@lay:3008.more

Second-Order Nonlinear Control System

A block diagram of an instrument servo system containing a saturating amplifier
is shown in Figure 7. It is desired to obtain the output phase trajectory
for a reference input of 5 units.

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@fig:Figure 7. Block diagram of an Instrument Servo

Every block of the control system can be modeled by the four-terminal device
as shown in Figure 8. The zero current source JI is artificially
introduced to permit the voltage between nodes I and GI, VJI, to be used as the
input variable for the amplifiers and transfer functions and presents
infinite load impedance to the preceding stage.
The voltage source EO results in zero source impedance for the following
stage. These considerations allow the devices to be interconnected either with
other models or with ordinary electrical circuits.

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@fig:Figure 8. General Four-Terminal Model, Input-Output Characteristic of Saturating
Amplifier

The output source EO performs the actual task of the device.
For the ideal amplifier Ks with a gain of 1 its value is EO = VJI.
The model is temporarily stored in the MODEL DESCRIPTION with the name GAIN.
The characteristic of the saturating amplifier is shown in Figure 8
and is described by TABLE 1 in model GSAT. Thus EO is just TABLE 1 (VJI).
To model a general transfer function, the following approach is used (detailed
information may be found in Resources 1 and 3).

Let G(s) be the n-th order transfer function:

\[ G(s)=\frac{a_{m}s^{m}+a_{m-1}s^{m-1}+\cdots+a_{1}s+a_{0}}
             {b_{n}s^{n}+b_{n-1}s^{n-1}+\cdots+b_{1}s+b_{0}}   \]

The voltage source EO must be given as:

\[ EO=a_{m}P_{m+1}+a_{m-1}P_{m}+\cdots+a_{1}P_{2}+a_{0}P_{1}.\]

The n-th derivative of the transfer function is supplied under DEFINED
PARAMETERS as:

\[ DP_{n}=\frac{V_{JI}-b_{n-1}P_{n}-\cdots-b_{0}P_{1}}{b_{n}}.\]

The remaining (n-1) derivatives are also supplied under DEFINED PARAMETERS
in the repetitive form:

\[ DP_{n-1}=P_{n} \] 
\[ DP_{n-2}=P_{n-1} \] 
\[ \vdots \] 
\[ DP_{1}=P_{2}.\]

In this example a 2nd order model is used:

\[ G(s)=\frac{3}{s^{2}+s} \]

The model is stored with the name 2ORDER.
Finally all models must be connected according to fig. 9
observing the correct polarity for the summing junction.
The SCEPTRE input reads as follows:

MODEL GAIN (TEMP) (I-O-GI-GO)
ELEMENTS
     JI, I-GI = 0.0
      EO, GO-O = X1 (VJI)
  MODEL GSAT (TEMP) (I-O-GI-GO)
    ELEMENTS
      JI, I-GI = 0.0
      EO, GO-O = TABLE 1 (VJI)
    FUNCTIONS
      TABLE 1 = -100, -1, -1, -1, 1,1, 100,1
  MODEL 2ORDER (TEMP) (I-O-GI-GO)
    ELEMENTS
      JI, I-GI = 0.0
      EO, GO-O = X0 (3.0*P1)
    DEFINED PARAMETERS
      DP2 = X2 (VJI-P2)
      DP1 = X3 (P2)
      P2 = 0
      P1 = 0
CIRCUIT DESCRIPTION
  ELEMENTS
    EIN, 9-1 = 5
    A1, 1-4-8-5 = MODEL GAIN
    A2, 4-6-5-7 = MODEL GSAT
    G1, 6-8-7-9 = MODEL 2ORDER
  DEFINED PARAMETERS
    PDEO = X4 (3.0*P2G1)
  OUTPUTS
    PDEO(RATE),EOG1(ANGLE)
    VJIA1,EOA1,EOA2
  RUN CONTROLS
    STOP TIME = 10
END

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@fig:Figure 9. Block diagram in SCEPTRE form

The output phase trajectory is a plot of the time derivative of the output
signal versus the output signal and is shown in fig. 10.
Remember to use the correct suffixes from
the CIRCUIT DESCRIPTION. Therefore the wanted output signal is the voltage
EOG1. For the time derivative PDEO yields:

\[ EOG1 = a_{0}P_{1} \]
\[ PDEO = \frac{d}{dt}(EOG1)=a_{0}DP_{1}=a_{0}P_{2}=3P_{2}.\]

@lay:Please insert figure 10 here 3008f10.eps

@fig:Figure 10. Phase trajectory of servo system with saturation

5.4 Nonlinear Pendulum

This example solves the elementary equation of a pendulum, which is given as

\[ \ddot{x}+\sin x = 0.\]

The substitution

\[ y = \dot{x} \]

results in two differential equation

\[ \dot{x} = y \]
\[ \dot{y} = \sin x \]

which can be easily solved in SCEPTRE:

CIRCUIT DESCRIPTION
DEFINED PARAMETERS
  DPX = X1 (PY)
  DPY = X2 (-SIN(PX))
  PX  = 0.5
  PY  = 0
OUTPUTS
  PX,PY
RUN CONTROLS
  INTEGRATION ROUTINE = IMPLICIT
  STOP TIME = 20
RERUN DESCRIPTION (3)
DEFINED PARAMETERS
  PX=1,2,3
RERUN DESCRIPTION (3)
DEFINED PARAMETERS
  PX=0.5, 0.5, 0.5
RUN CONTROLS
  INTEGRATION ROUTINE = XPO, TRAP, RUK
END

The simulation is initially started with a small amplitude (PX=0.5), this
initial amplitude is increased in three following reruns (PX=1,2,3).
Another 3 reruns keep the initial amplitude constant (PX=0.5,0.5,0.5) but
change the integration routines (XPO,TRAP,RUK).

The output of the master run and the first 3 reruns is shown in fig. 11 .
For small initial conditions (i. e. amplitudes) the phase trajectories are
almost circles (equal scaling for x- and y-axes assumed), the system behaves
linear. With increasing amplitudes the circles are deformed, the pendulum
becomes nonlinear.

The second set of reruns is used to check the quality of the integration
routines. As the system does not loose energy, the phase trajectories must
represent closed curves (to be seen in fig. 11).
For these equations the IMPLICIT and RUK (Runge Kutta) integration routines
result in a stable solution, whereas XPO (exponential) and TRAP (trapezoidal)
diverge. But this is not a general statement. The stability of the various
integration routines depends strongly on the type of equations to be solved.

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@fig:Figure 11. Phase Trajectory of the Nonlinear Pendulum with different initial
Amplitudes