State space modeling_introduction
- 1. State-Space Modeling David W. Graham EE 327
- 2. 2 State-Space Modeling • Alternative method of modeling a system than – Differential / difference equations – Transfer functions • Uses matrices and vectors to represent the system parameters and variables
- 3. 3 Motivation for State-Space Modeling • Easier for computers to perform matrix algebra – e.g. MATLAB does all computations as matrix math • Handles multiple inputs and outputs • Provides more information about the system – Provides knowledge of internal variables (states) ⇒Primarily used in complicated, large-scale systems
- 4. 4 Transfer Functions vs. State-Space Models • Transfer functions provide only input and output behavior – No knowledge of the inner workings of the system – System is essentially a “black box” that performs some functions • State-space models also represent the internal behavior of the system H(s)X(s) Y(s)
- 5. 5 Definitions V – Input vector • Can be multiple inputs • Written as a column vector ( ) ( ) ( ) ( ) = tv tv tv tv R 2 1 Y – Output vector • A function of the input and the present state of the internal variables ( ) ( ) ( ) ( ) = ty ty ty ty M 2 1
- 6. 6 Definitions X – State vector • Information of the current condition of the internal variables • N is the “dimension” of the state model (number of internal state variables) ( ) ( ) ( ) ( ) = tx tx tx tx N 2 1 X – “Next state” vector • Derivative of the state vector • Provides knowledge of where the states are going – Direction (+ or -) – How fast (magnitude) • A function fo the input and the present state of the internal variables ( ) ( ) ( ) ( ) = tx tx tx tx N 2 1
- 7. 7 State-Space Equations General form of the state-space model Two equations – ( ) ( )ty tx ( ) ( ) ( )( ) ( ) ( ) ( )( )ttvtxgty ttvtxftx ,, ,, = =
- 8. 8 Linear State-Space Equations ( ) ( ) ( ) ( ) ( ) ( )tDvtCxty tBvtAxtx += += ( ) ( ) ( ) ( ) vector vector vectors 1 1 1, ×→ ×→ ×→ Mty Rtv Ntxtx RMD NMC RNB NNA ×→ ×→ ×→ ×→ system matrix input matrix output matrix matrix representing direct coupling from system inputs to system outputs If A, B, C, D are constant over time, then the system is also time invariant → Linear Time Invariant (LTI) system
- 9. 9 Construction of State Equations from a Differential Equation (Let there be no derivatives of the input) • The dimension of the state equations (number of state variables) should equal the order of the differential equation • Let one state variable equal the output (y(t)) • Let one state variable equal the derivative of the output • Let one state variable equal the (N-1)-th derivative of the output (where N is the order of the differential equation) • Find the derivative of each of the newly defined state equations – In terms of the other state variables and the outputs • Write the state equations
- 10. 9 Construction of State Equations from a Differential Equation (Let there be no derivatives of the input) • The dimension of the state equations (number of state variables) should equal the order of the differential equation • Let one state variable equal the output (y(t)) • Let one state variable equal the derivative of the output • Let one state variable equal the (N-1)-th derivative of the output (where N is the order of the differential equation) • Find the derivative of each of the newly defined state equations – In terms of the other state variables and the outputs • Write the state equations