Matrix Integrators For Real-Time Simulation

MATRIX INTEGRATORS FOR REAL -TIME SIMULATIONTom T. HartleyDepartment of Electrical EngineeringThe University of AkronAkron, OH 44325G. O. BealeDepartment of Electrical and Computer EngineeringGeorge Mason UniversityFairfax, VAABSTRACTThis paper contains a discussion of numerical integration methods with regression coeffi-cients that are matrices rather than the usual scalars. Methods of this type will be shownto be particularly useful for real -time simulations as they allow arbitrary placement of thesimulation closed loop poles. This property is generally very useful, but especially so forthe real -time simulation of stiff systems. Linear systems will be considered initially,followed by obvious extensions to nonlinear systems. Adaptive integration using thesemethods will also be introduced.1. INTRODUCTIONThe simulation of physical systems is known to be a cost effective method for the designand testing of control systems. Furthermore, real -time simulation is very useful in testingcontrol system hardware as well as in personnel training. The hardware constraints as-sociated with real -time simulations, however, can sometimes restrict the timesteps used inthe simulation to be larger than are allowable for a stable simulation, while using smallerones would not allow enough time for all of the required computations to performed. In asituation such as this, the only possible solutions are to either use a faster computer forthe simulation, or to use an algorithm that allows a larger timestep to be used while main-taining stability. The latter solution will be the topic of this paper.Most methods that have historically been used for numerical integration are scalarmethods, although they can be used for vector systems of differential equations. Essen-tially, the same integrator is used for each component of the vector system. This paper willpresent methods that will replace the scalar components of the integrators by matrices thathave the same dimensions as the system they are integrating. For a linear system of dif-ferential equations, it is then possible to use the SRP method developed by the authors [1]to place the principle roots of the resulting discrete system at the exact mappings of thes -plane poles, while placing all the spurious roots at the origin in the z- plane. Thisdesign approach (Matrix SRP, MSRP) will be shown to be especially useful for stiff systemswhich have large negative eigenvalues in the s- plane. For linear time -invariant systems withpiecewise constant inputs, the resulting discrete system will give exact time responsesafter one timestep while having zero steady state error in the simulation.For nonlinear systems, custom integrators can also be designed; however, the nonlinearsystem must first be linearized about the desired steady state operating point. Once thesystem Jacobian matrix has been found for this operating point, the SRP method can again beused to derive the matrix integrators. The resulting simulations will be reasonably accurateas long as the system states are close "enough" to steady state. The larger the transients,the larger the expected transient error will be. As a potential solution to this problem,techniques for adaptively implementing this method will be considered. The idea will be toadaptively track the system Jacobian matrix as the system evolves. This will then allow theintegrator matrices to track the system Jacobian as it changes, while keeping the timestepconstant. An approach such as this would obviously support multiprocessing.Also included will be several illustrative examples of industrial systems. Design andimplementation software which uses the PC- MATLAB software package will be discussed, as itis a natural environment for this approach. A discussion of general multistep implementa-tions and closed loop root sensitivity will be included. Some consideration will also begiven to possible scalar integrators that still allow arbitrary placement of the closed loop