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# State equations for physical systems

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### State equations for physical systems

1. 1. STATE EQUATIONS FOR PHYSICAL SYSTEMS
2. 2. State-Space - referred to as the modern, or time-domain approach. - unified method for modelling, analyzing, and designing a wide range of system. - represents nonlinear (e.g. backlash, saturation, and dead zone. - represents time-varying systems (e.g. Missiles with varying fluid levels or lift in an aircraft flying through a wide range of altitudes.
3. 3. State-Space1. Select a particular subset of all possible system variables and call the variables in this subset state variable.
4. 4. State-Space2) For an nth-order system, write n simultaneous, first order differential equations in terms of the state variables. We call this system of simultaneous differential state equations.
5. 5. State-Space3) If we know all the initial condition of all of the state variables at to as well as the system input for t≥to ,we solve the simultaneous differential equations for the state variables for t≥to
6. 6. State-Space4) We algebraically combine the state variables with the systems input and find all of the other system variables for t≥to. We call this algebraic equation the output equation.
7. 7. State-Space5) We consider the state equations and the output equations a viable representation of the system. We call this representation of the system a state-space representation
8. 8. The General State-Space RepresentationLinear CombinationA linear combination of n variables, xi, for i=1 to n, is given by the following sum:S=Knxn + Kn-1xn-1 + ... +K1x1 NOTE: where K is constant
9. 9. The General State-Space RepresentationLinear IndependenceA set of variables is said to be linearly independent, if none of the variables can be written as a linear combination of the others.
10. 10. The General State-Space RepresentationLinear CombinationA linear combination of n variables, xi, for i=1 to n, is given by the following sum:S=Knxn + Kn-1xn-1 + ... +K1x1 NOTE: where K is constant
11. 11. The General State-Space RepresentationSystem variableAny variable that responds to an input or initial conditions in a system
12. 12. The General State-Space RepresentationState VariablesThe smallest set of linearly independent system variables such that the values of the members of the set at time to along with known forcing functions completely determine the value of all system variables for all t≥to
13. 13. The General State-Space RepresentationState SpaceThe n-dimension space whose axes are the state variables.(e.g. vR and vC)
14. 14. The General State-Space RepresentationState EquationA set of n simultaneous, first order differential equations with n variables, where n variables to be solved are the state variables.
15. 15. The General State-Space RepresentationOutput EquationThe algebraic equations that expresses the output variables of a system as linear combinations of the state variables and the inputs.
16. 16. The General State-SpaceRepresentation ẋ=Ax + Bu --- STATE EQUATION y=Cx + Du ---OUTPUT EQUATION for t≥to and initial conditions x(to), where x=state vector ẋ=derivative of the state vector with respect to time y=output vector u=input or control vector A=system matrix B=input matrix C=output matrix D= feedforward matrix
17. 17. Representing an electrical NetworkProblemGiven the electrical network, find a state-space representation if the output is the current through the resistor. Lv(t) R C
18. 18. Representing an electrical NetworkStep 1Label all the branches currents in the network. L iL(t)v(t) iR(t) R iC(t) C
19. 19. Representing an electrical NetworkStep 2Select the state variables by writing the derivative equation forall energy storage elements, that is, the inductor and thecapacitor.
20. 20. Representing an electrical NetworkStep 3Apply network theory, such as Kirchhoffs voltage and currentlaws
21. 21. Representing an electrical NetworkStep 4Substitute the results of Eqs. (3.24) and (3.25) into Eqs.(3.22) and (3.23) to obtain the following state equations:
22. 22. Representing an electrical NetworkStep 5Find the output equation.
23. 23. Representing a TranslationalMechanical System
24. 24. Representing a TranslationalMechanical System
25. 25. Representing a TranslationalMechanical System
26. 26. Representing a TranslationalMechanical System
27. 27. Converting a Transfer Function toState Space Converting a Transfer Function with Constant Term in Numerator
28. 28. Converting a Transfer Function toState Space Converting a Transfer Function with Constant Term in Numerator
29. 29. Converting a Transfer Function toState Space Converting a Transfer Function with Constant Term in Numerator
30. 30. Converting a Transfer Function toState Space Converting a Transfer Function with Constant Term in Numerator
31. 31. Converting a Transfer Function toState Space Converting a Transfer Function with Polynomial in Numerator
32. 32. Converting a Transfer Function toState Space Converting a Transfer Function with Polynomial in Numerator
33. 33. Converting a Transfer Function toState Space Converting a Transfer Function with Polynomial in Numerator
34. 34. Converting a Transfer Function toState Space Converting a Transfer Function with Polynomial in Numerator
35. 35. State-Space Representation toTransfer Function
36. 36. State-Space Representation toTransfer Function
37. 37. State-Space Representation toTransfer Function
38. 38. State-Space Representation toTransfer Function
39. 39. Linearization
40. 40. Linearization
41. 41. Linearization
42. 42. Linearization