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Category | Engineering |
---|---|

Subject | Electrical Engineering |

Difficulty | Graduate |

Status | Solved |

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## Short Assignment Requirements

## Assignment Description

Application Studies for Control System Design

1/21/13

### DCS – DC Servomotor

Dynamics

*θ*^{˙ }= *ω*

*ω*˙ = −*αω *+ *βu y *= *θ*

Let

*x*_{1 }= *y, x*_{2 }= *ω, α *= 4*, β *= 1

Application studies

1. Simulate open-loop system

2. Calculate plant
transfer function *G*(*s*) = *C*(*sI *− *A*)^{−1}*B*

3. Closed-loop
proportional control: *u *= −*G*_{1}(*θ *− *θ _{r}*) (See block diagram above.)

(a) Analytically show that closed-loop system is stable for all values of *G*_{1}.

(b) Obtain Nyquist plot using Matlab.

(c) Obtain Bode plot using Matlab.

(d) Sketch Root locus and verify using Matlab.

(e) Develop
closed-loop simulation and simulate performance for several values of *G*_{1}.

### PEN – Motor-Driven Pendulum

Dynamics

*θ*^{˙ }= *ω*

*ω*˙ = −*αω *+ (*g/L*)sin*θ *+ *βu
y *= *θ*

Let

*x*_{1 }= *y, x*_{2 }= *ω, α *= 4*, β *= 1*, g/L *= 12

Application studies

1. Simulate open-loop system

2. Obtain the plant
transfer functions *G*(*s*) = *C*(*sI *− *A*)^{−1}*B *for

• *θ *=
0

• *θ *= *π*

Do this analytically and using the Matlab function linmod

3. Closed-loop
proportional control for *θ *= *π *and *u *=
−*G*_{1}(*θ *− *θ _{r}*)

(a) Analytically show that closed-loop system is stable for all values of *G*_{1}.

(b) Obtain Nyquist plot using Matlab

(c) Obtain Bode plot using Matlab

(d) Sketch Root locus and verify using Matlab.

(e) Develop
closed-loop simulation and simulate performance for several values of *G*_{1}.

4. Closed-loop
proportional control for *θ *= 0 and *u *= −*G*_{1}(*θ *− *θ _{r}*)

(a) Find the range of *G*_{1 }for which the system is
stable.

(b) Obtain Nyquist plot.

(c) Obtain Bode plot.

(d) Sketch Root locus and verify using Matlab.

(e) Develop
closed-loop simulation and simulate performance for several values of *G*_{1}.

5. Full-state
feedback, *θ _{r }*= 0, Pole placement For each equilibrium
state (

*θ*= 0

*,θ*=

*π*determine the full state feedback gains that place the closed-loop poles at

(a) *s *= −6 ± 4*j*

(b) *s *=
−8*,*−10

For each equilibrium state and set of control gains, simulate the closed loop behavior, and comment on the results.

6. Full-state
feedback with reference input. For *θ _{r }*= 0

*.*5 compute the gain

*G*

_{0 }for the exogenous input for each equilibrium state in Part 5, and simulate the performance. Verify that the steady-state error goes to zero.

### TH3 – Third-Order Heat Conduction

Dynamics

*x*˙_{1 }= −3*x*_{1 }+ *x*_{2 }+ *u x*˙_{2 }= *x*_{1}− 2*x*_{2}− *x*_{3 }*x*˙_{3 }= *x*_{2}− 3*x*_{3 }+ *x*_{0 }*y *= *x*_{3}

Application Studies

1. Simulate open-loop system

2. Calculate plant
transfer function *G*(*s*) = *C*(*sI *− *A*)^{−1}*B*

3. Closed-loop
proportional control: *u *= −*G*_{3}(*x*_{3}− *x _{r}*)

(a) Find the range of *G*_{3 }for which the closed-loop
system is stable.

(b) Obtain Nyquist plot using Matlab(c) Obtain Bode plot using Matlab.

(d) Sketch Root locus and verify using Matlab.

(e) Develop a closed-loop simulation and simulate performance for several
values of *G*_{3}.

(f) Determine
the steady-state error for a unit step reference input as a function of *G*_{3}. 4. Closed-loop proportional + integral control
For the closed-loop control law

*U*(*s*)
= −(*G _{p }*+

*G*)(

_{I}/s*Y*(

*s*) −

*Y*(

_{r}*s*))

(a) Show that the steady state error is zero for *G _{I }> *0

(b) Find the region in the *G _{p},G_{I }*plane for which the system is stable.

(c) Develop a closed-loop simulation and simulate performance for several
values of *G _{P},G_{I }*within
the range of stability.

5. Design a
full-state feedback control law that places the closed-loop poles at *s *=
−10 ± 5*j,*−15 and accounts for a non-zero nonzero reference value,
and for a nonzero exogenous constant input. Simulate closed-loop performance.

6. Design a full-order observer for *x*_{0 }= 0 ( and for *y *= *x*_{3})
with poles at *s *= −20 + 10*j,*−30. Using this
observer with the gain matrix of Part 5, design the full-order compensator and
simulate performance

7. Augment the observer of Part 6 by the additional state variable to represent the reference and exogenous inputs.

8. Design (and simulate performance) of a compensator using a reduced-order
observer, with poles at*s *= −20 ± 10*j*

9. Design (and simulate performance) of a compensator with gains designed for a linear-quadratic regulator with

*Q *=
diag[0*,*0*,q*_{3}]*, R *= 1

and a full-order Kalman filter using

*F *= *B, V *= *v,, W *= 1

Choose *q*_{3 }and *v *to make the step response five times faster than open-loop with
overshoot less than 5% . This is to be done by simulation.

### PCA – Pendulum on Cart

Linearized Dynamics

For an inverted pendulum on a cart, driven by a d-c motor, the linearized dynamics are

*x*¨ = −*αx*˙ − (*mg/M*)*θ *+ *βu*

*θ*^{¨ }= (*α/L*)*x*˙ + ((*M *+ *m*)*g/ML*)*θ *− (*β/L*)*u*

The state and output variables are

*x*_{1 }= *x,
x*_{2 }= *θ, x*_{3 }= *x, x*˙ _{4 }= *θ, y*^{˙ }_{1 }= *x*_{1}*,
y*_{2 }= *x*_{2}

Numerical values of the parameters are:

*α *= 4*, β *= 1*, L *=
0*.*25*M, m *= 0*.*4*Kg, M *= 1*Kg*

### Application Studies

1. Develop a simulation of the system

2. Calculate the transfer functions from the input *u *to the outputs *y*_{1 }and *y*_{2}. (Do this analytically–with
the aid of Matlab’s Symbolic Toolbox, if necessary, and verify using the
Control System Toolbox.)

3. Find the open loop poles and the zeros of each transfer function.

4. Determine whether the system can be stabilized by the control law

*u *=
−*G*_{1}*x*_{1}

5. Determine whether the system can be stabilized by the control law

*u *=
−*G*_{1}*x*_{1}− *G*_{2}*x*_{2}

6. Determine the full state feedback gains that place the closed-loop poles at

(a) *s *= −10 ± 5*j,*−20 ± 10*j*

(b) *s *= −10*,*−15*,*−20*,*−30

Simulate the transient response using this gain matrix. (Investigate whether the poles farther from the origin are very important.)

(a) For an initial position offset (*x*_{1 }= *.*1), all other state variables initially zero,
for both set of gains.

(b) For an initial
angular error (*x*_{2 }= *.*1), all other state variables zero, for both
sets of gains.

7. Design a full-order observer having poles at *s *= −40 ±
30*j,*−60 ± 20*j*. Simulate the performance with this observer
in place and using the gain matrix determined in Part 6. Compare the actual
state and the state estimate produced by the observer.

8. Design a reduced-order observer with poles at *s *= −40 ±
30*j*. Simulate the performance with this observer in place and
using the gain matrix determined in Part 6. Compare the actual state and the
state estimate produced by the observer.

9. Design a full-state feedback control law using quadratic optimization (LQR) with matrices given by

*Q *=
diag[*q*_{1}*,q*_{2}*,*0*,*0]*, R *=
1

Choose *q*_{1 }and *q*_{2 }to meet the following design
goal (if possible). For an initial cart position error of

.1 m, and with the pendulum vertical,

• Reduce
the position error to zero in about 2 sec with overshoot *<*.01 m

• Keep the pendulum angle *< *.05 rad.

The values of *q*_{1 }and *q*_{2 }are to be determined by simulation.

10. Design a compensator by use of the gains obtained in Part 9 and the reduced-order observer of Part 8,and simulate performance.

11. Determine the
gains of a Kalman-filter based full-order observer. Use *F *= *B *and *W *= *I*_{2 }(2 by 2
identity matrix). To pick the value of *V *, try to match the pole locations of Part 6.