EE 422

EE 422

Exam II Objectives

At the point of the 2nd Exam in EE422, you should be able to perform the following objectives:

EXAM II: State Variables and Feedback Control

· Do all Exam I objectives (Lectures #1-8 and HWs #1-8)

· Obtain the State-Variable model for linear circuits and systems (Lectures #8-9 and HWs #8-9)

· Use the Laplace Transform to solve the State-Variable model (Lecture #10)

· Find transfer functions from state variable models (Lecture #10 and HW#10)

· Find block diagrams and state variable models from transfer functions using (Lecture #10 and HW#10):

- Parallel decomposition

- Cascade decomposition

- Phase variables

· Understand and determine eigenvalues and eigenvectors of a square matrix A.( Lecture #11 and HW#11)

· Understand the concepts of asymptotic stability, marginal stability and instability and determine the stability of state variable models ( Lecture #11 and HW#11)

· Form the matrices P and S and find A^k =PS^kP^-1 ( Lecture #11 and HW#11)

· Find the state transition matrix, e^At =Pe^sTP^-1 , and solve the zero-input state variable model ( Lecture #11 and HW#11)

· Use Laplace transforms to find the state transition matrix, e^At (Lecture #12 and HW#12)

· For an nxn matrix A, prove and apply the Cayley Hamilton theorem to find (Lecture #12 and HW#12):

- A^k =b₀A⁰+b₁A¹+...+b_n-1A^n-1

- e^At = c₀A⁰+c₁A¹+...+c_n-1A^n-1

· Use two methods to find, e^At , when A has repeated eigenvalues (Lecture #12 and HW#12)

· Use the convolution theorem of the Laplace transform to derive and apply the time domain solution to the state variable model (Lecture #13 and HW#13):

· Use the similarity transformation, x=Pz, to decouple a given state variable model (Lecture #13 and HW#13)

· From a decoupled state variable model, determine: (Lecture 13 and HW#13)

- the solution, x(t)=Pz(t)

- controllable and uncontrollable eigenvalues

- observable and unobservable eigenvalues

· Find a state feedback control, w=-K_pvx, to set the eigenvalues of a single input system in phase variable form( Lecture #14 and Lab 1, HW#16)

· Find the controllability matrix, M=[B AB ... A^n-1B] and find a similarity transformation, T_pv = MM_pv^-1 which will put a controllable state model into phase variable form (Lecture #14 and Lab 1, HW#16)

· Design a state feedback control, w= -K_pvT_pv^-1x, to force real systems to meet settling time specifications (Lecture #14 and Lab 1)

· Implement your state feedback control using op-Amps (Lecture #14 and Lab 1)