week_1 State Variable Control, TF, Block Diagram

week_1 State Variable Control, TF, Block Diagram

1. ECE 582 State Variable Control

2. Mathematical Modeling of Dynamic Systems
•Outline
–Review of Classical Control Systems
•Transfer Function (3.2)
•Block Diagram and Mason’s Gain Formula(3.3)
–State Variable Modeling (3.4)
–Using Matlab(3.6)

3. Classical Control
•System Model
–Transfer function
•Obtained by converting a system’s differential equation
to a transfer function
•It relates a representation of the output to a
representation of the input
????????????????????????=
ℒ????????????????????????????????????????????????????????????????????????????????????(????????????)
ℒ????????????????????????????????????????????????????????????????????????(????????????)
=
????????????(????????????)
????????????(????????????)
•Rapidly provides stability and transient response
information
•Frequency-domain approach
–Limits: applied only to linear, time-invariant
systems

4. Modern Control Theory
•System Model: State Space Equation
–Boosted by the increasing computing capability of
digital computers
–It can be used to model and analyze
•Nonlinear (backlash, saturation),
•Time-varying (missiles with varying fuel levels)
•Multi-input multi-output systems (i.e. an airplane)
–What is it?
•N-thorder differential equation =>
N 1
st
-order differential equations

5. Review of Classical Control Systems
•A linear time-invariant system with input x(t)
and output y(t) is defined by

6. Transfer Function

7. Mason’s Gain Formula
????????????????????????=
∑????????????
????????????Δ
????????????
Δ
•Can be applied to signal flow graph or block diagram to
evaluate transfer functions
•Reference:
–Section 3-9 of Ogata (Page 104)
–Section 3-3 of Ogata (Page 58)

8. Block Diagram

9. Block Diagram (Cont.)

10. Example of Mason’s Formula
(a)Evaluate C(s)/R(s); (b) C_1(s)/R(s); (c) C(s)/C_1(s)

11. Example of Mason’s Formula (Cont.)
Note: Mason’s Formula can be applied regardless of what node is
output, as long as the input is an independent node (external source)

12. System Realization
•Construct a signal flow graph or block diagram from system transfer function
•Using integrators, amplifiers, and summing points
•Check with ECE455 note page 137.

13. Example
Given the 3 equations
(a)Solve for A, B, & C using determinants

14. Example (Cont.)
(a)Draw a signal flow graph with 1 as the input and A, B, C as outputs
(b)Using Mason’s gain formula solve for A/1, B/1, and C/1.