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SYSTEM DYNAMICS & CONTROL CHAPTER STATE SPACE MODELING Dr Vo Tuong Quan HCMUT - 2011 State Space Modeling 2011 – Vo Tuong Quan State Space Modeling 2011 – Vo Tuong Quan State Space Modeling 2011 – Vo Tuong Quan State Space Modeling 2011 – Vo Tuong Quan State Space Modeling 2011 – Vo Tuong Quan State Space Modeling Block diagram for the system 2011 – Vo Tuong Quan State Space Modeling Find the state equations for the translational mechanical system shown in Figure 2011 – Vo Tuong Quan State Space Modeling 2011 – Vo Tuong Quan State Space Modeling In Vector-matrix from 10 2011 – Vo Tuong Quan State Space Modeling State space equation in controllable canonical form 25 2011 – Vo Tuong Quan State Space Modeling State space equation in controllable canonical form 26 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions 27 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions 28 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions 29 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions Example 2: Find the state-space representation in phase-variable form for the transfer function shown in Figure below: 30 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions Solution: Step 1: Find the associated differential equation C s 24 R s s s 26 s 24 Cross-multiplying yields ( s s 26 s 24 )C s 24 Rs The corresponding differential equation is found by taking the inverse Laplace transform, assuming zero initial conditions 31 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions Select the state variables: x1 x2 The combined state and output equations are x x3 x3 24 x1 26 x2 x3 24 r y c x1 Vector-matrix form: 32 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions 33 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions 34 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions Example 4: Find the state-space equation and output equation for the system defined by Y (s) 2s s s U ( s ) s s 5s 35 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions Solution: From the given transfer function, the differential equation for the system is Comparing this equation with the standard equation given We find: 36 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions Example 5: Find the state-space model of the system shown in Figure below: 37 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions Solution: We obtain Which can be written as 38 2011 – Vo Tuong Quan State Space Modeling Convert from state equations to transfer functions By taking the inverse Laplace transforms of the previous four equations, we obtain: x1 5 x1 10 x2 x2 x3 u x3 x1 x3 y x1 A state-space model of the system in the standard form is given by 39 2011 – Vo Tuong Quan .. .State Space Modeling 2011 – Vo Tuong Quan State Space Modeling 2011 – Vo Tuong Quan State Space Modeling 2011 – Vo Tuong Quan State Space Modeling 2011 – Vo Tuong Quan State Space Modeling. .. Quan State Space Modeling 2011 – Vo Tuong Quan State Space Modeling In Vector-matrix from 10 2011 – Vo Tuong Quan State Space Modeling 11 2011 – Vo Tuong Quan State Space Modeling The state- space. .. Quan State Space Modeling State space equation in controllable canonical form Establish the state equations describing the system below: 24 2011 – Vo Tuong Quan State Space Modeling State space
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