Oper Ability 2
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Transcript of Oper Ability 2
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Introduction
Ziegler-Nichols (1943):
„„In the application of automatic controllers, it is important to
realize that controller and process form a unit; credit or discreditfor results obtained are attributable to one as much as theother. A poor controller is often able to perform acceptably on aprocess which is easily controlled. The finest controller made,when applied to a miserably designed process, may not deliverthe desired performance. True, on badly designed processes,
advanced controllers are able to eke out better results thanolder models, but on these processes, there is a definite endpoint which can be approached by instrumentation and it fallsshort of perfection.‟‟
Always keep in mind:
The power of control is limited. Control quality depends not onlyon the controller but also on the plant/process itself.
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The objective of a control system is to make the output y behave in a desired way by manipulating the plant input u.
– The regulator problem is to manipulate u to counteract theeffect of a disturbance d .
– The servo problem is to manipulate u to keep the output y close to a given reference input r .
If the process can be represented as:
The controller:
is designed such that the control error e = r − y is small.
Control Problem
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Control Problem
A major source of difficulty is that the perfect models (G,Gd )
are never available (i.e. the models may be inaccurate or
may change with time).
In particular, inaccuracy in G may cause problems because
the plant will be part of a feedback loop.
Therefore, when designing the control system, it should be
noted that the control system should be robust. If the "true“ process can be represented as:
where:
E = “uncertainty” or “perturbation” (unknown)
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Control Problem
The controller is designed for the following specifications:
Nominal stability (NS):
The system is stable with no model uncertainty Nominal Performance (NP):
The system satisfies the performance specifications with nomodel uncertainty
Robust Stability (RS):
The system is stable for all perturbed plants about the nominalmodel up to the worst case model uncertainty
Robust performance (RP):
The system satisfies the performance specifications for allperturbed plants about the nominal model up to the worst case
model uncertainty.
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Process Representation
A linear time invariant process can be represented in
either its state space representation or transfer function.
A continuous time-invariant linear system can be
represented in state space form as:
This representation can be used for both SISO and
MIMO systems.
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Process Representation
Taking Laplace transform of the state space representation yields(with the assumption of zero initial conditions):
For a MIMO case, G(s) is a transfer function matrix rather than atransfer function. We usually use the notation of:
to denote that the transfer function matrix G(s) has a state spacerealization given by ( A, B, C , D) .
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Process Representation
Transfer function usually takes the form of:
where: n = order of denominator (or pole polynomial) or order of the system
nz = order of numerator (or zero polynomial)
n − nz
= pole excess or relative order
Definition: – G(s) is strictly proper if G(s) 0 as s ∞. – G(s) is semi-proper or bi-proper if G(s) D ≠ 0 as s ∞. – G(s) which is strictly proper or semi-proper is proper.
– G(s) is improper if G(s) ∞ as s ∞.
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Feedback Control System
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Feedback Control System
Since the input to the plant is:
The plant model can be written as:
And the closed-loop system can be written as:
The objective of control is to manipulate u (by design K )
such that the control error e remains small in spite of
disturbances d .
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Feedback Control System
The following notation and terminology are used:
– L = GK : loop transfer function
– S = (I + GK )−1 = (I + L)−1 : sensitivity function
– T = (I + GK )−1GK = (I + L)−1L : complementary sensitivity
function
Notes:
– Function S is the closed-loop transfer function from the outputdisturbances to outputs, while T is the closed-loop transfer
function from the reference signals to outputs.
– The term complementary sensitivity for T follows from the
identity: T + S = I.
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Closed-Loop Stability
1. Evaluate the poles of the
closed-loop system.
2. Use the frequency response of
the open loop system (L( jω)).
One of the main issues in designing feedback controllers
is stability. To determine closed-loop stability, we may:
Im
Re
Closed-loop instability
occurs if L( jω) encircles
the critical point −1
stable unstable
0
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Closed-Loop Performance
Step response:
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Closed-Loop Performance
Time domain performance:
– Rise time (t r ): the time it takes for the output to first reach 90%
of its final value, which is usually required to be small.
– Settling time (t s): the time after which the output remains
within ±5% of its final value, which is usually required to be
small.
– Overshoot: the peak value divided by the final value, which
should typically be 1.2 (20%) or less.
– Decay ratio: the ratio of the second and first peaks, which
should typically be 0.3 or less.
– Steady-state offset: the difference between the final value and
the desired final value, which is usually required to be small.
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Closed-Loop Performance
Notes:
The rise time and settling time are measures of the speed of the
response, whereas the overshoot, decay ratio and steady-stateoffset are related to the quality of the response.
Another way of quantifying time domain performance is in terms
of some norm of the error signal. For example, one might use the
integral squared error (ISE) or its square root which is the 2-norm
of the error signal. In this case the various objectives related toboth the speed and quality of response are combined into one
number. In most cases minimizing the 2-norm gives a reasonable
trade-off between the various objectives listed above.
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Closed-Loop Performance
Frequency response (Bode Plot):
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Closed-Loop Performance
For:
Then:
Gain margin , where
– GM is the factor by which the loop gain |L( j ω)| may be increased
before the closed-loop system becomes unstable.
– GM is thus a direct safeguard against steady-state gain uncertainty.
Phase margin , where – PM tells how much phase lag we can add to L(s) at frequency ωc
before the phase becomes -180o (related to closed-loop instability).
– PM is thus a direct safeguard against time delay uncertainty
– Decreasing the value of ωc (lower closed-loop bandwidth, slower
response) means that we can tolerate larger time delay errors.
bjaW
abW baW 122 tan and
)(/1GM 180 j L o j L 180)( 180
oc j L 180)(PM 1)( c j L
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Closed-Loop Performance
Frequency response (Bode Plot):
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Closed-Loop Performance
Frequency domain performance:
– Maximum peak criteria:
Let maximum peaks of sensitivity and complementarysensitivity functions be given by:
Typically we require:
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Closed-Loop Performance
Notes:
MS or MT represents the worst case performance. A large value of
MS or MT indicates poor performance as well as poor robustness. There is a close relationship between these maximums peaks
and the gain and phase margins.
For a given MS we are guaranteed:
with MS = 2, we are guaranteed GM ≥ 2 and PM ≥ 29.0o.
For a given MT we are guaranteed:
with MT = 2, we have GM ≥ 1.5 and PM ≥ 29.0o
.
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Closed-Loop Performance
Frequency domain performance (continued):
– Bandwidth and crossover frequency:
Bandwidth is defined as the frequency range [ω1, ω2 ] over whichcontrol is “effective”. Usually ω1 = 0, and then ω2 = ωB is the
bandwidth.
Closed-loop bandwidth ωB is the frequency where |S( jω)| first
crosses 1/√2 = 0.707(≈ −3dB) from below.
Closed-loop bandwidth in terms of T , ωBT , is the highestfrequency at which |T ( jω)| crosses 1/√2 = 0.707(≈ −3dB) fromabove.
Crossover frequency ωc is defined as the frequency where
|L( jωc )| first crosses 1 from above.
For systems with PM < 90o we have: ωB < ωc < ωBT
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Controller Design
1. Online tuning methods – Ziegler-Nichols
–Cohen and Coon – etc.
2. Shaping of transfer functions
– Loop shaping. This is the classical approach in which the magnitude
of the open-loop transfer function, L( jω), is shaped. Usually no
optimization is involved and the designer aims to obtain |L( jω)| withdesired bandwidth, slopes etc. However, classical loop shaping is
difficult to apply for complicated systems.
– Shaping of closed-loop transfer functions, such as S, T and KS.
Optimization is usually used, resulting in various H ∞ optimal control
problems such as mixed weighted sensitivity.
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Controller Design
3. The signal-based approach.
– This involves time domain problem formulations resulting in the
minimization of a norm of a transfer function. – Here one considers a particular disturbance or reference change
and then one tries to optimize the closed-loop response.
– The "modern" state-space methods from the 1960‟s, such as LinearQuadratic Gaussian (LQG) control, are based on this signal-orient
approach.
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Controller Design
4. Numerical optimization
– This often involves multi-objective optimization where one attempts
to optimize directly the true objectives, such as rise times, stabilitymargins, etc.
– Computationally, such optimization problems may be difficult to
solve, especially if one does not have convexity.
– The numerical optimization approach may also be performed on
line, which might be useful when dealing with cases with constraints
on the input and outputs. Online optimization approaches such asmodel predictive control likely to become more popular as faster
computers and more efficient and reliable computational algorithms
are developed.
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Input-Output Controllability
Input-output controllability is the ability to achieve acceptable control
performance; that is, to keep the outputs (y ) within specified bounds
displacements from their references (r ), in spite of unknown butbounded variations, such as disturbances (d ) and plant changes
(including uncertainties), using available inputs (u) and available
measurements (y m or d m).
Controllability is independent of the controller, and is a property of theplant (or process) alone. It can only be affected by:
– changing the apparatus itself, e.g. type, size, etc.
– relocating sensors and actuators
– adding new equipment, extra sensors and actuators
– changing the control objectives
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Perfect Control and
Plant Inversion
Internal Model Control (IMC)
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Perfect Control and
Plant Inversion
The simplest way to understand how process properties may
limit the achievable control performances is by using the
internal model control (IMC) framework. The corresponding closed-loop response can be written as:
If a perfect model of the process is available (i.e. GM = G), the
closed-loop response can be further simplified as follows:
It can be seen that the perfect control performance can be
achieved when Q = G−1
.
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Perfect Control and
Plant Inversion
Based on this framework, Morari argues that any feedback
controllers are somehow trying to invert the process directly or
indirectly in order to find suitable inputs to keep the process outputsat the desired set-points in the presence of any disturbances
affecting the process.
Any process characteristics that pose a constraint to this inversion
represent inherent limitations to the achievable control performance.
Perfect control cannot be achieved if: – G contains time delay (since then G−1 contains a prediction)
– G contains RHP-zeros (since then G−1 is unstable)
– G has more poles than zeros (since then G−1 is unrealizable)
– |G−1R | and |G-1Gd | is large (due to input constraint)
– G is uncertain (since then G−1 cannot be obtained exactly)
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Performance Limitation for
SISO System
Limitation imposed by time delays (θ )
The maximum gain crossover frequency (ωc ) of the closed-loop
system is bounded from above by:
The larger the time delay, the more detrimental the effect to
process operability.
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Performance Limitation for
SISO System
Limitation imposed by RHP zeros (z )
The maximum gain crossover frequency (ωc ) of the closed-loop system
is bounded from above by: – For a real RHP zero:
– For a complex pair of RHP zeros:
RHP zeros located close to the origin are bad for process operability.
For a complex pair of RHP zeros, the effects of the zeros are worse if
they are located closer to the real axis than to the imaginary axis.
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Performance Limitation for
SISO System
Limitation imposed by relative degree
– If the process has a relative degree larger than 0 (i.e. has
more poles than zeros), the inverse of the process transferfunction is improper hence not realizable in practice.
– A proper IMC controller can only be obtained by using a
suitable order low-pass filter.
– This results in reduction of achievable performance at high
frequencies.
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Performance Limitation for
SISO System
Limitation imposed by input constraints
– All practical systems have constraints to the changes that can be
made to their manipulated variables. For example, a control valveis limited by its fully open and fully close position.
– These constraints impose limitations on implementing the inverse
of the process transfer function as the IMC controller Q to achieve
perfect control performance.
– To achieve perfect control performance, it is required that:
Disturbance rejection:
Set-point tracking:
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Unstable Processes
A process that has right-half-plane (RHP) poles is unstable. This
process needs feedback control for stabilization.
While the presence of RHP zeros and time delays poses anupper bound to the controller gain, hence limiting the achievable
control performance, the presence of RHP poles on the other
hand imposes a lower bound on the controller gain to stabilize
the process.
– For a real RHP pole:
– For a pair of pure imaginary RHP poles:
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Combined RHP Poles and
RHP Zeros
For a system with a single RHP pole and a single
RHP zero is required that:
in order to achieve acceptable performance and
robustness.
The location of pole and zero cannot be too closed.