1 / 27

Realization of State Space Equations

Understand different methods to convert transfer functions to state space equations, including Frobenius, Observer, and Canonical forms, for various system configurations. Learn relevant algebraic equivalences and transformation.

spikes
Download Presentation

Realization of State Space Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related