1. Chapter 11 Digital Control Systems
Topics to be covered:
- Discrete-time systems
- z-Transforms
- State Equation models of discrete systems
(continuous  differential equations)
(discrete  difference equations)

2. Digital Control Systems
Simulation Diagrams and Flow Graphs:
(How to make an electrical simulation of a system.)
For a discrete-time simulation, the main operation is the delay operator. (Recall that for a continuous-time system, we work with integrators. A delay operator is simpler, and, as a result, cheaper.)
We can also represent the delay as a z-1 operator.
ECE 4923/ Chapter 11 Part 2

3. Digital Control Systems
Example:
Draw a simulation diagram for the difference equation:
m(k) = e(k) – e(k-1) – m(k-1)
Components required:
2 delay elements and 1 summer
(This is a special purpose computer, which can only solve the given equation.)
Alternate representation (in SFG form):
Note that the z-1 and the
-1 paths can be combined in a “self-loop”
ECE 4923/ Chapter 11 Part 2

4. Digital Control Systems
Consider a general nth order difference equation of the following form:
By using the real translation theorem, we get:
This leads to the transfer function:
The following two pages show a simulation diagram and a flow graph for this system.
ECE 4923/ Chapter 11 Part 2

5. Digital Control Systems
For a general nth order difference equation, we can draw the following simulation diagram:
ECE 4923/ Chapter 11 Part 2

6. Digital Control Systems
As we have seen before, the SFG form is generally much more compact and is easier to draw...
ECE 4923/ Chapter 11 Part 2

7. Digital Control Systems
State Variables - Consider the following TF and the corresponding simulation diagram and SFG:
ECE 4923/ Chapter 11 Part 2

8. Digital Control Systems
Alternative
representations:
ECE 4923/ Chapter 11 Part 2

9. Digital Control Systems
State Variables – Let the outputs of each delay be a state, and we can write state variable equations as:
and
(Phase Variable form of the State Equations) xn
xn-1 x1
ECE 4923/ Chapter 11 Part 2

10. Digital Control Systems
Note that these state equations can be more compactly written as:
(Note that these give a linear, time-invariant discrete system description.)
A somewhat more general form is the linear time-varying form of the state equations:
ECE 4923/ Chapter 11 Part 2

11. Digital Control Systems
Example: Write State Equations for a system with the TF
(Technique: generate a SFG, label outputs of delays as states, and then write equations.)
(see board for details…)
ECE 4923/ Chapter 11 Part 2

12. Digital Control Systems
Solution of State Equations:
Plug in each value of k:
x(1) = A x(0) + B u(0)
x(2) = A x(1) + B u(1) = A2 x(0) + AB u(0) + B u(1)
x(3) = A3 x(0) + A2B u(0) + AB u(1) + B u(2)
. . .
x(n) = An x(0) + An-1B u(0) + An-2B u(1)
+ AB u(n-2) + B u(n-1)
So,
ECE 4923/ Chapter 11 Part 2

13. Digital Control Systems
Solution of State Equations:
By z-Transforms:
x(n+1) = A x(n) + B u(n)
The term multiplying the initial condition is the z-transform of the state transition matrix and
ECE 4923/ Chapter 11 Part 2

14. Digital Control Systems
Example:
Draw a simulation diagram:
And assign states
(from right to left):
ECE 4923/ Chapter 11 Part 2

15. Digital Control Systems
Example, continued
With states assigned, we can then write out the specific state equations:
In matrix form, they
end up as:
ECE 4923/ Chapter 11 Part 2

16. Digital Control Systems
Example, continued
To solve, we can use
the sequential technique:
(assume that x(0)=0 and u(k)=1, for all k.
ECE 4923/ Chapter 11 Part 2

17. Digital Control Systems
Example, continued
. . .or, by z-transforms:
Recall that:
So, with x(0)=0, we get
and
Since we get
ECE 4923/ Chapter 11 Part 2

18. Digital Control Systems
Transfer Functions (TF’s) and State Variables
For the SISO case, we have
So, with zero IC’s (since we are looking at TF’s):
and
So, we end up with
ECE 4923/ Chapter 11 Part 2

19. Digital Control Systems
Example: Using a previous example setup, we get Since D = 0, the transfer function is:
ECE 4923/ Chapter 11 Part 2

20. Digital Control Systems
Note: If you don’t really like computing matrix inverses, you can alternatively use the following: This method allows us to compute the TF without the need to determine a matrix inverse. For our previous example,
ECE 4923/ Chapter 11 Part 2

21. Digital Control Systems
Finally, we can use MATLAB to automate much of the work: Example: num=[ ]; den=[ ]; printsys(num,den,’z’) [A,B,C,D]=tf2ss(num,den) Results: Then, we can use: dstep(A,B,C,D) (MATLAB chooses # of points) dimpulse(A,B,C,D) dstep(A,B,C,D,IU,50) (we specify 50 points) dimpulse(A,B,C,D,IU,50) (IU is # of input, only one in this case)
ECE 4923/ Chapter 11 Part 2

22. Digital Control Systems
More MATLAB: Still using: But note that if we specify: which is different from before, we find that: [num,den]=ss2tf(A,B,C,D) gives num=[ ] den = [ ] which is the SAME TF as before! What gives???? Answer: These are two different forms of the state equations, but are equivalent. Note that there are an infinite number of valid state equations for any given system.
ECE 4923/ Chapter 11 Part 2