- A block diagram for a two loop, closed-loop control system composed of
a plant, actuator, sensor, and two compensators, G1(s) and G2(s), is shown
below. The plant is represented by the block 1/Ms^2. The value
of M is 1,175,000. The value of the sensor time constant T is 3. The
actuator, 1/s, converts the control signal U(s) into the force acting on the
plant. The plant output C(s) represents position. The signal V(s) is the
measured velocity signal produced by the sensor s/(Ts+1). A disturbance D(s)
also acts on the plant through the block 1/s. Specifications are imposed
on both the rate loop and the position loop.
- The purpose of the rate loop is to provide acceptable transient and
steady-state performance for velocity under the assumption that the position
loop is opened. The reference input signal for the rate loop is R1(s),
E1(s) is the rate error signal, and V(s) is the output signal for the rate
loop. Only the compensator G2(s) is involved in the rate loop.
The
following specifications apply to the rate loop:
- the absolute value in steady-state for E1(s) due to a unit ramp
disturbance input D(s) must be 0.05;
- the phase margin and gain crossover frequency must be such that the time
delay associated with the phase margin is at least 7 seconds;
- the gain margin must be at least 18 db;
- the settling time for the closed-loop rate control system for a step
input at R1(s) must be less than 85 seconds;
- the magnitude of the control signal u(t) may not exceed 9,000 at any
point in time.
- The purpose of the position loop is to provide acceptable transient and
steady-state performance for positions under the assumption that both the
rate and position loops are closed. The reference input signal for the
position loop is R(s), E(s) is the position error signal, and C(s) is the
output signal for the position loop. The design of G1(s) assumes that the
rate loop is closed.
The following specifications apply to the
position loop:
- the phase margin and gain crossover frequency must be such that the time
delay associated with the phase margin is at least 90 seconds;
- the gain margin must be at least 13 db;
- the settling time for the closed-loop position control system for a step
input at R(s) must be less than 250 seconds.
TASKS:
- Design compensators G1(s) and G2(s) such that all specifications are
satisfied. Verify that the specifications are indeed all satisfied. Bode
plots of the uncompensated and compensated systems for both rate and
position loops are required. Step responses of the closed-loop rate and
position control systems are also required, including a plot of u(t) for the
closed-loop rate control system. Discuss your design process in terms of how
your compensators were designed to meet the specifications.
- MATLAB is the preferred software for the design process. MATLAB stores
each variable as an array; for example, a polynomial is represented by an
array of the polynomial coefficients in descending powers. In MATLAB, the
command "help function_name" provides information on how to use that
function. Some useful MATLAB functions are:
- conv (multiplies polynomials), bode (generates magnitude and phase vs.
frequency), logspace (create a logarithmically spaced frequency array),
semilogx (makes Bode plots on semilog scale), log10 (common logarithm),
step (generates the step response), roots (computes the roots of
polynomials), series (combines blocks in series), feedback (combines
blocks in feedback configuration), rlocus (makes root locus plots).
System Block Diagram
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Lastest revision on
Wednesday, June 7, 2006 9:18 PM
