analysis and control design. The paper follows closely the approach [28]. Models of chemical processes, for example, typically consist of differential equations describing the dynamic balances of mass and energy while additionally algebraic equations account for thermodynamic equilibrium relations, steady-state assumptions, empirical correlations, etc. [7, 14, 31]. Also electrical networks can be considered to be composed by subsystems of network elements (like resistors, capacitors, inductors described by different types of equations including differential equations) and by coupling due to Kirchhoff's laws (described by algebraic equations) [12, 20]. In mechanical systems the differential equations usually describe the dynamics of the subsystems and the algebraic equations characterize couplings by constraints such as joints. A general approach to handle mechanical systems as an interconnected set of dynamic modules has been given in [35]. In the following three examples of descriptor modeling are shortly dealt with for illustration. 1.1 Lagrange's Equations of First Kind Lagrange's equations of ®rst and second kind are well established in analytical mechanics, cf. [34]. They describe the dynamic behavior of discrete systems, particularly of multibody systems. The difference of the two kinds consists in the manipulation of the kinematic constraints. If a kinematic description of the system has been performed by generalized coordinates consistent with the constraints the Lagrange's equation of second kind can be applied leading to MODELING DYNAMICAL SYSTEMS BY DAE 135 a set of differential equations only. But if a redundant set of coordinates is used to describe kinematically the system regarding still some constraints explicitly then Lagrange's equations of ®rst kind hold. In case of holonomic constraints f…q†ˆ 0 …1† we have d @L @L ÿ ˆ Q‡ F …2† dt @q @q where the Lagrangian function L ˆ T ÿ U consists of the kinetic and potential energies T and U, Q represents the nonconservative forces acting on the system, F ˆ F…q†ˆ @f=@q is the Jacobian matrix of the constraints and is the vector of Lagrange's multipliers. They represent the constraint forces if the column vectors of F are normalized. While the variables q describe the motion of the system, the Lagrange's multipliers give some information on the load of the mechanical structure. Therefore, critical loads due to the motion may be considered simultaneously. Equations (1, 2) represent a system of differential-algebraic equations. If Q includes some actuator forces to control the multibody system then a descriptor system is under consideration. 1.2 Subsystem Modeling If the interconnection-oriented modeling approach is applied [22], usually the dynamics of N subsystems are described by sets of differential equations x ˆ a…x ; u†; i ˆ 1; .. . ; N …3† i i i i where x are the internal state vectors and u the control vectors of the i i corresponding subsystems. The couplings among the subsystems may be obtained kinematically by constraints or kinetically by forces leading to N N X X x _ ˆ a…x ; u†‡ a …x ; x†‡ L …x† …4† i i i i ij ij j i j j jˆ1 jˆ1 0 ˆ f …x†; i ˆ 1; .. . ; N: …5† ij j jˆ1 The additional terms compared to (3) are the kinetic couplings a between ij subsystems no. i and j and the kinematic couplings (5) which have to be È 136 P.C. MULLER considered in the dynamic balance equation (4) by some Lagrange's multipliers with some input matrices L due to the coupling requirements. j ij How L is de®ned more precisely depends on the physical principles behind ij the system discipline; equations (1, 2) show an example of mechanical systems. All over, equations (4, 5) represent again a descriptor system. 1.3 Tracking Control In control engineering often the problem of tracking control arises, e.g., the prescribed path control of a robot. In this case the process dynamics may be described in the state space by x ˆ a…x; u; t†…6† and it is asked for the control u which guarantees that some output variables y ˆ c…x; u; t† follow a prescribed reference path y …t† ref 0 ˆ c…x; u; t†ÿ y …t†: …7† ref This descriptor system (6, 7) can be described by _ a…x; x; t† I 0 x ˆ …8† c…x; x; t†ÿ y …t† 00 x ref which de®nes explicitly the desired tracking control x…t† u…t†ˆ‰Š 0I …9† x…t† where I , I are identity matrices of dim…x† and dim…u† respectively. For linear x u time-invariant systems (6, 7) with a…x; u; t†ˆ Ax‡ Bu; c…x; u; t†ˆ Cx‡ Du …10† and dim…u†ˆ dim…y† the descriptor system (8) reads I 0 x _ AB x ˆ ÿ …11† y …t† 00 x CD x ref |‚‚‚‚‚{z‚‚‚‚‚} |‚‚‚‚‚{z‚‚‚‚‚} E A Then the tracking control problem is solvable if and only if the matrix pencil …E; A† is regular, cf. (15). MODELING DYNAMICAL SYSTEMS BY DAE 137 1.4 Pros and Cons With respect to the tasks of system modeling the descriptor system approach has many advantages. It is a very natural way to model process dynamics. It refers much more to the physical behavior of the system and gives more physical insight. The interpretation of results is also more simple than in case of the more abstract description by state space models. In the opposite the state space system approach was mainly required by the mathematic tools available until 1980 to simulate, to analyse and to design such systems. 2 SIMULATION As long as it was not possible to simulate descriptor systems very ef®cient and very accurate still the state space approach was superior according to the well- established tools of numerical integration of ordinary differential equations. But in the 1970's the simultaneous numerical solution of differential and algebraic equations was ®rstly considered [10]. Step by step numerical system solvers were developed. For index-1-problems (see below) the code DASSL has been presented [32], stimulating more research also for higher index problems. A ®rst code for mechanical index-3-systems has been presented in [8]. In the meantime a lot of ef®cient solvers for DAE s have been developed [3, 11], especially for mechanical descriptor systems [17, 38, 39]. In a more recent Ph.D. thesis [36] on the modular simulation of mechatronic systems several solvers have been compared resulting in the recommendation of the codes SDOP853 and SDOPRI5 which are modi®ed versions of Runge±Kutta solvers for ordinary differential equations including projection steps with respect to the constraints of the algebraic equations. A recent survey on solvers of higher index DAE s is given by [1]. With respect to these results today a number of stable and ef®cient DAE solvers exist and can be applied as naturally as ODE solvers for state space models. Such solvers are included in many program packages to generate and to simulate the equations of motion of dynamical systems, e.g., in DIVA [13] for chemical processes or in ADAMS and SIMPACK, cf. [37], for multibody systems. 3 ANALYSIS AND SYNTHESIS The tools for the analysis and synthesis of descriptor systems have been developed enormously in the last two decades. As usual, linear theory has È 138 P.C. MULLER been in the foreground of the discussion, but ®rst results on nonlinear problems have been reported, too. 3.1 Linear Systems Linear time-invariant descriptor systems are presented by Ex _…t†ˆ Ax…t†‡ Bu…t†; …12† y…t†ˆ Cx…t†‡ Du…t†…13† where x is an n-dimensional descriptor vector, u denotes the r-dimensional control input vector, and y characterizes the m-dimensional measurement output vector. The matrices E, A are n n-matrices, and B, C, D have dimensions n r; m n; m r, respectively. The essential property of descriptor systems is that E is a singular matrix, rank E < n; …14† such that (12) consists of differential and algebraic equations. The basic tool in discussing (12) is the theory of the matrix pencil…sEÿ A† by Weierstrass and Kronecker in the 19th century, cf. [6], separating the system into a few subsystems with different properties. Assuming unique behavior of (12) for all control inputs, i.e., assuming that the matrix pencil is regular, p…s† det…sEÿ A†6 0 …15† then system (12) is strictly equivalent to the Weierstrass-Kronecker form x _ …t†ˆ A x …t†‡ B u…t†; …16† 1 1 1 1 N x …t†ˆ x …t†‡ B u…t†; …17† k 2 2 2 y…t†ˆ C x …t†‡ C x …t†: …18† 1 1 2 2 Equation (16) represents the slow subsystem of dimension n , and the n - 1 2 dimensional fast subsystem is described in (17). The n n -matrix N is 2 2 k kÿ1 k nilpotent of degree k (N 6ˆ 0; N ˆ 0 ) de®ning the index k of the linear k k descriptor system. According to the separation into the two subsystems controllability and observability investigations split off into at least two different concepts of so- called R / I-controllability and -observability guaranteeing different properties of a feedback control, cf. [6, 16]. The results of many investigations in the MODELING DYNAMICAL SYSTEMS BY DAE 139 1980's have been summarized by Lewis [16] and Dai [6]. Stability can be discussed by the eigenvalues of the matrix pencil…sEÿ A†, i.e., by the roots of the characteristic polynomial (15). Another approach is based on the generalized Lyapunov matrix equation T T A PE‡ E PAˆÿ Q …19† where de®niteness properties of P and Q with respect to certain subspaces assure stability [21]. First results on the design of linear feedback control by pole placement have been presented in [6]. But the main problem of the synthesis of feedback control consists in the possibility of non-proper system behavior. This can be seen immediately by the solution of the fast subsystem (17), cf. [6], jÿ1 … jÿ1† x …t†ˆÿB u…t†ÿ N B u _…t†ÿ :::ÿ N B u …t†; …20† 2 2 k 2 2 which includes generally higher-order time-derivatives of the control input. The two cases have to be distinguished where the solution of (12) (or (16 and … jÿ1† (17)) depends either only on u…t† but not on its derivatives u _…t†; .. . ; u …t† … jÿ1† or on u…t† and its derivatives u _…t†; .. . ; u …t† according to the general case (20). In the ®rst case the system is called proper, in the second case non-proper according to related proper and non-proper transfer matrix functions. The system (12) is proper if and only if in the representation (16, 17) the equation N B ˆ 0 …21† k 2 holds. The distinction between proper and non-proper descriptor systems and its consequences for the control design has been discussed just recently [25, 26, 27, 29]. Regarding proper and non-proper systems in different ways, the linear quadratic optimal regulator problem [25, 26, 29] and the descriptor state estimation problem [27] has been discussed in detail and properly solved. Therefore, the standard design methods are available for linear descriptor systems, too. Also ®rst results on robust control design exist. The H -control problem has been considered in [15, 19, 33, 40, 43]. In the case of stabilizable, detectable, impulse-controllable and impulse-observable descriptor systems necessary and suf®cient conditions for the solvability of the control problem are given. But what happens in case of descriptor systems which are not impulse-controllable like mechanical systems (1, 2)? Therefore, research work is still necessary to loosen such conditions for more general descriptor È 140 P.C. MULLER systems. For example, a linear descriptor system (1, 2) is never I-controllable but it may be R-controllable and a H -control design is still of interest. 3.2 Nonlinear Systems The analysis and the design of nonlinear descriptor systems is more or less an open ®eld. A survey on the mathematical theory has been presented in [5]. Essential results with respect to the stability problem can be found in [2, 23, 24]. Some steps have been taken for the optimal control design [24], but especially for non-proper systems many problems still have to be solved. First results on the method of nonlinear decoupling and exact linearization by state feedback have been presented is [9, 30]. But both papers are restricted by some assumptions on the nonlinear descriptor systems: it is assumed either that the algebraic variables only appear in linear form [9] or that the descriptor system is proper [30]. Still some aspects are open and the problem is under consideration. 3.3 Pros and Cons For the analysis and synthesis of linear descriptor systems the usual theoretical tools are available (more or less). Related program packages are being developed, [4, 42]. In Varga [42], mainly the analysis of linear time-invariant descriptor systems is considered; [4] will start in 2001 to improve the algorithms. Usually they are based on van Dooren's algorithm characterizing the eigenstructure problem of the matrix pencil (sE ± A) [41]. Therefore, some research groups are partly provided with computer algorithms for the analysis and design of linear descriptor systems, but still we do not have the standard or the comfort of a MATLAB toolbox. But it is expected that this is only a matter of time. In a few years the linear descriptor toolbox will be available as linear state space algorithms today. For non-linear descriptor systems still a lot of research work has to be done. Here, we are only on the beginning to understand nonlinear system behavior and to develop control design methods. 4 CONCLUSIONS In this contribution an effort has been made to characterize the state of the art of modeling, analyzing and designing dynamical processes by the descriptor MODELING DYNAMICAL SYSTEMS BY DAE 141 system approach. 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Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis
Published: Jun 1, 2001
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