# System Transfer Functions

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```					Review of Feedback Control                     JR Boston                    January 8, 2003   page 1 of 4

System Transfer Functions
Consider a linear, time-invariant system with impulse response h(t), input u(t) and output
y(t).

u(t)               h(t)              y(t)

We often describe the system in terms of its transfer function, which is the ratio of the Laplace
transform of the output to the Laplace transform of the input. In general, the transfer function of
a system can be written as a ratio of polynomials in s:
Y(s)      b (s - z1)(s - z2 ) . . . (s - zm )
H(s) =        = m
U(s)      a n (s - p1)(s - p2 ) . . . (s - pn )
where zi are the zeros and pi are the poles of the transfer function H(s).

Poles and zeros:        are complex numbers
determine H(s) and h(t), the impulse response.
determine the stability of the system (poles in the LHP).

These same ideas can be developed for discrete signals, using z-transforms in place of Laplace
transforms. If poles are inside the unit circle in the complex z-plane, the system is stable.
Otherwise it is unstable.

State Variable Formulation
If a (single-input, single-output) system is defined with input u(t), output y(t), and state
vector x = (x1 x2 x3 . . . xn)T , then
x=Ax+Bu
y=Cx+Du
where A, B, C, and D are matrices that characterize the system. This system can be solved by
direct integration to obtain
t
x ( t ) = φ ( t ) x ( 0) + ∫ φ ( t − λ ) B u ( λ ) dλ
0
The first term represents the natural response and the second term, which is the convolution of
the input with the state transition matrix, represents the forced response.

The state transition matrix φ(t) can be evaluated using Laplace transform techniques, yielding
L{φ(t)} = (sI – A)-1
We can use this state variable formulation to obtain a good discrete approximation to a linear
continuous system.

Linear Feedback
Consider a negative feedback path H(s) applied to a plant with transfer function G(s)
Review of Feedback Control                         JR Boston                 January 8, 2003             page 2 of 4

+        e(t)
u(t)                        G(s)                   y(t)
-

H(s)

The closed-loop transfer function is
Y (s )       G (s)
L(s) =  =
U (s) 1 + G (s) H (s)
The zeros of the closed-loop transfer function L(s) are the same as those of the original plant, but
the poles are now the roots of 1 + G(s)H(s). Feedback changes the poles (natural modes) of the
system.

We often evaluate controllers in terms of their step responses (response to a step input):
1.5

overshoot
1

0.5                                               settling time

risetime
0
0   50       100          150     200    250    300             350      400        450   500
time (dimensionless)

Designers typically aim for about 10% overshoot (damping = 0.707 for a second-order system).

A common approach to feedback design is to use unity feedback (H(s) = 1) and introduce a
compensator network in series with the plant Gp(s) [G(s) = plant + compensator]:
+                                                       y(t)
u(t)                 compensator            Gp(s)
-

The compensator network (such as a lag-lead network or a PID compensator) can move the poles
around in the s-plane to achieve desired responses.
An advantage of this approach is that, if 1 + G(s) is large (still assuming that H(s) = 1), the
sensitivity of the closed-loop transfer function L(s) to changes in G(s) is reduced. Since
∂L       1       G            1
=       −           =
∂G 1 + G (1 + G )     2
(1 + G ) 2
ΔL   1 ΔG
we can define sensitivity       =      .
L 1+ G G
Review of Feedback Control                     JR Boston                      January 8, 2003   page 3 of 4

Comparison of PD and PI control
PD (lead)     speed                                               sensitive to noise
stabilizes system                                   expensive

PI (lag)         eliminates steady-state error                    sluggish
smooths data                                     destabilizes system

Summary of Discrete Control
Essentially nonlinear
Discrete control can be superior to linear control (e.g., deadtime control - a form of
bang-bang control).
Discrete versions of continuous control tend to be less stable.
Need to sample at 10-30 samples per period of highest natural mode in the system to
avoid problems (stability, hidden oscillations in continuous system between controller
sample points).

State Feedback Control
Can place system poles anywhere in the s-plane.
Need to know all states.
Costs
Need a computer (not much of an issue anymore)
Moving poles requires actuator power.
State knowledge requires expensive measurements.

plant
e(t)                       x(t)
+
r(t)                   x = Ax + Be           C          y(t)
- -

^
x
KT             Observer

1. Determine performance requirements
2. Find plant model
Determine appropriate level of detail in model.
If model is too complex
Too many parameters to measure accurately.
High computational requirements.
If the model is too simple
Can’t describe all performance characteristics.
Can’t describe all natural modes of system (especially high-frequency modes).
Often involves linearizing about an operating point.
3. Select approach
PID can be implemented without a detailed system model
PID parameters are usually tuned to achieve acceptable performance.
Review of Feedback Control                    JR Boston        January 8, 2003          page 4 of 4

PID parameters often change as operating point changes.
State feedback
Must determine a “cost function” to select pole positions.
Develop an observer (worry about convergence, etc.).

Intelligent Control
Phrase coined by K.S.Fu in 1950’s. It is now an official NSF buzzword.

What is intelligent control?
“We can’t state precisely what it is, but we know one when we see one.”
K.S. Narendra, Center for System Science, Yale
Narendra identified 3 categories of difficulties in designing control systems:
1) Computational complexity (environment, sensors)
2) Nonlinear systems (multiple degrees of freedom)
3) Uncertainty
The better a system deals with these factors, the more intelligent it is.

Some characteristics of intelligent control
1) Sets its own goals
2) Utilizes past experience
3) Assesses its environment
4) Makes decisions in unstructured environments
5) Identifies model limitations / changes
6) Fails gracefully
7) Reasons under uncertainty (shows judgement)
8) Often incorporates hierarchies, as illustrated below:

Supervisor         Coordinator    Executor

meta-knowledge
control signals
Plant
sensor signals

In intelligent control, concern shifts from dynamic response of system (stability) to
1) Finding set point / reference
2) Robustness to
Model errors
Measurements errors
System failure
3) System identification / fault detection / system reconfiguration

Generally assumes a computer-based system.

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