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State Space Models - Stanford University

MUS420 Introduction to Linear State Space ModelsJulius O. Smith III for Computer Research in Music and Acoustics (CCRMA)Department of Music, Stanford UniversityStanford, California 94305 February 5, 2019 Outline State Space Models Linear State Space Formulation Markov Parameters (Impulse Response) Transfer Function Difference Equations to State Space Models Similarity Transformations Modal Representation (Diagonalization) Matlab Examples1 State Space ModelsEquations of motionfor any physical system may beconveniently formulated in terms of itsstatex(t):ftInput Forcesu(t)RModel Statex(t) x(t) x(t) =ft[x(t), u(t)]wherex(t) =stateof the system at timetu(t) =vector ofexternal inputs(typically driving forces)ft=general function mapping the current statex(t)andinputsu(t)to the State time-derivative x(t) The functionftmay be time-varying, in general This potentially nonlinear time-varying model isextremely general (but causal) Even thehuman braincan be modeled in this form2 State - Space History1.

solving the ODE • ODE can be nonlinear and/or time-varying • The sampling interval Tn may be fixed or adaptive 7 State Definition We need a state variable for the amplitude of each physical degree of freedom Examples: • Ideal Mass: Energy = 1 2 mv2 ⇒ state variable =v(t) Note that in 3D we get three state variables (vx,vy,vz ...

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