Modern Control Theory Solution 1c49
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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
Dept. of EEE, SJBIT
Page 1
Modern Control Theory
10EE55
3) Determine state model for given transfer function (June 2007) (Dec 2012)
Taking in LT
Dept. of EEE, SJBIT
Page 2
Modern Control Theory
10EE55
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
Dept. of EEE, SJBIT
Page 3
Modern Control Theory
10EE55
5) Construct the state model using phase variables if a the system is described by the differential equation (Dec 2012)
Dept. of EEE, SJBIT
Page 4
Modern Control Theory
10EE55
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
Dept. of EEE, SJBIT
Page 5
Modern Control Theory
10EE55 UNIT 4
1) What is STM? Obtain the state transition matrix using power series method (Dec 09) (Jan 2010)
Dept. of EEE, SJBIT
Page 6
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 7
Modern Control Theory
10EE55
2) What is STM? Compute the state transform matrix eAT . (Dec 2008) (June 2007)
Dept. of EEE, SJBIT
Page 8
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 9
Modern Control Theory
10EE55
(Jan 2006)
Dept. of EEE, SJBIT
Page 10
Modern Control Theory
10EE55
4 ) What is STM? Compute the state transform matrix eAT using Cayley Hamilton theorem. (Dec 2008) (June 2007) (Dec 2012)
Dept. of EEE, SJBIT
Page 11
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 12
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 13
Modern Control Theory
10EE55
(July 2008)
Dept. of EEE, SJBIT
Page 14
Modern Control Theory
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.
Dept. of EEE, SJBIT
Page 15
Modern Control Theory
10EE55
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,
Dept. of EEE, SJBIT
Page 16
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 17
Modern Control Theory
10EE55
2) Check the controllability of the system (Jan 2008)
Dept. of EEE, SJBIT
Page 18
Modern Control Theory
10EE55
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)
Dept. of EEE, SJBIT
Page 19
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 20
Modern Control Theory
10EE55
5) Test the observablity using Kalmans method (Dec 2005)
Dept. of EEE, SJBIT
Page 21
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 22
Modern Control Theory
10EE55 UNIT 6
Pole Placement Techniques 1) Design a controller K for the state model ( Dec 2009)
Dept. of EEE, SJBIT
Page 23
Modern Control Theory
10EE55
2) Design controller K which places the closed loop poles at -4 ± j4 for a system using Acermanns formula. (Dec 2007)
Dept. of EEE, SJBIT
Page 24
Modern Control Theory
10EE55
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)
Dept. of EEE, SJBIT
Page 25
Modern Control Theory
10EE55
4) Design a full order state observer. Assume the eigen values of the observer matrix at and -5 (June 2010) (Jan 2010) (Dec 2012)
Dept. of EEE, SJBIT
-2 ± j 3.464
Page 26
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 27
Modern Control Theory
10EE55
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)
Dept. of EEE, SJBIT
Page 28
Modern Control Theory
10EE55 UNIT 7
NON LINEAR SYSTEM 1) What is phase- plane plot ? Describe delta method of drawing phase- plane trajectories (Jan 2010) (Dec 2012)
Dept. of EEE, SJBIT
Page 29
Modern Control Theory
10EE55
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)
Dept. of EEE, SJBIT
Page 30
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 31
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 32
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 33
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 34
Modern Control Theory
10EE55
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.
Dept. of EEE, SJBIT
Page 35
Modern Control Theory
10EE55
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
Dept. of EEE, SJBIT
Page 36
Modern Control Theory
10EE55
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.
Dept. Of EEE, SJBIT
Page 37
Modern Control Theory
10EE55 UNIT 8 STABILITY
1) Determine the stability of the system. (Jan 2009) (Dec 2012)
Dept. Of EEE, SJBIT
Page 38
Modern Control Theory
Dept. Of EEE, SJBIT
10EE55
Page 39
Modern Control Theory
10EE55
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.
Dept. Of EEE, SJBIT
Page 40
Modern Control Theory
10EE55
3) Explain PID controller (June, Dec 2010) ( Jan 2010) (Dec 2012)
Dept. Of EEE, SJBIT
Page 41
Modern Control Theory
Dept. Of EEE, SJBIT
10EE55
Page 42
Modern Control Theory
10EE55
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− ൩࢞൩ −− ࢞ 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
Dept. Of EEE, SJBIT
Page 43
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
Dept. of EEE, SJBIT
Page 1
Modern Control Theory
10EE55
3) Determine state model for given transfer function (June 2007) (Dec 2012)
Taking in LT
Dept. of EEE, SJBIT
Page 2
Modern Control Theory
10EE55
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
Dept. of EEE, SJBIT
Page 3
Modern Control Theory
10EE55
5) Construct the state model using phase variables if a the system is described by the differential equation (Dec 2012)
Dept. of EEE, SJBIT
Page 4
Modern Control Theory
10EE55
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
Dept. of EEE, SJBIT
Page 5
Modern Control Theory
10EE55 UNIT 4
1) What is STM? Obtain the state transition matrix using power series method (Dec 09) (Jan 2010)
Dept. of EEE, SJBIT
Page 6
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 7
Modern Control Theory
10EE55
2) What is STM? Compute the state transform matrix eAT . (Dec 2008) (June 2007)
Dept. of EEE, SJBIT
Page 8
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 9
Modern Control Theory
10EE55
(Jan 2006)
Dept. of EEE, SJBIT
Page 10
Modern Control Theory
10EE55
4 ) What is STM? Compute the state transform matrix eAT using Cayley Hamilton theorem. (Dec 2008) (June 2007) (Dec 2012)
Dept. of EEE, SJBIT
Page 11
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 12
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 13
Modern Control Theory
10EE55
(July 2008)
Dept. of EEE, SJBIT
Page 14
Modern Control Theory
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.
Dept. of EEE, SJBIT
Page 15
Modern Control Theory
10EE55
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,
Dept. of EEE, SJBIT
Page 16
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 17
Modern Control Theory
10EE55
2) Check the controllability of the system (Jan 2008)
Dept. of EEE, SJBIT
Page 18
Modern Control Theory
10EE55
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)
Dept. of EEE, SJBIT
Page 19
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 20
Modern Control Theory
10EE55
5) Test the observablity using Kalmans method (Dec 2005)
Dept. of EEE, SJBIT
Page 21
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 22
Modern Control Theory
10EE55 UNIT 6
Pole Placement Techniques 1) Design a controller K for the state model ( Dec 2009)
Dept. of EEE, SJBIT
Page 23
Modern Control Theory
10EE55
2) Design controller K which places the closed loop poles at -4 ± j4 for a system using Acermanns formula. (Dec 2007)
Dept. of EEE, SJBIT
Page 24
Modern Control Theory
10EE55
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)
Dept. of EEE, SJBIT
Page 25
Modern Control Theory
10EE55
4) Design a full order state observer. Assume the eigen values of the observer matrix at and -5 (June 2010) (Jan 2010) (Dec 2012)
Dept. of EEE, SJBIT
-2 ± j 3.464
Page 26
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 27
Modern Control Theory
10EE55
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)
Dept. of EEE, SJBIT
Page 28
Modern Control Theory
10EE55 UNIT 7
NON LINEAR SYSTEM 1) What is phase- plane plot ? Describe delta method of drawing phase- plane trajectories (Jan 2010) (Dec 2012)
Dept. of EEE, SJBIT
Page 29
Modern Control Theory
10EE55
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)
Dept. of EEE, SJBIT
Page 30
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 31
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 32
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 33
Modern Control Theory
Dept. of EEE, SJBIT
10EE55
Page 34
Modern Control Theory
10EE55
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.
Dept. of EEE, SJBIT
Page 35
Modern Control Theory
10EE55
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
Dept. of EEE, SJBIT
Page 36
Modern Control Theory
10EE55
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.
Dept. Of EEE, SJBIT
Page 37
Modern Control Theory
10EE55 UNIT 8 STABILITY
1) Determine the stability of the system. (Jan 2009) (Dec 2012)
Dept. Of EEE, SJBIT
Page 38
Modern Control Theory
Dept. Of EEE, SJBIT
10EE55
Page 39
Modern Control Theory
10EE55
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.
Dept. Of EEE, SJBIT
Page 40
Modern Control Theory
10EE55
3) Explain PID controller (June, Dec 2010) ( Jan 2010) (Dec 2012)
Dept. Of EEE, SJBIT
Page 41
Modern Control Theory
Dept. Of EEE, SJBIT
10EE55
Page 42
Modern Control Theory
10EE55
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− ൩࢞൩ −− ࢞ 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
Dept. Of EEE, SJBIT
Page 43
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