oper ability 2

8/12/2019 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.



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



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)



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.



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.




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) .




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 ∞.



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 .




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.



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



Closed-Loop Performance

Step response:




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.




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.




Frequency response (Bode Plot):




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




Frequency response (Bode Plot):




Frequency domain performance:

– Maximum peak criteria:

Let maximum peaks of sensitivity and complementarysensitivity functions be given by:

Typically we require:




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

.




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



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.



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.



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.



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



Perfect Control and

Plant Inversion

Internal Model Control (IMC)



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

.



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)



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.




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.




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.




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:



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:



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.

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