Control Systems--The Last Basic Course, Pt I

8/3/2019 Control Systems--The Last Basic Course, Pt I

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Dorf & Bishop, Modern Control Systems

(Editorialized by Me)

Mathematical Models of System

Dynamic systems are described by differential equations. Laplace work only in case of

linear systems.

Based upon the analysis of relation between system variables.

Linear Approximtions of Physical Systems: Within some range of variables, a great range

of physical systems are linear. A system is defined as linear in terms ofexcitation andresponse

Linear systems follow superposition principle: when the excitation is x1(t), response isy1(t); again when excitation is x2(t), response is y2(t). Now when excitation is x1(t) + x2(t),

response is y1(t) + y2(t). Or, the responses are directly mappable to the respective

excitationsthe excitations do not affect eachother in such a way that responses might be

influenced.Homogeniety is also followedax(t) gives ay(t) in response.

Although y = mx + b doesn't satisfy the condition of homogeniety, y = mx satisfiesbecause of constant offset. Here y must be seen as a fuction itself, and not a differential

change.

Small signal analysis: the slope at the operating point is a good approximation of the

function in small interval about the deviation (x x0). Or, y = g(x0) + dg/dx(x x0) => y =

y0 + m(x x0) => y y0 = m(x x0) which is nothing but y = mx.

To use laplace transform, the linearized differential equation is used.

The lower limit of the integral in the condition of convergence of f for it to have a

mathematically valid Laplace Transform takes care of any discontinuityone like delta

function. The 1 is abscissa of absolute convergence.

|f(t)| < Mexp(t), which implies the function converges for all 1 > , also gives the regionof convergence.

Poles and zeroes are critical frequencies.

Steady state/final value is found by final value theorem stated as

Simple pole of Y(s) at origin is allowed, but repeated poles at origin, in right half plane and

on imaginary axis are excluded.

Damping Ratio : is always in multiplication with natural frequency and arises from the

coefficient of y' (degree and order 1). The equations below is laplace transform of second

order equation of oscillations of y(t).


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= b/{2 sq root(kM)} decides how the system oscillations damp: >1 means over dampedand real roots;


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With Y(s) given, y(t) can be found to look like familiar exponential form

Effect of position of poles and zeros in s-plane over the response: adjusting value in

power of e above alters the envelope and hence the response of the transient.

So, apparantly, as the moves further towards left, s moves to left implying a faster

damping.

Remark: systems are designed from these mathematical resultsthese are the options we

are provided with to design. For example, one will now think of something which will be

capable of moving roots to left to form a system which performs variation of damping.

The case of many complex poles: the overall response is a combination of all these. The

magnitude of each pole is represented by respective residues in s-plane graphical analyses.


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The Transfer Function of Linear Systems

Definition: ratio of laplace transform of output variable to the input variable, assuming all

the initial conditions zero.

This represents the relationship describing dynamics of system(or element).

This maybe defined only linear and stationary system.

Transfer function is an input-output relationship, implying it gives no information about theinternal design details of the systems.Inference: the black-box study is best done in these

terms with internal design being independent of care for it.

The impedance model is based upon laplace only and it lets you treat every element in

lumped manner with simple rules of parallel and series connections applicable. So,

eventhough the sysmbols like R, Cs show up in transfer function, they still convey little

information about the actualinternal design.

Any output response is sum ofnatural response(due to initial conditions) andforcedresponsedetermined by the input. Y(s) = m(s)/q(s) + R(s)p(s)/q(s); q(s) = 0 is characteristic

equation. The transient response has q(s), while the steady response will be free of q(s).

the transfer function (s)/V(s) is of a DC Motor linearized by considering armature

current Ia constant.

Transfer function means linear functionnot only superposition, but also homogeinity

can be small signal too. It is a choice for analysis of blackbox, across its terminals, its ports.

In the transfer function above, id does not even show...

An amplidyne is a special type ofmotor-generatorwhich uses regeneration to increase its

gain. Energy comes from the motor, and the power output is controlled by changing the

field current of the generator. In a typical generator the load brushes are positioned

perpendicular to the magnetic field flux. To convert a generator to an amplidyne you

connect what would be the load brushes together and take the output from another set of

brushes that are parallel with the field. The perpendicular brushes are now called the'quadrature' brushes. This simple change can increase the gain by a factor of 10,000 or

more. Vacuum tubes of reasonable size were unable to deliver enough power to control
http://en.wikipedia.org/wiki/Motor-generatorhttp://en.wikipedia.org/wiki/Gainhttp://en.wikipedia.org/wiki/Vacuum_tubeshttp://en.wikipedia.org/wiki/Gainhttp://en.wikipedia.org/wiki/Vacuum_tubeshttp://en.wikipedia.org/wiki/Motor-generator


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large motors, but vacuum tube circuits driving the input of an amplidyne could be used to

boost small signals up to the power needed to drive large motors.

Block Diagram Models

Dynamic systems which comprise automatic control systems are given by a series of

differential equations. One thing to always keep in mind is that order of differntial equation

has got nothing to do with its linearity or even systems. Keeping the power of any term in

the differential equation to 1 only helps avoiding the convolution which would thenintroduce composite frequencies.

Control systems are concerned with controlling few variables, and hence controlled

variables must relate to controlling ones.

Block diagram consist of unidirectional operational blocks which represent transfer

function of variable of interest.

For J inputs and I outputs we have a transfer fucntion with IxJ dimention matrix. Y = GR

refer to table 2.6 on page 81.

When systems are cascaded, there can be loading or coupling of the blocks. In that case,

right transfer functions need to be calculated by the engineer. Block diagrams discretize the

effects into lumped unitsmake it easy to understand how to reduce and where to add more

units.

Signal Flow Control Graphs

block diagrams fail to help when systems grow beyond a certain complexity. The advantage

of line path method isflow graph gain formula which renders all the reduction methods


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useless.

Signal flow graph too is a representation of linear relations.

A branch relates the dependency of an output over input in the same way as the block does.

Summation of all the signals entering a node(signal) is equal to node variable.

A path is a branch or a set of branches between a node to another.A loop is a closed path

with no node met twice except the one at which it ends; two loops are non-touchingif they

have no common node. Linear Dependence: Mason's signal flow gain formula for xi independent and xj

dependent variables is given as


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Design of a lowpass filter: This is how to implement it.

Pg 128 has system analysis of Disk drive.

State Variable ModelsPsychology of Systems

Time variable techniques, alike these, are used for non-linear, time-varying and

multivariable systems. We are considering these only because wegot computers now!

Time Domain is that mathematical domain which incorporates the response and the

description of the system in terms of time variable t.Time domain analysis uses concept ofstate of system.

State of system: first of all, it is a set of variables. The values of these variables alongwith

the equations describing the system and input signals will provide thefuture state and

output of the system.

Each dynamic system can be described completely usingstate variables(all function of time)which provide thefuture state of the system ifpresent state and inputs(excitation signals, so

to speak) are known.

For RLC circuits, number of state variables is same as the independent number of energy

storage elements.

One thing to note is that the set of state variables chosen is not unique and several other

choices can be made.

Response of system is given by set ofdifferential equations written in terms of state

variables and input variables.


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A is nxn matrix while B is nxm.

The differential equation related the rate of change of state to the present state and inputs.

The outputs of a linear system can be related to state variables and inputs through output

equation y = Cx + Du, where y is output column vector.

Taking laplace transform of eq (3.16) and then taking inverse laplace to solve for x(t) gives

(t) = exp(At) is what gives unforced response and is called fundamental orStatetransition Matrix. It converges for afinite tand all A.


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Signal Flow Graph and Block Diagram Models

State of the system gives the dynamics of it only when it can be explained with a set ofdifferential equations.

Oftentimes, finding the correct set of differential equations is tough.

There is a signal flow diagram and block diagram corresponding to each set of state

variables. This puts the very fact clearly that there can be more than one graphs for a singletransfer function.

An n order transfer function means n state variables.

This method, needless to mention, creates only the graphs with all loops touching. While

designing a system, this constraint might seem way too much. Still, let's continue....

This designing based upon deciding state variables from transfer function, signal flow

diagram or the block diagram revolve around Mason's Signal Flow Gain formula's

propertieswhich are forPath factor and determinant in denominator.

General form of flow graph state model and block diagram model below is phase variable


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canonical form.

Where the transfer function is

Another direct method is


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x1, x2, x3 ... are phase variables.

The very fact that there are more than one state models possible and hence signal flow

graphs and block diagrams only means that no signal flow graph can be seen as a rigid

universal formone can distort then for analysis until the transfer function doesn't change.

Transfer Function From State Equation

Columns in matrix B of a state equation tells the number of inputs. For SISO, it is always

nx1.

G(s) = C(s)B + D, where (s) = [sI A]^(-1). This can be obtained through simplemanipulation of laplace of State Equations in matrix form.

Designs can be sequential, parallel etc. Disk drive read system is a sequential design.

Feedback Control Systems Characteristics

Principle: the principle offeedback system lies within the very basic reason of creating acontrol systemdesigning a system with desired response in mind. When we already know

the response, what is controlled and used as variable is the error signalthat is the difference

between desired and actual received signal in output. A closed loop uses measurement of output signal and comparision with desired output

to generate an error signal which is used by the controller to adjust actuator.


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E(s) = R(s) Y(s) is tracking error. Also, Sensitivity function S(s) is defined as 1/{1 + L(s)}

and Complimentary sensitivity function C(s) is defined as L(s)/(1 + L(s)).For tracking error

to be minimum, we need both S(s) and C(s) to be minimum. Td is disturbance and N is

measurement noise.

Also note that S(s) + C(s) = 1

Tracking error cannot be analyzed without understanding of the very meaning of a

transfer function's being small or large. |L(j)| for the range of frequencies, , ofinterest defines modulous of Loop Gain L(s).

Range of frequecies of interest: the E(s) equation makes it an apparent conflictthat toreduce the effect of disturbance Td L(s) must be made large for a given G(s) by selecting

apt Control Gc(s) and to reduce the effect of Noise N(s), L(s) cannot be soaring too high.

Fortunately, this conflictis resovled by making L(s) high for lowfrequencies which

affect disturbance Td and making it low for high frequencies which affect N(s).

Next, we need to address howfeedback can be helpful in reducing sensitivity of the system

to the variations in the parameters in process G(s).

Sensitivity of Control Systems to Parameter Variation

Supposing Td(s) = N(s) = 0 and making Gc(s)G(s)>>1 for all frequencies of operation gives

Y(s) ~= R(s), or say input somewhat equal to the output. This is the case with buffer circuit

of Opamp.

Although doing this makes the system very oscillatory, yet the very result that

increasing Loop Gain somewhat reduces the effect of G(s) is exceedingly useful.

We are still relying on the principle of superposition.

Larger L(s) also means a reduced sensitivity. Although. Question remains how we define


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sensitivity.


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Example 4.1 on page 241(the book) or pg 269(.pdf)

Disturbance Signals in a feedback Control System

The smaller the sensitivity, the lesser the effect of signal.

For disturbance rejection, we need large loop gain over the frequencies of interest.

This means designing a Gc(s) for low sesitivity for low frequencies

Measurement Noise Attenuation

for L(s)


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Control Systems--The Last Basic Course, Pt I

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