Modern Control Theory Solution

Modern Control Theory

10EE55

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)  Frequency domain analysis & Design(Transfer function based)  Based on SISO models  Deals with input and output variables  Initial conditions are assumed to be zero.  Restricted to linear timeinvariant systems

Modern Control (Linear)  Time domain analysis and design(Differential equation based)  Based on MIMOmodels  Deals with input, output and state variables  Initial conditions are taken into consideration  Applicable to nonlinear time variant system also

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

s3C + 9s2C + 26sC + 24 c = 24 R Take 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 ẋ1(t) 0 1 ቮẋ2(t)ቮ=൥ 0 0 ẋ3(t) −4 − 14

The output is Y(t) = x1(t) |1

0

0 ‫ݔ‬1 0 1 ൩อ‫ݔ‬2อ+ อ0 อu(t) 8 ‫ݔ‬3 10

‫ݔ‬1 0 | อ‫ݔ‬2อ+ [0] u(t) ‫ݔ‬3

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10EE55 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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10EE55 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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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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10EE55 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 and -5 (June 2010) (Jan 2010) (Dec 2012)

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-2 ± j 3.464

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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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10EE55 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 nonlinear 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

are

Nonlinearities:

The common examples of physical

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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Figure 6.3

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.

Figure 6.4

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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 torquespeed characteristics of a servomotor, transistor characteristics etc., are functions of more than one variable. Such nonlinearities are called multivariable nonlinearities.

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10EE55 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 is asymptotically stable in the large.

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

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4 ) Check the sign definiteness of the following quadratic equation : V(x) = 8x1 2 + x22 + 4x32 + 2x1x2 – 4x1x3 – 2x2x3. (Dec 2012)

Sol.

ૡ૚− ૛ ࢞૚ V(x)= X PX = [x x2 x3] ൥ 1૚− ૚ ൩൥࢞૛൩ −૛− ૚૝ ࢞૜ T

Applying Sylvesters criterion 8>0

8 ቚ 1

8 1 1 ቚ= 7 > 0 ൥ 1 1 1 −2 − 1

−2 − 1 ൩= 20 > 0 4

As the successive principle minors of matrix P are +ve Therefore V(x) is +ve definite

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Modern Control Theory Solution

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