Adaptive control system

A method of adaptive control for use in a controlled system having an adjustable forward loop gain and a control system apparatus for carrying out the method are disclosed. The control system measures the value of a parameter which is to be controlled and generates an error signal by comparing the measured valve of the parameter with a desired valve. The error signal is squared and the time derivative of the squared error signal is obtained. The control system compares the length of time that the derivative so obtained has a positive value to the time it has a negative value to obtain a gain control signal, and applies the gain control signal to the forward loop gain. The gain control signal reduces the adjustable forward loop gain until a preselected margin of stability in the controlled system is obtained.

BACKGROUND OF THE INVENTION 
The present invention relates to adaptive control systems. 
Adaptive control systems are closed loop systems in which the gain of the 
system is varied according to the system's response to events that disturb 
it. While most controlled processes have response characteristics which 
vary only slightly over their entire range of operation, there are other 
systems where the operating characterics vary substantially. For example, 
a crane or the arm of a robot has different response characteristics 
depending on the weight of an object being moved. Many other controlled 
systems or processes have changing response characteristics in different 
portions of their operating range. One consequence is that a forward gain 
which provides a stable response in one portion of the operating range may 
make the system unstable in another portion of the operating range. For 
these types of controlled systems an adaptive control rather than a fixed 
gain control is desirable. 
Most adaptive control systems have depended on knowledge of the controlled 
system's response characteristic. Some adaptive controllers have applied 
white noise to the controlled system input to determine the controlled 
system's response characteristics. Other adaptive controllers have 
superimposed discrete interval binary noise on a controlled system's input 
in order to determine the controlled system's response characteristics. 
Still other adaptive controls have used a mathematical model of the 
controlled system. The model is fed control signals at the same time that 
the real system is fed the same control signals. The actual system's 
response and the model system's response are compared and any difference 
is used to adjust the actual system's forward gain until the actual system 
behaves just like the model. 
SUMMARY OF THE INVENTION 
The present invention provides an adaptive control system which does not 
require detailed knowledge of the process or mechanism being controlled. 
In fact, the only assumption about the controlled process or system is 
that reducing the forward gain of the control loop will increase system 
stability. Moreover, the adaptive control system of the present invention 
is not responsive to the magnitude of an error signal but only to the 
stability of the system's overall response. 
In carrying out the present invention an error signal of a closed loop 
control system is processed and used to vary the overall gain of the 
system. Specifically, the error signal is first squared and the resulting 
signal is differentiated. The differentiated signal is used as a trigger 
for a switching device which produces a selectable positive output 
whenever the differentiated signal is positive and a selectable negative 
output whenever the differentiated signal is negative. The output of the 
switching device is integrated and used to vary the forward gain of the 
system. 
The system may be embodied in electronic circuitry in which case the 
switching device may be a pulse width modulation circuit. By selecting the 
ratio between the positive and negative outputs of the pulse width 
modulation circuit, the damping ratio and thus the stability of the system 
may be controlled. The present invention adjusts the gain of the 
controlled system or process to maintain a predetermined margin of 
stability. Further, the pulse width modulation circuit may have a 
threshold so that it produces no output, either high or low, until the 
magnitude of the derivative signal exceeds a minimum or threshold level. 
The invention then comprises the features hereinafter fully described and 
particularly pointed out in the claims, the following description and the 
annexed drawings setting forth in detail an illustrative embodiment of the 
invention, this being indicative, however, of but one of the various ways 
in which the principles of the invention may be carried out.

DESCRIPTION OF PREFERRED EMBODIMENT 
The present invention utilizes Liapunov's Second or Direct Method of 
determining a control system's stability and corrects the gain of the 
forward loop of the controlled system to maintain a preselected margin of 
stability. This description will proceed first with an explanation of how 
and why the present invention works from a theoretical approach and then 
will describe one embodiment of an apparatus for practicing the invention. 
It will be clear from this theoretical discussion that many other 
embodiments of the present invention are possible. Specifically, although 
the embodiment illustrated and described is an electronic control system, 
it is contemplated that the present invention could be utilized in fluidic 
control systems, mechanical control systems, optical control systems, or 
any other type of control system. 
Liapunov functions have the characteristic that their derivative with 
respect to time can be used to measure the stability of any closed loop 
control system. The time derivative is negative for systems that are 
stable and positive for systems that are unstable. (Liapunov functions are 
well known to those skilled in the art and a full explanation of the 
criteria for determining that the square of the difference between the 
measured and desired values of a controlled parameter of a closed loop 
system, E.sup.2, is such a function would unnecessarily lengthen this 
application.) Liapunov's work is broadly applicable to all closed loop 
systems, however it can most easily be understood in terms of a system 
which behaves like a damped harmonic oscillator. In such a system the 
magnitude of the error signal following a step or unit input is 
EQU E=E.sub.0 e.sup.-.zeta..omega.t sin .omega.t (1) 
where: 
E=controlled system error 
E.sub.0 =initial error in controlled system 
.zeta.=damping ratio 
.omega.=natural frequency 
t=time. 
It is known that .zeta. is positive for stable systems and negative for 
unstable systems and that in designing a control system the choice of 
.zeta. controls the responsiveness of the system. If the error signal (E) 
is squared (E.sup.2) the resulting expression meets the requirements of a 
Liapunov function. 
In the case of a damped harmonic oscillator 
EQU (d(E.sup.2)/dt)=E.sub.0.sup.2 e.sup.-2.zeta..omega.t [-2.zeta..omega. 
sin.sup.2 .omega.t+2.omega. sin .omega.t cos .omega.t (2) 
An examination of this expression discloses why it may be used to measure 
the stability of a system. The terms outside the bracket on the righthand 
side of equation (2) provide an exponential envelope that decreases with 
time, making the system stable, when the damping ratio is positive. The 
envelope increases with time when the system is unstable, i.e. when the 
damping ratio is negative. The term in equation (2) inside the brackets 
has two components, the first of which is always negative when .zeta. is 
positive, i.e. when the system is stable. The second term inside the 
brackets oscillates evenly above and below the time axis. When these two 
terms are added and averaged over time, the sum is negative for .zeta.&gt;0. 
Thus whenever a system described by equation (1) is stable, the time 
integral of equation (2) will be negative and proportional to the damping 
ratio. From the above it is clear that the time averaged value of equation 
(2) is a measure of system stability. 
Again it is worth noting that for any controlled system whether linear or 
nonlinear, the difference between actual output and desired output, when 
squared, meets the requirements of a Liapunov function. Therefore although 
the above explains the operation of the Liapunov function for a damped 
harmonic oscillator, it will be appreciated that the result is the same 
for other systems, including nonlinear systems. 
It is considered desirable for an adaptive gain control to be sensitive 
only to the changing stability of the system as a whole and not to the 
magnitude of an error signal. Evaluating the lefthand side of equation (2) 
one finds that 
EQU (d(E.sup.2)/dt)=2E(dE/dt) (3) 
This expression is clearly directly proportional to the magnitude of the 
error signal. Therefore using the time integral of equation (2) to control 
the overall system gain has the disadvantage of being dependent on the 
amplitude of the error signal, E. 
This disadvantage can be overcome by utilizing a pulse width modulation 
circuit. Such a circuit has two output states, one positive and one 
negative. Such a circuit may be switched between output states as the 
value of equation (2) changes sign. The result is a series of square wave 
pulses alternating between predetermined positive and negative values and 
whose width is proportional to the length of time that the time derivative 
of the error signal squared, i.e. equation (2), is positive or negative, 
respectively. The resulting train of pulses is independent of the 
magnitude of the error signal. 
If the output train of square waves from the pulse width modulation circuit 
is integrated over time, the result is a signal either positive or 
negative proportional to the time integral of the time derivative of the 
error signal squared (.intg.d(E.sup.2)/dt dt) but completely independent 
of the magnitude of the error signal (E). This signal is a measure of 
system stability, i.e. it is proportional to .zeta., the damping ratio, 
and it can be used to adjust the gain of the controlled process to 
maintain stability. For example, if in response to some input to the 
controlled process, the integrated output of the pulse width modulation 
circuit is positive, indicating that the system is unstable, the gain of 
the controlled process is adjusted downward until the integrated output of 
the pulse width modulation circuit is zero. 
A system operating as just described would always operate on the very 
margin of stability, with a damping ratio of zero. It is usually 
preferable to operate controlled processes with some margin of stability, 
i.e. with the damping ratio having some relatively small positive value, 
usually between 0.5 and 1.5. This can be accomplished in a number of ways. 
One way would be to add a selectable amount to the integrated output of 
the pulse width modulation circuit. This would provide a margin of 
stability related to the selected additional amount. Another way, as will 
be discussed more fully below, is to vary the ratio between the amplitudes 
of the positive and negative outputs of the pulse width modulation 
circuit. This provides any desired specific margin of stability, although 
the damping ratio and the ratio between the outputs are related not 
linearly but by a transcendental function. 
The invention described above may be carried out in the embodiment shown in 
FIG. 1. The system 10 includes a controlled system or process 12 which may 
be a mechanical system such as a position control, possibly a wing surface 
on a missile or aircraft or the arm of a robot which must work in varying 
environments such as in and out of water. The controlled system 12 could 
also be a process such as a chemical process where the chemical properties 
at the output of the process are measured and the controlled variable may 
be a flow control valve, a reactant temperature or reaction mixer speed 
among others. 
The controlled system 12 has input 14 which controls some input parameter 
of the system 12 to establish or maintain an output parameter or response 
at a selected value. This value may represent a particular physical 
relationship between two parts, a chemical composition or concentration of 
an effluent from a chemical process, or almost any other measurable 
physical quantity. The value of the parameter to be controlled is measured 
and fed back to the input through back line 16. 
A comparator 18 (shown as a summing junction) measures the difference 
between input 14, representing the desired value of the controlled 
parameter and the actual value of the controlled parameter transmitted 
from feed back line 16. The resultant error signal, E, is fed through 
amplifier 20 to operate system 12. 
The amplifier 20 may be a conventional electronic amplifier in which the 
output is generally proportional to the input. However the term amplifier, 
as used in this specification and the claims that follow also encompasses 
the broader concept of any system or group of components which receive an 
input signal and produce a larger or modified output signal. One example 
is a hydraulic system which includes a source of fluid under pressure and 
a proportional flow control valve. Such a system, taken as a whole, has a 
gain which may be controlled according to the precepts of the present 
invention. In the example given, the gain could be varied by adjusting the 
pressure upstream of the proportional flow control valve. 
The response of system 12, as indicated by the error signal, E, to a 
unitary or step input may take the shape of curve A in FIG. 2. The 
controlled system 12 and the feed back loop 16, comparator 18 and 
amplifier 20 together form a conventional feed back control system where 
the gain of amplifier 20 controls the damping ratio of the system and thus 
the system stability. The only requirement to guarantee applicability of 
the present invention to controlling system 12 is that the stability of 
system 10 may be increased by decreasing the gain of amplifier 20. Very 
many systems of practical interest meet this criterion. 
The adaptive control of the present invention varies the gain of amplifier 
20 in response to the error signal (E) so that the system 10 is maintained 
at a consistent, selectable margin of stability. To this end the system 10 
also includes an amplifier 22 which receives the error signal through 
wires 24 and 26 and squares the error signal E. The resultant (E.sup.2) is 
shown, for example, as curve B in FIG. 2. The E.sup.2 signal is then 
differentiated with respect to time by differentiator 28 which also 
inverts the signal. The differentiator is selected with a break frequency 
one or two orders of magnitude greater than that of any other element of 
the controlled system 12 so that the differentiator's own response does 
not affect system performance. Curve C of FIG. 2 shows the negative of the 
derivative of the error signal squared with respect to time. The output of 
the differentiator 28 is a signal which, as shown in FIG. 2, passes above 
and below zero and which spends more time on the positive side of zero 
when the system 10 is stable and therefore has a positive damping ratio. 
The output of differentiator 28 is fed to a pulse width modulation (PWM) 
circuit 30. This circuit has two levels of output, one a positive voltage 
B and the other a negative voltage A. The output of the PWM circuit 30 is 
A whenever the output of the differentiator 28 is negative, and the output 
of the PWM circuit 30 is B whenever the output of the differentiator 28 is 
positive. The resulting train of square wave pulses is shown as curve D in 
FIG. 2. 
The PWM circuit 30 also has a dead band during which its output is zero. 
The PWM circuit does not respond at all unless the absolute value of the 
signal received from differentiator 28 exceeds some minimum value. The 
dead band is shown by the phantom lines crossing the curve C in FIG. 2 and 
by the horizontal portions of curve D which coincide with the X axis of 
that curve. This dead band is effective to prevent changes in system gain 
at amplifier 20 unless the stability of the system has changed more than a 
predetermined minimum amount. 
The output from the PWM circuit 30 is fed to an integrator 32 whose output 
represents the sum of the area under the curve D of FIG. 4, i.e., positive 
areas minus negative areas. The integrator 32 is selected with a break 
frequency approximately one-half the break frequency of the controlled 
system 12. This assures that the integration occurs over a sufficient 
length of time to reflect accurately the average of the train of square 
wave pulses. When the output of integrator 32 is positive, the system 10 
is stable, and when the output of integrator 32 is negative, the system is 
unstable. The degree above or below zero of the output of integrator 32 is 
a measure of the degree of stability or instability of the system 10. The 
output of the integrator 32 is fed through a limiter 34 which limits the 
maximum and minimum values of the gain of amplifier 20. 
In response to a change in input 14, an error signal will be generated. The 
oscillations of the error signal may indicate that the stability of the 
system is decreasing. In this case the output of integrator 32 will 
decrease because of the increased amount of time the output of 
differentiator 28 spends below the zero axis. This decrease in turn will 
decrease the gain of amplifier 20. The gain of amplifier 20 will continue 
to fall until the time derivative output spends as much time in the 
positive region as in the negative region. If the oscillations of the 
error signal indicate increasing stability, the gain of amplifier 20 would 
be increased. 
The PWM circuit 30 also serves to establish a margin of stability for the 
system 10. Specifically, the ratio of the values of A and B as outputs of 
the PWM circuit 30 controls the margin of stability by controlling the 
damping ratio of the system 10. As can be seen from FIG. 2 the output of 
the PWM circuit 30 has a wave form that is repeated every 180.degree. of 
rotation of the vector .omega.t. This repeating unit has two portions, the 
positive one having a width 90.degree.+.delta., where .delta. is an offset 
angle and the negative one having a width 90.degree.-.delta.. For a single 
repeating unit, the area above the curve is B(90.degree.+.delta.), and the 
area below the curve is A(90.degree.-.delta.). The adaptive control 
circuit changes the gain of amplifier 20 until 
EQU B(90.degree.+.delta.)=A(90.degree.-.delta.) (4) 
The relationship between .delta. and .zeta. can be relatively easily 
derived. When 
EQU .omega.t=90.degree.-.delta. (5) 
equation (2)=0. So at that time 
EQU 2.zeta..omega. sin.sup.2 .omega.t=2.omega. sin .omega.t cos .omega.t (6) 
This reduces to 
EQU .omega.t=tan.sup.-1 1/.zeta. (7) 
Substituting Equation (5) into Equation (7) and solving for yields 
EQU .delta.=tan.sup.-1 .zeta. (8) 
Therefore, when .delta.=0, .zeta.=0, which represents a condition of zero 
stability and which is therefore usually very undesirable. Equations (4) 
and (8) can be combined to obtain 
EQU tan.sup.-1 .zeta.=90.degree.(A-B/A+B) (9) 
Equation (9) shows that any desired damping ratio can be obtained by 
varying A and B. The gain of amplifier 20 will still be changed until the 
areas above and below the zero axis of curve D (FIG. 2) are equal, but 
this equality will occur at some desired stability margin represented by a 
selected positive damping ratio. 
As noted above the control system 10 was described as an electronic system. 
However it should be clear that components of other types may be used. For 
example, but not by way of limitation, fluidic components which carry out 
each of the functions required by the block diagram of FIG. 1 are known, 
and the system could be carried out using such fluidic components. Further 
the system could be carried out using mechanical components, or optical 
components or any combination of such components as may be suited to the 
particular system 12 being controlled and its environment. 
Thus the adaptive control system of the present invention varies the 
forward gain of a controlled system 12 to maintain a desired margin of 
stability. Unlike other adaptive control systems, no detailed knowledge of 
the frequency response of controlled system 12 is required. Instead, the 
error signal squared (E.sup.2), which satisfies the requirements of a 
Liapunov function, is used to measure system stability. Specifically, the 
error signal squared is differentiated with respect to time, and the 
derivative so obtained is used to trigger a pulse width modulation (PWM) 
circuit 30 which has a fixed positive output whenever the time derivative 
is positive and a fixed negative output whenever the time derivative is 
negative. The output of the PWM circuit 30 is integrated by integrator 32, 
and the result is a signal representing the difference between the length 
of time the time derivative of E.sup.2 is positive and the length of time 
the time derivative of E.sup.2 is negative, and thus under the Liapunov 
theory is a measure of system stability. The output of integrator 32 is 
used to control the forward loop gain, returning the controlled system 12 
to a margin of stability determined by the values of the fixed positive 
and negative outputs of the PWM circuit 30.