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Equivalent State Equations

Chapter 4. State Space Solutions and Realizations. Equivalent State Equations. State variables: : inductor current i L : capacitor voltage v C. Chapter 4. State Space Solutions and Realizations. Homework 2: Equivalent State Equations.

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Equivalent State Equations

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  1. Chapter 4 State Space Solutions and Realizations Equivalent State Equations State variables: • : inductor currentiL • : capacitor voltagevC

  2. Chapter 4 State Space Solutions and Realizations Homework 2: Equivalent State Equations 1. Prove that for the same system, with different definition of state variables, we can obtain a state space in the form of: State variables: • : current of leftloop • : current of rightloop

  3. Chapter 4 State Space Solutions and Realizations Homework 2: Equivalent State Equations State variables: • : loop current left • : loop current right

  4. Chapter 4 State Space Solutions and Realizations Homework 2: Equivalent State Equations 2. Derive a state-space description for the following diagram

  5. Chapter 4 State Space Solutions and Realizations Homework 2: Equivalent State Equations

  6. Chapter 4 State Space Solutions and Realizations Equivalent State Equations • Consider an n-dimensional state space equations: • Let P be an nn real nonsingular matrix, and let x = Px. Then, the state space equations ~ where is said to be algebraically equivalent with the original state space equations. ~ • x = Px is called an equivalence transformation.

  7. Chapter 4 State Space Solutions and Realizations Equivalent State Equations • Proof: Substituting

  8. Chapter 4 State Space Solutions and Realizations Equivalent State Equations • From the last electrical circuit, State variables: • : inductor currentiL • : capacitor voltagevC State variables: • : loop current left • : loop current right • The two sets of states can be related in the way: or

  9. Chapter 4 State Space Solutions and Realizations Transfer Function and Transfer Matrix • Consider a state space equations for SISO systems: • Using Laplace transform, we will obtain: • For zero initial conditions, x(0) = 0, Transfer Function

  10. Chapter 4 State Space Solutions and Realizations Realization of State Space Equations • Every linear time-invariant system can be described by the input-output description in the form of: • If the system is lumped (i.e., having concentrated parameters), it can also be described by the state space equations • The problem concerning how to describe a system in state space equations, provided that the transfer function of a system, G(s), is available, is called Realization Problem. G(s) A,B,C,D.

  11. Chapter 4 State Space Solutions and Realizations Realization of State Space Equations • Three realization methods will be discussed now: • Frobenius Form • Observer Form • Canonical Form

  12. Chapter 4 Realization of State Space Equations Frobenius Form • Special Case: No derivation of input

  13. Chapter 4 Realization of State Space Equations Frobenius Form • We now define: Frobenius Form,Special Case

  14. Chapter 4 Realization of State Space Equations Frobenius Form • General Case: With derivation of input • If m = n–1 (largest possible value), then

  15. Chapter 4 Realization of State Space Equations Frobenius Form • If m = n–1 (largest possible value), then • But

  16. Chapter 4 Realization of State Space Equations Frobenius Form • The state space equations can now be written as: Frobenius Form,General Case

  17. Chapter 4 Realization of State Space Equations Observer Form

  18. Chapter 4 Realization of State Space Equations Observer Form

  19. Chapter 4 Realization of State Space Equations Observer Form • The state space equations in observer form: Observer Form

  20. Chapter 4 Realization of State Space Equations Canonical Form • To construct state space equations in canonical form, we need to perform partial fraction decomposition to the respective transfer function. • In case all poles are distinct, we define:

  21. Chapter 4 Realization of State Space Equations Canonical Form • The state space equations in case all poles are distinct: Canonical Form,Distinct Poles • The resulting matrix A is a diagonal matrix. • The ODEs are decoupled, each of them can be solved independently.

  22. Chapter 4 Realization of State Space Equations Canonical Form • In case of repeating poles, for example λ1 is repeated for p times, the decomposed equation will be: • We define: • x1(t) coupled with x2(t) • xp–1(t) coupled with xp(t)

  23. Chapter 4 Realization of State Space Equations Canonical Form

  24. Chapter 4 Realization of State Space Equations Canonical Form • The state space equations in case of repeating poles: Canonical Form,Repeating Poles

  25. Chapter 4 Realization of State Space Equations Canonical Form • The state space equations in case of repeating poles: Canonical Form,Repeating Poles

  26. Chapter 4 Realization of State Space Equations Homework 3: Transfer Function to State Space • Find the state-space realizations of the following transfer function in Frobenius Form, Observer Form, and Canonical Form. • Hint: Learn the following functions in Matlab and use the to solve this problem: roots, residue, conv.

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