Modern Control Theory Solution

Modern Control Theory 10EE55

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Question Bank Solutions

UNIT 1 & 2

1) Compare modern control theory with conventional control theory (Jan 2010)

Comparison: Conventional vs. Modern Control

Conventional Control (Linear) Modern Control (Linear)

Frequency domainanalysis & Design(Transferfunction based) Based on SISO models Deals with input andoutput variables Initial conditions areassumed to be zero. Restricted to linear time-invariant systems

Time domain analysisand design(Differentialequation based) Based on MIMOmodels Deals with input, outputand state variables Initial conditions aretaken into consideration Applicable to nonlineartime variant system also

2) Determine state model for given transfer function (Jan 2007)

s3C + 9s2C + 26sC + 24 c = 24 RTake in LT


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3) Determine state model for given transfer function (June 2007) (Dec 2012)

Taking in LT


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C(S) = s2X1 +7s X1 + 2 X1

.. .C(t) = X1 + 7 X1 + 2 X1

= X3 + 7 X2 + 2 X1

4.) Develop a state model in Cascading form (June 2009)

The denominator of TF is to be in factor form


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5) Construct the state model using phase variables if a the system is described by the

differential equation (Dec 2012)


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Select variables x1(t) = y(t)

x2(t) = ẏ (t) = ẋ1(t) = dy(t)/dt

x3(t) = Ÿ(t) = ẋ2(t) = d2y(t)/dt2

ẋ1(t) = x2(t) ...........1

ẋ2(t) = x3(t) ........ ..2

To obtain ẋ3(t)

ẋ3(t) = -4x1(t) -14x2(t) -8 x3(t) + 10u(t)

from equation

ቮẋ1(t)

ẋ2(t)

ẋ3(t)ቮ=

0 1 00 0 1

−4 − 14 8൩อ1ݔ2ݔ3ݔอ+ อ

00

10อu(t)

The output is Y(t) = x1(t)

|1 0 0 |อ1ݔ2ݔ3ݔอ+ [0] u(t)


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UNIT 4

1) What is STM? Obtain the state transition matrix using power series method (Dec 09) (Jan 2010)


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2) What is STM? Compute the state transform matrix eAT . (Dec 2008) (June 2007)


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(Jan 2006)


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4 ) What is STM? Compute the state transform matrix eAT using Cayley Hamilton theorem. (Dec

2008) (June 2007) (Dec 2012)


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(July 2008)


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UNIT 5

Controllability and Observability

1) Explain the concept of Controllability and observability, with the condition for complete

controllability and observability in the S- plane (Dec 2010) (June 2010)

Concept:

Consider the typical state diagram of a system. The system has two state variables. X1(t) and X2(t).

The control input u(t) effects the state variable X1(t) while it cannot effect the effect the state variable

X2(t). Hence the state variable X2(t) cannot be controlled by the input u(t). Hence the system is

uncontrollable, i.e., for nth order, which has ‘n’ state variables, if any one state variable is uncontrolled by

the input u(t), the system is said to be UNCONTROLLABLE by input u(t).

Definition:

For the linear system given by

Y (t) = CX (t) + Du (t)

X (t ) = AX (t) + Bu(t)

is said to be completely state controllable. If there exists an unconstrained input vector u(t), which transfers

the initial state of the system x(t0) to its final state x(tf) in finite time f(tf-t0) i.e. ff. It can be seen if all the

initial states are controllable the system is completely controllable otherwise the system the system

uncontrollable.

Methods to determine the Controllability:

1) Gilbert’s Approach

2) Kalman’s Approach.


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Observability:

Concept:A system is completely observable, if every state variable of the system effects some of the outputs. In

other words, it is often desirable to obtain information on state variables from the measurements of outputs

and inputs. If any one of the states cannot be observed from the measurements of the outpits and inputs,

that state is unobservable and system is not completely observable or simply unobservable. Consider the

state diagram of typical system with state variables as x1 and x2 and y and u(t) as output and inputs

respectively,


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Dept. of EEE, SJBIT

2) Check the controllability of the system (Jan 2008)


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3) Check the controllability of the system by Kalman's method (Dec 2009)

4) Determine the state controllability of the system by Kalmans approach. (June 2006) (Dec 2012)


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5) Test the observablity using Kalmans method (Dec 2005)


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UNIT 6

Pole Placement Techniques

1) Design a controller K for the state model ( Dec 2009)


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2) Design controller K which places the closed loop poles at -4 ± j4 for a system using Acermanns

formula. (Dec 2007)


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3) Design a full order state observer. Assume the eigen values of the observer matrix at -2 ± j 3.464

and -5 (June 2010) (Jan 2010)


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4) Design a full order state observer. Assume the eigen values of the observer matrix at -2 ± j 3.464

and -5 (June 2010) (Jan 2010) (Dec 2012)


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Dept. of EEE, SJBIT

5) Design controller to take place closed loop poles -1± j1, -5. Also design an observer such that

observer poles are at -6, -6, -6. (Jun 2009) (Jan 2007)


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UNIT 7

NON LINEAR SYSTEM

1) What is phase- plane plot ? Describe delta method of drawing phase- plane trajectories (Jan 2010)

(Dec 2012)


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2) What are singular points? Explain the different singular points with respect to stability of non-

linear system (Jan 2010) (Dec 2009) (June 2009) (June 2010) (Jan 2009)( Dec 2010) (Dec 2012)


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3) Explain the common physical Non-linearities (Jan 2010) (Dec 2010) (Jan 2008) (June 2010)

Common Physical Nonlinearities: The common examples of physical

nonlinearities are saturation, dead zone, coulomb friction, stiction, backlash,

different types of springs, different types of relays etc.

Saturation: This is the most common of all nonlinearities. All practical systems,

when driven by sufficiently large signals, exhibit the phenomenon of

saturation due to limitations of physical capabilities of their components.

Saturation is a common phenomenon in magnetic circuits and amplifiers.

Dead zone: Some systems do not respond to very small input signals. For a

particular range of input, the output is zero. This is called dead zone existing in a

system. The input-output curve is shown in figure.


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Backlash: Another important nonlinearity commonly occurring in physical systems is

hysteresis in mechanical transmission such as gear trains and linkages. This

nonlinearity is somewhat different from magnetic hysteresis and is commonly referred

to as backlash. In servo systems, the gear backlash may cause sustained

oscillations or chattering phenomenon and the system may even turn unstable for

large backlash.

Control Theory 10EE55

Figure 6.3

Another important nonlinearity commonly occurring in physical systems is



servo systems, the gear backlash may cause sustained


Figure 6.4

Control Theory 10EE55

Another important nonlinearity commonly occurring in physical systems is



servo systems, the gear backlash may cause sustained



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Relay: A relay is a nonlinear power amplifier which can provide

large power amplification inexpensively and is therefore deliberately

introduced in control systems. A relay controlled system can be switched

abruptly between several discrete states which are usually off, full

forward and full reverse. Relay controlled systems find wide

applications in the control field. The characteristic of an ideal relay is as

shown in figure. In practice a relay has a definite amount of dead zone as

shown. This dead zone is caused by the facts that relay coil requires a

finite amount of current to actuate the relay. Further, since a larger coil

current is needed to close the relay than the current at which the relay

drops out, the characteristic always exhibits hysteresis.

Multivariable Nonlinearity: Some nonlinearities such as the torque-

speed characteristics of a servomotor, transistor characteristics etc., are

functions of more than one variable. Such nonlinearities are called

multivariable nonlinearities.


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UNIT 8

STABILITY

1) Determine the stability of the system. (Jan 2009) (Dec 2012)


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2) Determine the stability of the system described by (Dec 2010)

23 > 0 and det(P) > 0. Therefore, P is positive define. Hence, the equilibrium state at origin isasymptotically stable in the large.


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3) Explain PID controller (June, Dec 2010) ( Jan 2010) (Dec 2012)


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Dept. Of EEE, SJBIT

4 ) Check the sign definiteness of the following quadratic equation :

V(x) = 8x1 2 + x22 + 4x32 + 2x1x2 – 4x1x3 – 2x2x3. (Dec 2012)

Sol. V(x)= XTPX = [x x2 x3] ૡ��− 1��−

−��− ��൩൩

Applying Sylvesters criterion

8 > 0 ቚ8 11 1

ቚ= 7 > 0 8 1 − 21 1 − 1

−2 − 1 4൩= 20 > 0

As the successive principle minors of matrix P are +ve

Therefore V(x) is +ve definite

Modern Control Theory Solution

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