EE451_Chapter7_Notes_F09

# EE451_Chapter7_Notes_F09 - EE451/551: Digital Control...

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EE451/551: Digital Control Chapter 7: State Space Representations

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State Variables Definition 7.1: The system state is a minimal set of } { 0 variables ( ), 1,2, , needed together with the i xt i n = " ) 0 input ( ), and in the interval , to uniquely determine the behaviour of the system in the f ut t t t interval de e e e be v ou o e sys e e 0 ev , ; where is the order of the system f tt n   The set of state variables for the state vector is written: x [] 1 T 2 12 ( ) , , n x x x x  == " n x #
State Variables he state space is the dimensional vect r space where The state-space is the - dimensional vector space where the ( ) from ( ) represent the coordinate axes i n xt x t For a second-order system, the state space is two- imensional and known as the ; for ate -plane e special dimensional and known as the ; for state plane the special case where the state-variables are proportional to the i t i f t h t t t h t t lil l d t h h derivatives of the output, the state plane is called the and the state variables are called phase plane phase variables Curves in state space are known as the , d a plot of the trajectories in the pl e is the state trajectories ate and a plot of the trajectories in the plane is the (or for the phase plane) state portrait phase portrait

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Example 7.1: Motion of a Pt. Mass onsider the motion of a point mass dri en by a force : u Consider the motion of a point mass driven by a force mu u my u y =⇒ = ±± where is the displacement of the point mass; defining the m y system states as the phase variables: 11 2 ( ) ( ) ( ) ( ) ( ) xt y t y t x t ⇒= = ² 22 ( ) ( ) ( ) ( ) u x t yt m = ± ± ²
Example 7.1: Motion of a Pt. Mass the state vector is given by: 1 2 If the state vector is given by: () , () T xt t x t x t  == [ ] 12 2 then the system dynamics in state-variable form is:   2 11 0 01 xx = + ± ± A x B u ² 22 1 00 x u mm ± cc uA + [] 1 2 1 0 x yC x D u x =+ ²

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Questions?
The General State Space Forms he general form of a linear state-space quation is: The general form of a linear state space equation is: ( ) ( ) ( ) cc xt Axt But =+ ± () where ( ) is a 1 real vector, yt C D ut n × state ( ) is a vector, m × input is a vector, y t p × output and the matrices are defined as: is a real matrix n n ate matrix, is a real matrix, c c An Bn m × × state input is a real matrix, i c c Cp n D × output s a real matrix pm × feedforward

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The General State Space Forms he general form of a nonlinear state-sp e equation is: () The general form of a nonlinear state space equation is: ( ) ( ), ( ) xt f xt ut = ± ( ) ( ) ( ), ( ) where () and () are real 1 and 1 vectors of yt g xt ut fg n p = × × ii functions that guarantee the existence and uniqueness of solution to the system dyanmics a solution to the system dyanmics The nonlinear system can be approximated at an ( ) 00 equilibrium point , using a linear model, specified previously xu sing the concept of a as specified previously, using the concept of a system's Jacobians, as defined on the next slide
System Jacobians ( ) 00 System Jacobians are defined for equilibrium point , as: xu 11 1 1 1 m ff f f

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## This note was uploaded on 12/20/2009 for the course ECE 451 taught by Professor Staff during the Fall '09 term at Clarkson University .

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EE451_Chapter7_Notes_F09 - EE451/551: Digital Control...

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