Introduction

Continuous system behavior can be well represented by differential equations that describe behavior evolution in time [3]. A convenient and exhaustive analysis applies state space techniques, a space that has the system state variables as dimensions and the gradient of behavior as their function. This clearly shows trajectories along which the system may evolve. Often the differential equations are supplemented by a set of algebraic equations that may constrain the state variables. The result is a system of differential and algebraic equations, DAEs, where behavior in state space is confined to a manifold by the algebraic equations.

To solve a system of DAEs, the algebraic equations that constrain state variables need to be explicitly expressed in terms of time-differentials of state variables to eliminate and obtain a set of independent differential equations. Typically, this requires a subset of the algebraic equations to be differentiated. For example, consider the two parallel capacitors in Fig. 1. The system of equations for this circuit is  $$\left[\begin{array} {ccc} C_1 & 0 & 0 \\ 0 & C_2 & 0 \\ 0 & ... ...ight] \left[\begin{array} {c} v_1 \\ v_2 \\ i\end{array}\right]$$
The bottom row in the system of equations puts a constraint on the state variables v₁ and v₂ and this can be solved by differentiating the particular row to obtain $$\left[\begin{array} {ccc} C_1 & 0 & 0 \\ 0 & C_2 & 0 \\ 1 & ... ...ight] \left[\begin{array} {c} v_1 \\ v_2 \\ i\end{array}\right]$$
which results in a system of explicit differential equations.

  

Figure 1: Two parallel capacitors.

Pantelides has presented an efficient algorithm to find the minimum set of algebraic equations that need to be differentiated in order to find consistent state derivatives, $\dot{x}$, that enforce system behavior evolution on the prescribed manifold in state space [13]. However, initial state values that are not on the manifold have to first be projected onto it. Inconsistent initial state values may be due to user specification or they may arise when the system configuration changes dynamically.

Configuration changes may occur in mixed continuous/discrete, hybrid, systems [7] where behavior evolution in a mode of operation is governed by a system of DAEs. At well-defined points in time, discrete events occur that change the structure of the DAEs. This may cause algebraic equations to become active and the initial values in the newly inferred mode have to satisfy these constraints, requiring a projection from the state variable values just before mode switching occurred to values that obey additional constraints in the newly found mode.

For example, the state spaces for an open and closed switch in the capacitor circuit in Fig. 1 are given in Fig. 2. When the switch is open, the entire state space can be covered. When the switch is closed, system behavior is constrained to the manifold v_C1 = v_C2 and the initial state has to be projected onto it.

  

Figure 2: Projection onto a manifold has to satisfy physical principles.

Such a projection results in discontinuous changes at well-defined points in time. Because of the derivative nature of the state variables, this may result in impulses on system variables. When integrated at the point of discontinuity, i.e., over an infinitesimal interval of time, variables that contain impulses return nonzero values. This paper shows how the areas of these impulses have to satisfy physical conservation constraints, and, therefore, require a specific consistent projection. In other work [5], minimum norm constraints are used to find initial values on the manifold. However, these methods have no physical interpretation and in general violate physical conservation principles.

The difficulty in deriving a physically consistent projection stems from the observation that the algebraic equations impose constraints on the state variables with respect to each other, but no absolute measure is established. Therefore, one may know how two values relate to one another but not what their value with respect to a reference value is. For example, in case of the two parallel capacitors (Fig. 1), the constraint requires v₁ = v₂, but it is undetermined what the value of either v₁ or v₂ is.

Previous work [7,8,9], established the principle of conservation of state to govern the additional initialization constraint. This principle relies on the observation that in physical systems during mode changes there may be discontinuity in power and even instantaneous loss of energy, but a generalized quantity is conserved. Physically this quantity embodies the extensive variables such as charge, momentum, and mass [1]. In the two capacitor example (Fig. 1), the charge is conserved when the switch closes.

This paper derives a consistent projection for a class of DAE systems using equations derived from physical conservation principles. Once consistent initial values are obtained, behavior evolution as specified by solving the rate dependencies in the system of DAEs remains on the manifold. Section 2 describes the principle and how it affects DAEs across mode changes. Section 3 formally derives the algorithm to find the state projection for semi-explicit DAEs with linear constraints. Section 4 illustrates the algorithm for a number of dynamic physical system examples. Section 5 concludes and discusses future research.