Week_2.pdf State Variable Modeling, State Space Equations

Week_2.pdf State Variable Modeling, State Space Equations

1. ECE 582 State Variable C ontrol
Week_2

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)

3. State Variable Modeling
•State
–The smallest set of variables, called state variables such that the
knowledge of these variables at t=????????????
0, together with the knowledge
of input for ????????????<????????????
0, determines the system behavior for ????????????≥????????????
0
•State Variable
–The variables that determine the state of the dynamic
system,????????????
1,????????????
2,…,????????????
????????????. The number of state variables is dependent on
the order of the system, number of inputs and outputs.
•State Vector
–If n variables ????????????
1,????????????
2,…,????????????
????????????describe the system, then these ????????????variables
are considered as the n-components of a vector ????????????(????????????). This vector is
called the state vector
????????????????????????=
????????????
1
⋮
????????????
????????????

4. More Definitions
•State Space
–The n-dimensional space whose coordinate axes consist of the ????????????
1axis,
????????????
2axis, …, ????????????
????????????axis is called a state space.
•State Space equations
–
Consider multiple input multiple output system. The inputs are denoted
by????????????
1
????????????,????????????
2????????????,…,????????????
????????????(????????????). The outputs are denoted by ????????????
1????????????,…,????????????
????????????(????????????).
The state variables are denoted by ????????????
1
????????????,…,????????????
????????????(????????????).
–State equations: Derivatives of the state variables as function of state
variables and inputs
–Output equations: outputs as function of state variables and inputs

5. State Space Equation – general form
•State equation
•Output equation

6. State Space Equation – general form

7. State Space Equation – Time-invariant
•Time-invariant systems
Here A, B, C, D are constants.
•Block Diagram:

8. Example of Linear Systems
•Input: u(t)
•Output: y(t)
Matrix Form:
̇????????????
1=????????????
2
̇????????????
2=−
????????????
????????????
????????????
1−
????????????
????????????
????????????
2+
1
????????????
????????????
????????????=????????????
1

9. Example of Linear Systems
•Block Diagram:
̇????????????
1=????????????
2
̇????????????
2=−
????????????
????????????
????????????
1−
????????????
????????????
????????????
2+
1
????????????
????????????
????????????=????????????
1

10. Example of Linear Systems
•Input: v(t)
•Output: i(t)
????????????=????????????
1

11. Example of Linear Systems
•Matrix Form
•Block Diagram

12. Multi-Input Multi-Output Systems
•The procedure to obtain state variable equations is not unique
•The state variable equations are not unique
•In previous example
•We may also choose
????????????
1????????????=
1
????????????????????????
????????????????????????,????????????
2????????????=
????????????
????????????
????????????????????????
????????????????????????
Or
????????????
1????????????=
1
????????????????????????
????????????
????????????+
????????????
????????????
????????????????????????
????????????????????????
,????????????
2=
????????????????????????
????????????????????????

13. Multi-Input Multi-Output Systems
•More generally, a system may consists of multiple input multiple
output.

14. Multi-Input Multi-Output Systems (Cont.)
•Example: write the differential equations for
the system shown below and then represent
them in state variable form

15. Multi-Input Multi-Output Systems (Cont.)
̈????????????
1+2̇????????????
1+3????????????
1−????????????
2=????????????
1
4̈????????????
2+5̇????????????
1+3????????????
2−????????????
1=????????????
2
Define
????????????
1=????????????
1,????????????
2=̇????????????
1,????????????
3=????????????
2,????????????
4=̇????????????
2
Then,
̇????????????
1=????????????
2;̇????????????
3=????????????
4
̇????????????
2=−3????????????
1−2????????????
2+3????????????
3+????????????
1
̇????????????
4=
3
4
????????????
1−
3
4
????????????
3−
5
4
????????????
4+
1
4
????????????
2
We can now write the above equations in matrix form:
̇????????????
1
̇????????????
2
̇????????????
3
̇????????????
4
=
01 0 0
−3−2 3 0
00 0 1
3/40−3/4−5/4
????????????
1
????????????
2
????????????
3
????????????
4
+
00
10
00
01/4
????????????
1(????????????)
????????????
2(????????????)

16. Transfer Function to State Variable Models
•Consider the following n-thorder differential equation in
which the forcing function does not involve derivative terms
????????????
????????????
????????????
????????????????????????
????????????
+????????????
????????????−1
????????????
????????????−1
????????????
????????????????????????
????????????−1
+⋯+????????????
1
????????????????????????
????????????????????????
+????????????
0????????????=????????????
0????????????
????????????????????????
????????????????????????
=
????????????
0
????????????
????????????
+????????????
????????????−1????????????
????????????−1
+⋯+????????????
1????????????+????????????
0
•Choosing the state variables

17. Transfer Function to State Variable Model (Cont.)
•Matrix Form
•Block Diagram

18. Transfer Function to State Variable Model(Cont.)
•Consider the following n-thorder differential equation in
which the forcing function involves derivative term
•If we choose the state variables the same way as in the
previous case, the last state equation will contain derivatives
of the input.
????????????
????????????
????????????
????????????????????????
????????????
+????????????
????????????−1
????????????
????????????−1
????????????
????????????????????????
????????????−1
+⋯+????????????
1
????????????????????????
????????????????????????
+????????????
0????????????
=????????????
????????????
????????????
????????????
????????????
????????????????????????
????????????
+⋯+????????????
1
????????????????????????
????????????????????????
+????????????
0????????????
????????????????????????
????????????????????????
=
????????????
????????????????????????
????????????
+⋯+????????????
1????????????+????????????
0
????????????
????????????
+????????????
????????????−1????????????
????????????−1
+⋯+????????????
1????????????+????????????
0

19. Transfer Function to State Variable Model(Cont.)
•A general approach:
1)Realize system in block diagram or signal flow graph
2)Name the integrator output as state variables
3)Write the equations for inputs of integrators
•Show the method on an example of order 3

20. Transfer Function to State Variable Model(Cont.)
•Realization 1

21. Transfer Function to State Variable Model(Cont.)
•Realization 2

22. Transfer Function to State Variable Model(Cont.)
•Another Special Case (diagonal realization)
–Transfer function has a partial fraction decomposition of the form
where are distinct real numbers.
One of these terms: block diagram :