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Week_2.pdf State Variable Modeling, State Space Equations
- 1. ECE 582 StateVariable Control Week_2
- 2. Mathematical Modeling ofDynamic Systems • Outline – Review of Classical Control Systems • Transfer Function (3.2) • Block Diagram and Mason’s Gain Formula(3.3) – State Variable Modeling (3.4)
- 3. State Variable Modeling •State – The smallest set of variables, called state variables such that the knowledge of these variables at t = 𝑡𝑡0, together with the knowledge of input for 𝑡𝑡 < 𝑡𝑡0, determines the system behavior for 𝑡𝑡 ≥ 𝑡𝑡0 • State Variable – The variables that determine the state of the dynamic system,𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑛𝑛. The number of state variables is dependent on the order of the system, number of inputs and outputs. • State Vector – If n variables 𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑛𝑛 describe the system, then these 𝑛𝑛 variables are considered as the n-components of a vector 𝑥𝑥(𝑡𝑡). This vector is called the state vector 𝑥𝑥 𝑡𝑡 = 𝑥𝑥1 ⋮ 𝑥𝑥𝑛𝑛
- 4. More Definitions • StateSpace – The n-dimensional space whose coordinate axes consist of the 𝑥𝑥1 axis, 𝑥𝑥2 axis, …, 𝑥𝑥𝑛𝑛 axis is called a state space. • State Space equations – Consider multiple input multiple output system. The inputs are denoted by 𝑢𝑢1 𝑡𝑡 , 𝑢𝑢2 𝑡𝑡 , … , 𝑢𝑢𝑟𝑟(𝑡𝑡). The outputs are denoted by 𝑦𝑦1 𝑡𝑡 , … , 𝑦𝑦𝑚𝑚(𝑡𝑡). The state variables are denoted by 𝑥𝑥1 𝑡𝑡 , … , 𝑥𝑥𝑛𝑛(𝑡𝑡). – State equations: Derivatives of the state variables as function of state variables and inputs – Output equations: outputs as function of state variables and inputs
- 5. State Space Equation– general form • State equation • Output equation
- 6. State Space Equation– general form
- 7. State Space Equation– Time-invariant • Time-invariant systems Here A, B, C, D are constants. • Block Diagram:
- 8. Example of LinearSystems • Input: u(t) • Output: y(t) Matrix Form: ̇ 𝑥𝑥1 = 𝑥𝑥2 ̇ 𝑥𝑥2 = − 𝑘𝑘 𝑚𝑚 𝑥𝑥1 − 𝑏𝑏 𝑚𝑚 𝑥𝑥2 + 1 𝑚𝑚 𝑢𝑢 𝑦𝑦 = 𝑥𝑥1
- 9. Example of LinearSystems • Block Diagram: ̇ 𝑥𝑥1 = 𝑥𝑥2 ̇ 𝑥𝑥2 = − 𝑘𝑘 𝑚𝑚 𝑥𝑥1 − 𝑏𝑏 𝑚𝑚 𝑥𝑥2 + 1 𝑚𝑚 𝑢𝑢 𝑦𝑦 = 𝑥𝑥1
- 10. Example of LinearSystems • Input: v(t) • Output: i(t) 𝑦𝑦 = 𝑥𝑥1
- 11. Example of LinearSystems • Matrix Form • Block Diagram
- 12. Multi-Input Multi-Output Systems •The procedure to obtain state variable equations is not unique • The state variable equations are not unique • In previous example • We may also choose 𝑥𝑥1 𝑡𝑡 = 1 𝐿𝐿𝐿𝐿 𝑖𝑖 𝑡𝑡 , 𝑥𝑥2 𝑡𝑡 = 𝑅𝑅 𝐿𝐿 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 Or 𝑥𝑥1 𝑡𝑡 = 1 𝐿𝐿𝐿𝐿 𝑖𝑖 𝑡𝑡 + 𝑅𝑅 𝐿𝐿 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 , 𝑥𝑥2 = 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
- 13. Multi-Input Multi-Output Systems •More generally, a system may consists of multiple input multiple output.
- 14. Multi-Input Multi-Output Systems(Cont.) • Example: write the differential equations for the system shown below and then represent them in state variable form
- 15. Multi-Input Multi-Output Systems(Cont.) ̈ 𝑦𝑦1 + 2 ̇ 𝑦𝑦1 + 3 𝑦𝑦1 − 𝑦𝑦2 = 𝑢𝑢1 4 ̈ 𝑦𝑦2 + 5 ̇ 𝑦𝑦1 + 3 𝑦𝑦2 − 𝑦𝑦1 = 𝑢𝑢2 Define 𝑥𝑥1 = 𝑦𝑦1, 𝑥𝑥2 = ̇ 𝑦𝑦1, 𝑥𝑥3 = 𝑦𝑦2, 𝑥𝑥4 = ̇ 𝑦𝑦2 Then, ̇ 𝑥𝑥1 = 𝑥𝑥2; ̇ 𝑥𝑥3 = 𝑥𝑥4 ̇ 𝑥𝑥2 = −3𝑥𝑥1 − 2𝑥𝑥2 + 3𝑥𝑥3 + 𝑢𝑢1 ̇ 𝑥𝑥4 = 3 4 𝑥𝑥1 − 3 4 𝑥𝑥3 − 5 4 𝑥𝑥4 + 1 4 𝑢𝑢2 We can now write the above equations in matrix form: ̇ 𝑥𝑥1 ̇ 𝑥𝑥2 ̇ 𝑥𝑥3 ̇ 𝑥𝑥4 = 0 1 0 0 −3 −2 3 0 0 0 0 1 3/4 0 −3/4 −5/4 𝑥𝑥1 𝑥𝑥2 𝑥𝑥3 𝑥𝑥4 + 0 0 1 0 0 0 0 1/4 𝑢𝑢1(𝑡𝑡) 𝑢𝑢2(𝑡𝑡)
- 16. Transfer Function toState Variable Models • Consider the following n-th order differential equation in which the forcing function does not involve derivative terms 𝑑𝑑𝑛𝑛𝑦𝑦 𝑑𝑑𝑡𝑡𝑛𝑛 + 𝑎𝑎𝑛𝑛−1 𝑑𝑑𝑛𝑛−1𝑦𝑦 𝑑𝑑𝑡𝑡𝑛𝑛−1 + ⋯ + 𝑎𝑎1 𝑑𝑑𝑦𝑦 𝑑𝑑𝑡𝑡 + 𝑎𝑎0𝑦𝑦 = 𝑏𝑏0𝑢𝑢 𝑌𝑌 𝑠𝑠 𝑈𝑈 𝑠𝑠 = 𝑏𝑏0 𝑠𝑠𝑛𝑛 + 𝑎𝑎𝑛𝑛−1𝑠𝑠𝑛𝑛−1 + ⋯ + 𝑎𝑎1𝑠𝑠 + 𝑎𝑎0 • Choosing the state variables
- 17. Transfer Function toState Variable Model (Cont.) • Matrix Form • Block Diagram
- 18. Transfer Function toState Variable Model(Cont.) • Consider the following n-th order differential equation in which the forcing function involves derivative term • If we choose the state variables the same way as in the previous case, the last state equation will contain derivatives of the input. 𝑑𝑑𝑛𝑛𝑦𝑦 𝑑𝑑𝑡𝑡𝑛𝑛 + 𝑎𝑎𝑛𝑛−1 𝑑𝑑𝑛𝑛−1𝑦𝑦 𝑑𝑑𝑡𝑡𝑛𝑛−1 + ⋯ + 𝑎𝑎1 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 + 𝑎𝑎0𝑦𝑦 = 𝑏𝑏𝑛𝑛 𝑑𝑑𝑛𝑛𝑢𝑢 𝑑𝑑𝑡𝑡𝑛𝑛 + ⋯ + 𝑏𝑏1 𝑑𝑑𝑢𝑢 𝑑𝑑𝑑𝑑 + 𝑏𝑏0𝑢𝑢 𝑌𝑌 𝑠𝑠 𝑈𝑈 𝑠𝑠 = 𝑏𝑏𝑛𝑛𝑠𝑠𝑛𝑛 + ⋯ + 𝑏𝑏1𝑠𝑠 + 𝑏𝑏0 𝑠𝑠𝑛𝑛 + 𝑎𝑎𝑛𝑛−1𝑠𝑠𝑛𝑛−1 + ⋯ + 𝑎𝑎1𝑠𝑠 + 𝑎𝑎0
- 19. Transfer Function toState Variable Model(Cont.) • A general approach: 1) Realize system in block diagram or signal flow graph 2) Name the integrator output as state variables 3) Write the equations for inputs of integrators • Show the method on an example of order 3
- 20. Transfer Function toState Variable Model(Cont.) • Realization 1
- 21. Transfer Function toState Variable Model(Cont.) • Realization 2
- 22. Transfer Function toState Variable Model(Cont.) • Another Special Case (diagonal realization) – Transfer function has a partial fraction decomposition of the form where are distinct real numbers. One of these terms: block diagram :