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03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
03 elec3114
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03 elec3114

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  • 1. 1 Modeling in the Time Domain• How to find a mathematical model, called a state-space representation, for a linear, time-invariant system• How to convert between transfer function and state-space models• How to linearize a state-space representation Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 2. 2 IntroductionClassical (frequency-domain) technique:• Modeling of systems using linear, time-invariant differential equations and subsequent transfer functionsState-space (time-domain) technique:• More powerful tool than classical techniques• Can be used to represent nonlinear systems, systems with nonzero initial conditions, time-varying systems, multiple-input multiple- output systems Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 3. 3 State-Space Approach1. We select a particular subset of all possible system variables and call the variables in this subset state variables.2. For an nth-order system, we write n simultaneous, first-order differential equations in terms of the state variables. We call this system of simultaneous differential equations state equations.3. If we know the initial condition of all of the state variables at t0 as well as the system input for t≥t0, we can solve the simultaneous differential equations for the state variables for t≥t0.4. We algebraically combine the state variables with the systems input and find all of the other system variables for t≥t0. We call this algebraic equation the output equation.5. 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. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 4. 4Example:We select i(t) and q(t), as the two state variables. We transform this equation into 2 first order equations in terms of i(t) and q(t) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 5. 5From these two state variables, we can solve for all other network variables. … output equation state-space representation. !!! State variables must be linearly independent which means that no state variable can be chosen if it can be expressed as a linear combination of the other state variables. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 6. 6The state equationscan be written in matrix form as where The output equation can be written in matrix form as where Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 7. 7 The General State-Space RepresentationA system is represented in state space by the following equations: • x = Ax + Bu y = Cx + Du Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 8. 8Example of a second order system: • x = Ax + Bu y = Cx + Du Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 9. 9 Applying the State-Space RepresentationState-vector selection: 1. A minimum number of state variables must be selected as components of the state vector. This minimum number of state variables is sufficient to describe completely the state of the system. (Note: Typically, the minimum number equals the order of the differential equation describing the system or to the number of independent energy-storage elements in the system) 2. The components of the state vector must be linearly independent. (Note: Variables related by derivative are linearly independent) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 10. 10Problem Given the electrical network, find a state-space representationif the output is the current through the resistor. Solution Select the state variables by writing the derivative equation for all energy- storage elements, that is, the inductor and the capacitor next step is to express ic and vL as linear combinations of the state variables vc and iL, and the input, v(t). Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 11. 11 State variablesAfter substitution we obtain: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 12. 12The output equation: Final state-space representation: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 13. 13Problem Find the state equations for the translational mechanical systemSolution• It is more convenient when working with mechanical systems to obtain the state equations directly from the equations of motion rather than from the energy-storage elements.• State variables will be the position and velocity of each point of linearly independent motion. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 14. 14We select x1, x2, v1, and v2 as state variables dx1 / dt = v1 dx2 / dt = v2 dv1 / dt = d 2 x1 / dt 2 dv2 / dt = d 2 x2 / dt 2State equations: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 15. 15Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 16. 16 Converting a Transfer Function to State Space• We will show how to convert a transfer function to a state-space representation• First. we select a set of state variables, called phase variables, where each subsequent state variable is defined to be the derivative of the previous state variable. Consider: - we choose output y(t) and its (n-1) derivatives as the state variables Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 17. 17Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 18. 18Converting a transfer function with constant term in numerator Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 19. 19Converting a transfer function with polynomial in numerator Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 20. 20Example: Converting a transfer function with polynomial in numerator Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 21. 21Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 22. 22 Converting from State Space to a Transfer FunctionTaking the Laplace transform of :Taking the Laplace transform : Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 23. 23 Linearization• state –space representation can be used to represent systems with nonlinearities• state –space representation can be linearized for small perturbations about an equilibrium point Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 24. 24Example Let Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 25. 25In order to find the transfer function we must linearize the nonlinear state-spacerepresentation Let equilibrium point x1 = 0, x2 = 0 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.

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