Chapter-9-State-space-model-Student

Chapter-9-State-space-model-Student - ELEC 215: Tim Woo...

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Unformatted text preview: ELEC 215: Tim Woo Spring 2009/10 Chapter 9 - 1 Chapter 9: State-space Models Spring 2009/10 Lecture: Tim Woo ELEC 215: Tim Woo Spring 2009/10 Chapter 9 - 2 Laplace Transform Where we are Differential equations State-space model CTFT Hardware Implementation System Characteristics System Responses Closed-loop Systems Continuous-time z-Transform Difference equations State-space model DTFT Hardware Implementation System Characteristics System Responses Closed-loop Systems Discrete-time Mapping Done in 211 To be covered In progress Done Will be covered if available Open-loop Systems Open-loop Systems ELEC 215: Tim Woo Spring 2009/10 Chapter 9 - 3 Expected Outline • On successful completion of this chapter, you will be able to – Obtain state-space model with a set of first-order differential (or difference) equations describing n th order ordinary differential (or difference) equation. – Determine the state responses and output responses. – Obtain the system function from state-space model. ELEC 215: Tim Woo Spring 2009/10 Chapter 9 - 4 Introduction • In the preceding chapter we used the Laplace transform (or z- transform) to obtain system function models representing LTI system described by ordinary differential (difference) equations. • In this chapter, we turn to an alternative method of system modeling: State-space models • This model provides an alternative description to a Single-Input- Single-Output (SISO) system, but not limit to, Multi-Input-Multi- Output system. • In addition, we only consider the LTI system in the following discussion. ELEC 215: Tim Woo Spring 2009/10 Chapter 9 - 5 Introduction • Consider block diagram for continuous-time system System (State variables) ) ( ) ( ) ( 2 1 t x t x t x m M ) ( , ), ( ), ( 2 1 t v t v t v n L ) ( ) ( ) ( 2 1 t y t y t y k M Inputs Outputs • Definition: • The state variables { } of a system are a set of independent variables whose initial values of state variables at the initial time t , together with inputs { } for all t ≥ t , determine the outputs of the system { } for all t ≥ t . This is called Multi-Input-Multi-Output (MIMO) system. ) ( , ), ( ), ( 2 1 t v t v t v n L ) ( , ), ( ), ( 2 1 t x t x t x m L ) ( , ), ( ), ( 2 1 t y t y t y k L ELEC 215: Tim Woo Spring 2009/10 Chapter 9 - 6 State-space model for continuous-time systems • The state of a system is described by the set of first-order differential equations written in terms of the state variables { } and input signals { }. ) ( , ), ( ), ( 2 1 t v t v t v n L ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 1 11 2 1 2 1 1 22 21 1 12 11 2 1 t x t x b b b b t v t v t v a a a a a a a a a t v t v t v m nm n m n nn n n n n n M L M M L M L M L M M L L & M & & ) ( , ), ( ), ( 2 1 t x t x t x m L ⎥ ⎥ ⎥...
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This note was uploaded on 03/27/2011 for the course ELEC 215 taught by Professor Prof.kamtimwo during the Spring '11 term at HKUST.

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Chapter-9-State-space-model-Student - ELEC 215: Tim Woo...

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