Effect of feedback on parameter variation
Effect of feedback on parameter variation
R(s)
G(s)
C(s)
Open loop system
Unity feedback system
C(s) = R(s) . G(s)
C(s) = R(s) . G(s)/[1 + G(s)]
Effect of variation of forward path gain: G(s)
Let G(s) G(s) + G(s)
O/P of open loop system: C(s) + C(s) = [G(s) + G(s)] . R(s)
C(s) = G(s). R(s)
O/P of closed loop system:
Assuming G(s) G(s)
In a closed loop system, effect of variation in G(s) is reduced by a
factor [1 + G(s).H(s)]
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Effect of feedback: Sensitivity analysis
System sensitivity: Relative variation in overall transfer function T(s) = C(s)/R(s)
due to variation of G(s), H(s) or other parameters.
For variation of G(s):
For Small incremental variation in G(s), sensitivity is:
Similarly, sensitivity of the open loop system is:
AS T = G
Sensitivity of a closed loop system w.r.t. variation in G is reduced by a factor
of (1 + GH) compared to open loop system
Effect of feedback: Sensitivity analysis
Sensitivity of T w.r.t. H, the feedback gain, is given by:
In general, |GH| 1 Sensitivity of T w.r.t. H tends to unity.
Selection of H is very important
A system is more sensitive to changes in H (feedback element)
compared to G (plant)
Effect of feedback: External disturbances
For calculating effect of disturbance
D(s), assume R(s) is zero. Let CD(s) is
the O/P corresponding to D(s).
For calculating effect of disturbance R(s), assume D(s) is zero. Let CR(s) is the O/P
corresponding to R(s).
Assuming,
CD(s)/D(s) becomes almost zero
CR(s) becomes almost 1/H(s)
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Effect of feedback: System bandwidth
A control system is a low pass filter
At system bandwidth freq. ,wb, system gain
reduces to 1/2 of its DC gain
A large bandwidth means system responds
fast during transients
Open-loop transfer fn.:
where k = k/, = 1/
Closed loop transfer fn.:
where c = /(1 + K); Time constant of closed loop system is reduced
Open loop system bandwidth: b & of closed loop system cb
b = 1/ & similarly, cb = 1/ c
cb / b = / c = (1 + K)
Closed loop system bandwidth is (1 + K) time
that of open loop system
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Linearising effect of feedback
Suppose g(t) = e2(t)
As an open loop: c = r2
As a closed loop: e = (r c)2
Classification of control systems
Linear/ Non-linear
Continuous/ Digital (Discrete)
Time invariant/ Time varying
Minimum phase/Non-minimum phase
Deterministic/Stochastic
Lumped parameter (ODE) / Distributed parameter (PDE)
SISO/MIMO
Etc.
Linear System
A linear system obeys the principle of superposition
For a function: y = f(x),
If inputs x1(t) y1(t) and x2(t) y2(t), then:
[1x1(t) + 2x2(t)] [1 y1(t) + 2y2(t)] where 1 , 2 are constants
Sinusoidal I/P to a linear system will produce a sinusoidal O/P of same
freq. but with different amplitude and phase shift
Stability is clearly defined in a linear system
Input signals and initial conditions have no effect on stability
Non-linear System
Does not follow principle of superposition
System response depends on I/P signal
System stability depends both on I/P signal and initial conditions
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Continuous-data System
Various parts of the control system like, input, plant O/P, feedback, error
etc. are all continuous function of time i.e. all signals are continuously
available/measurable over time.
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Discrete-data System (Sampled-data system)
In these systems, signals at one or more points of the system are either in
the form of a pulse train or in digital code.
Preferred in complex control system where a digital computer forms the
heart of a controller or where digital transmission over a long distance is
involved.
In general, a discrete-data system receives data only at uniformly spaced
discrete instances of time.
Cost of analog controller rises steeply with increase in control system
complexity.
Digital systems can be time-shared by a number systems
When signal transmission channel forms a part of a control loop, it
becomes necessary to use digital techniques
For a no. of control component, output is in discrete form: Shaft encoder,
image etc.
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Time-invariant & Time-varying systems
A system is time-invariant if its characteristics do not change with time.
The co-efficient of its equations do not change over time
Response of a time-invariant system is independent of the time at which
the input is applied
A system is time-varying if its characteristics change with time.
The co-efficient of its equations change over time
Response of a time-varying system is dependent on the time at which the
input is applied
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Linear Time-invariant control system (LTI)
All subsequent system analysis will be for
Linear Time-Invariant control systems
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Minimum phase/Non-minimum phase systems
A transfer function whose all poles and zeros are on ve side of s-plane is
called a minimum phase system
A system with at least one pole/zero on the RHS of s-plane is called a
non-minimum phase systems
Minimum phase system
Non-minimum phase system
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