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Introduction to State Variable Modeling

Learn about state variable models, differential equations, and simulating system responses using MATLAB. Explore the advantages of state variable models and input-output equations.

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Introduction to State Variable Modeling

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  1. Lecture 24 Introduction to state variable modeling Overall idea Example Simulating system response using MATLAB Related educational modules: Section 2.6.1, 2.6.2

  2. Overview of our approach to date • We wrote one differential equation in one unknown • This is called an input-output equation • The equation relates the output directly to the input • This approach is useful only for low-order systems • We applied the approach to first and second order systems • Allows us to obtain insight into the overall system behavior relative to input-output equation coefficients • We want to simulate higher-order systems

  3. Notes on generating input-output equations • I defined unknowns as voltages across capacitors and currents through inductors • Used KVL and KCL to write multiple equations in these unknowns • Combined them into a single equation in one unknown • Why did I recommend this approach?

  4. State-variable models • State-variable models consist of N first-order differential equations in N unknowns • Mathematically equivalent to a single Nth order equation in one unknown • The unknowns are the state variables • The state variables are the minimum amount of information necessary to completely describe the system’s state

  5. Definition of system state • The state of a system is the information necessary to completely characterize the system at some instant in time • Given the state at some time, and the input at all subsequent times, we can determine the output at any subsequent time • Comparable to initial conditions • The state uniquely determines the energy in all the system’s energy storage elements

  6. Choice of state variables • The choice of state variables is not unique • We will choose as our state variables: • Voltages across capacitors • Currents through inductors • As we have seen, these variables will describe the energy stored in all energy storage elements • They are a legitimate choice for state variables

  7. Form of the state variable model • State equations are in the form: • are the state variables and u is the input • “dot” notation denotes differentiation with respect to time

  8. Output equations • Any other circuit parameter can be determined from the states (this relation is called the output equation): • The previous equations can be written in matrix form:

  9. Matrix form – continued

  10. Example 1 • Determine state equations governing the circuit below

  11. Example 1 continuedMatrix form of state equations

  12. Example 1 (still continued)Determine output equation

  13. Simulating system response using MATLAB • Analytical solution of higher-order differential equations is generally not feasible • Typically, we resort to numerical (or computer-generated) solutions • Numerical simulation of the system response • MATLAB is a common tool for solving differential equations numerically • Contains a number of built-in simulation capabilities

  14. Prerequisite MATLAB background • It is assumed that you have access to: • MATLAB • MATLAB’s control systems toolbox • It is also assumed that you can use MATLAB to: • Set up arrays • Perform basic matrix operations • Perform basic plotting operations

  15. Natural and step responses using MATLAB • MATLAB’s control systems toolbox provides special-purpose commands to determine the natural and step responses of a given system • System is (for us) defined by a state variable model {A,b,c,d} • MATLAB commands and syntax: • step(A,b,c,d) plots the system’s unit step response • initial(A,b,c,d,x0) plots the system’s natural response to the initial conditions provided in the vector x0

  16. Example 1 – simulated step response • Use MATLAB to simulate the current through the 0.1H inductor • Choosing states as before results in previous state variable model • Output equation:

  17. MATLAB syntax • Type at MATLAB’s command prompt (>>): >> R = 100; L1 = 0.1; L2 = 0.5; C = 1e-3; >> A = [-R/L1 R/L1 -1/L1; R/L2 -R/L2 0; 1/C 0 0]; >> b = [1/L1; 0; 0]; >> c = [1 0 0]; >> d = 0; >> step(A,b,c,d)

  18. Simulated Response

  19. State variable models and input-output equations • Advantages of state variable models: • Fairly easy to write governing equations for higher-order systems • State variable models easy to solve numerically • Changing output does not affect state equations! • {A,b,c,d} can give direct information about system behavior • Advantages of input-output equations • Higher-order system responses are often interpreted in terms of first and second order system responses • Laplace transform based techniques often rely on input-output representation of a system

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