6. State space_upload

6. State space_upload - fig_05_02 fig_05_03 fig_05_05...

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State Space JIN KANG
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Review of how we got here Considering models of physical systems In particular, specific representations that are common in systems analysis and design of control systems 1. Functional block diagram Began course by laying out functional block diagram of heating system 2. ODE Considered diff eqs as math models that describe dynamic behavior of physical system 3. Laplace Created Laplace forms of ODE 4. Transfer function From Laplace formed transfer function which represents characteristics of physical system How system turns input to output 5. Block diagram reduction Realized that CL system could be reduced to a single CLTF
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State Space Representations What we’ve done is considered “classical” controls Since 1960s (age of computers), a new view has developed called “modern” controls Remain in time domain Allow for multi-input systems (MIMO) Open up option to develop controller that optimized system * Engineers in practice use both! * Different forms n th order diff eq gets modeled by n 1 st order diff eqs
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State Space Form State equation Output equation [ ] [ ] u B x x x A x x x x n n + = = 2 1 2 1 n×1 n×n n×r r×1 [ ] [ ] u D x x x C y n + = 2 1 m×1 m×n m×r r×1
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Definitions State State of system is smallest set of variables (state variables) Knowledge of variables at t=t 0 plus knowledge of input for t≥t 0 completely determines behavior of system for any t≥0 State variables Variables making up smallest set of variables that determine state of dynamic system Note: state variables need not be physically measurable or necessarily observable Gives much freedom in choosing or defining state of system State vector Vector of n state Determines uniquely system state x(t) for any time t≥t 0 State space n dimension space whose coordinate axes consist of x 1 axis, x 2 axis, … Any state can be expressed as point in state space
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n states r inputs m outputs n state equations m output equations Matrix form ) ( ) ( ) ( 1 1 1 t y y t u u t x x m r n ) ; ; ( ) ( ) ; ; ( ) ( 1 1 1 1 1 1 t u u x x f t x t u u x x f t x n n n n n n = = ) ; ; ( ) ( ) ; ; ( ) ( 1 1 1 1 1 1 t u u x x g t y t u u x x g t y n n m m n n = = ) , , ( ) ( ) , , ( ) ( t
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This note was uploaded on 07/27/2011 for the course MEM 230 taught by Professor Awerbuch during the Spring '08 term at Drexel.

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6. State space_upload - fig_05_02 fig_05_03 fig_05_05...

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