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Claims  |
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I claim:
1. Apparatus for determining open-loop parameters for a First
Configuration, multiple-input, multiple output, closed-loop control
system, the apparatus comprising:
a feedback loop having a difference module, a controller module, a plant
module and a sensor module arranged in that order in a feedback loop that
operates in its linear operating region, the sensor module having sensor
dynamics represented by a matrix H of dimensions n.sub.1 .times.n.sub.2,
the controller module having controller dynamics represented by a matrix C
of dimensions n.sub.3 .times.n.sub.1, and the plant module having physical
system dynamics represented by a matrix P of dimensions n.sub.2
.times.n.sub.3, where n.sub.1, n.sub.2 and n.sub.3 are selected positive
integers;
the difference module having a first input terminal that receives an array
R of n.sub.1 predetermined input signals, having a second input terminal
that receives an array Z of n.sub.1 input signals, and having an output
terminal, where the difference module forms and issues at its output
terminal a difference signal array E=R-Z;
the controller module receiving the vector array E of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
U=CE of n.sub.3 intermediate output signals at an output terminal thereof;
the plant module receiving the vector array U of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
Y=PU of n.sub.2 output signals at an output terminal thereof;
the sensor module receiving the vector array Y of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
Z=HY of n.sub.1 signals at an output terminal thereof;
measurement means connected to the feedback loop for measuring the array Z
of signals received in response to provision of the array R of n.sub.1
signals; and
computation means for receiving measurements of the array Z of n.sub.1
signals from the measurement means, for determining the entries of a
matrix T, of dimensions n.sub.1 .times.n.sub.1 and representing the closed
loop response of the feedback loop, by solving the matrix identity Z=TR,
for forming a quasi-inverse matrix (I-T).sup. , for determining the
entries of a matrix HPC or HP.
2. The apparatus of claim 1, wherein said measurement means and said
computation means comprise:
a control system signal analyzer, connected to said feedback loop, that
injects said array of signals R and measures said array of signals Z and
said array of signals R to form said matrix T; and
a multivariable work station, including a digital computer, a digital
signal processor, a mathematics module and an interface between the
control system signal analyzer and the remainder of the multivariable work
station, that receives and temporarily stores said arrays of signals R and
Z as measured by the control system signal analyzer, and that determines
the entries of said matrices T and (I-T).sup. , the entries of said matrix
C.sup. , and the entries of said matrix HP by the relation HP=(I-T).sup.
TC.sup. =T(I-T).sup. C.sup. .
3. Apparatus for determining open-loop parameters for a First
Configuration, multiple-input, multiple output, closed-loop control
system, the apparatus comprising:
a feedback loop having a difference module, a controller module, a plant
module and a sensor module arranged in that order in a feedback loop that
operates in its linear operating region, the sensor module having sensor
dynamics represented by a matrix H of dimensions n.sub.1 .times.n.sub.2,
the controller module having controller dynamics represented by a matrix C
of dimensions n.sub.3 .times.n.sub.1, and the plant module having physical
system dynamics represented by a matrix P of dimensions n.sub.2
.times.n.sub.3, where n.sub.1, n.sub.2 and n.sub.3 are selected positive
integers;
the difference module having a first input terminal that receives an array
R of n.sub.1 predetermined input signals, having a second input terminal
that receives an array Z of n.sub.1 input signals, and having an output
terminal, where the difference module forms and issues at its output
terminal a difference signal array E=R-Z;
the controller module receiving the vector array E of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
U=CE of n.sub.3 intermediate output signals at an output terminal thereof;
the plant module receiving the vector array U of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
Y=PU of n.sub.2 output signals at an output terminal thereof;
the sensor module receiving the vector array Y of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
Z=HY of n.sub.1 signals at an output terminal thereof;
measurement means connected to the feedback loop for measuring the array E
of signals received in response to provision of the array R of n.sub.1
signals; and
computation means for receiving measurements of the array E of n.sub.1
signals from the measurement means, for determining the entries of a
matrix S of dimensions n.sub.1 .times.n.sub.1 by solving the matrix
identity E=SR, for forming a quasi-inverse C.sup. of the matrix C and a
quasi-inverse matrix S.sup. of the matrix S, and for determining the
entries of a matrix HP or HPC.
4. The apparatus of claim 3, wherein said measurement means and said
computation means comprise:
a control system signal analyzer, connected to said feedback loop, that
injects said array of signals R and measures said array of signals E and
said array of signals R to form said matrix S; and
a multivariable work station, including a digital computer, a digital
signal processor, a mathematics module and an interface between the
control system signal analyzer and the remainder of the multivariable work
station, that receives and temporarily stores said arrays of signals R and
E as measured by the control system signal analyzer, and that determines
the entries of said matrices S.sup. , the entries of said matrix C.sup. ,
and the entries of said matrix HP by the relation HP=C.sup. .
5. Apparatus for determining open-loop parameters for a First
Configuration, multiple-input, multiple output, closed-loop control
system, the apparatus comprising:
a feedback loop having a difference module, a controller module, a plant
module and a sensor module arranged in that order in a feedback loop that
operates in its linear operating region, the sensor module having sensor
dynamics represented by a matrix H of dimensions n.sub.1 .times.n.sub.2,
the controller module having controller dynamics represented by a matrix C
of dimensions n.sub.3 .times.n.sub.1, and the plant module having physical
system dynamics represented by a matrix P of dimensions n.sub.2
.times.n.sub.3, where n.sub.1, n.sub.2 and n.sub.3 are selected positive
integers;
the difference module having a first input terminal that receives an array
R of n.sub.1 predetermined input signals, having a second input terminal
that receives an array Z of n.sub.1 input signals, and having an output
terminal, where the difference module forms and issues at its output
terminal a difference signal array E=R-Z;
the controller module receiving the vector array E of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
U=CE of n.sub.3 intermediate output signals at an output terminal thereof;
the plant module receiving the vector array U of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
Y=PU of n.sub.2 output signals at an output terminal thereof;
the sensor module receiving the vector array Y of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
Z=HY of n.sub.1 signals at an output terminal thereof;
measurement means connected to the feedback loop for measuring the array U
of signals received in response to provision of the array R of n.sub.1
signals; and
computation means for receiving measurements of the array U of n.sub.3
signals from the measurement means, for determining the entries of a
matrix V of dimensions n.sub.3 .times.n.sub.1 by solving the matrix
identity U=VR, for forming a quasi-inverse C.sup. of the matrix C and a
quasi-inverse (C.sup. V).sup. of the matrix C.sup. V, and for determining
the entries of a matrix HP or HPC or CHP.
6. The apparatus of claim 5, wherein said measurement means and said
computation means comprise:
a control system signal analyzer, connected to said feedback loop, that
injects said array of signals R and measures said array of signals U and
said array of signals R to form said matrix V; and
a multivariable work station, including a digital computer, a digital
signal processor, a mathematics module and an interface between the
control system signal analyzer and the remainder of the multivariable work
station, that receives and temporarily stores said arrays of signals R and
U as measured by the control system signal analyzer, and that determines
the entries of said matrix C.sup. , the entries of said matrix ((VC.sup.
).sup. , and the entries of said matrix HP by the relation HP=C.sup.
C.sup. .
7. Apparatus for determining open-loop parameters for a First
Configuration, multiple-input, multiple output, closed-loop control
system, the apparatus comprising:
a feedback loop having a difference module, a controller module, a plant
module and a sensor module arranged in that order in a feedback loop that
operates in its linear operating region, the sensor module having sensor
dynamics represented by a matrix H of dimensions n.sub.1 .times.n.sub.2,
the controller module having controller dynamics represented by a matrix C
of dimensions n.sub.3 .times.n.sub.1, and the plant module having physical
system dynamics represented by a matrix P of dimensions n.sub.2
.times.n.sub.3, where n.sub.1, n.sub.2 and n.sub.3 are selected positive
integers;
the difference module having a first input terminal that receives an array
R of n.sub.1 predetermined input signals, having a second input terminal
that receives an array Z of n.sub.1 input signals, and having an output
terminal, where the difference module forms and issues at its output
terminal a difference signal array E=R-Z;
the controller module receiving the vector array E of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
U=CE of n.sub.3 intermediate output signals at an output terminal thereof;
the plant module receiving the vector array U of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
Y=PU of n.sub.2 output signals at an output terminal thereof;
the sensor module receiving the vector array Y of signals at an input
terminal thereof and, in response thereto, forming and issuing an array
Z=HY of n.sub.1 signals at an output terminal thereof;
measurement means connected to the feedback loop for measuring the array Y
of signals received in response to provision of the array R of n.sub.1
signals; and
computation means for receiving measurements of the array Y of n.sub.2
signals from the measurement means, for determining the entries of a
matrix W of dimensions n.sub.2 .times.n.sub.1 by solving the matrix
identity Y=WR, for forming a quasi-inverse C.sup. of the matrix C and a
quasi-inverse (I-HW).sup. of the matrix I-HW, and for determining the
entries of a matrix HP or HPC.
8. The apparatus of claim 7, wherein said measurement means and said
computation means comprise:
a control system signal analyzer, connected to said feedback loop, that
injects said array of signals R and measures said array of signals Y and
said array of signals R to form said matrix W; and
a multivariable work station, including a digital computer, a digital
signal processor, a mathematics module and an interface between the
control system signal analyzer and the remainder of the multivariable work
station, that receives and temporarily stores said arrays of signals R and
Y as measured by the control system signal analyzer, and that determines
the entries of said matrices HW and (I-HW).sup. , the entries of said
matrix C.sup. , and the entries of said matrix HP by the relation
HP=HW(I-HW).sup. C.sup. .
9. Apparatus for determining open-loop parameters for a Second
Configuration, multiple-input, multiple output, closed-loop control
system, the apparatus comprising:
a feedback loop having a difference module, a controller module, a sum
module and a plant module arranged in that order in a feedback loop that
operates in its linear operating region, the controller module having
controller dynamics represented by a matrix C of dimensions n.sub.2
.times.n.sub.1, and the plant module having physical system dynamics
represented by a matrix P of dimensions n.sub.1 .times.n.sub.2, where
n.sub.1 and n.sub.2 are predetermined positive integers;
the difference module having a first input terminal that receives an array
R.sub.1 of n.sub.1 predetermined input signals and having a second input
terminal that receives an array Y.sub.2 of signals, with the difference
module forming and issuing at an output terminal thereof a difference
array of signals E.sub.1 =R.sub.1 -Y.sub.2 ;
the controller module receiving the array E.sub.l at an input terminal
thereof and, in response thereto, forming and issuing an array Y.sub.1
=CE.sub.1 of n.sub.2 intermediate output signals at an output terminal
thereof;
the sum module having a first input terminal to receive an array R.sub.2 of
n.sub.2 predetermined input signals and having a second input terminal to
receive the array Y.sub.1 of intermediate output signals, with the sum
module forming and issuing a sum array of signals E.sub.2 =R.sub.2
+Y.sub.1 ;
the plant module receiving the vector array of signals E.sub.2 at an input
terminal thereof and, in response thereto, forming and issuing an array
Y.sub.2 =PE.sub.2 of n.sub.1 output signals at an output terminal thereof;
measurement means connected to the feedback loop for measuring the arrays
Y.sub.1 and Y.sub.2 of signals received in response to provision of the
arrays R.sub.1 and R.sub.2 of signals; and
computation means for receiving measurements of the arrays Y.sub.1 and
Y.sub.2 of signals from the measurement means, for determining the entries
of at least one of the matrices T=T.sub.1, T.sub.2, T.sub.3 and T.sub.4 of
dimensions n.sub.2 .times.n.sub.1, n.sub.2 .times.n.sub.2, n.sub.1
.times.n.sub.1 and n.sub.1 .times.n.sub.2, respectively, by solving the
matrix identity
##EQU8##
for these entries, and for determining the entries of P or PC or CP from
the entries of the matrix T.
10. The apparatus of claim 9, wherein T=T.sub.1 and said measurement means
and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, Y.sub.1 and Y.sub.2 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, Y.sub.1 and
Y.sub.2 as measured by the signal analyzer, and that determines the
entries of said matrices T.sub.1 and T.sub.1.sup. , the entries of said
matrix C.sup. , and the entries of said matrix P by the relation
P=T.sub.1.sup. -C.sup. .
11. The apparatus of claim 9, wherein T=T.sub.2 and said measurement means
and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, Y.sub.1 and Y.sub.2 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, Y.sub.1 and
Y.sub.2 as measured by the signal analyzer, and that determines the
entries of said matrices T.sub.2 and (I+T.sub.2).sup. , the entries of
said matrix C.sup. , and the entries of said matrix P by the relation
P=-C.sup. T.sub.2 (I+T.sub.2).sup. =-C.sup. (I+T.sub.2).sup. T.sub.2.
12. The apparatus of claim 9, wherein T=T.sub.3 and said measurement means
and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, Y.sub.1 and Y.sub.2 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, Y.sub.1 and
Y.sub.2 as measured by the signal analyzer, and that determines the
entries of said matrices T.sub.3 and (I-T.sub.3).sup. , the entries of
said matrix C.sup. , and the entries of said matrix P by the relation
P=T.sub.3 (I-T.sub.3).sup. C.sup. =(I-T.sub.3).sup. T.sub.3 C.sup. .
13. The apparatus of claim 9, wherein T=T.sub.4 and said measurement means
and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, Y.sub.1 and Y.sub.2 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, Y.sub.1 and
Y.sub.2 as measured by the signal analyzer, and that determines the
entries of said matrices T.sub.4 and (I-CT.sub.4).sup. , the entries of
said matrix C.sup. , and the entries of said matrix P by the relation
P=T.sub.4 (I-CT.sub.4).sup. =(I-T.sub.4 C).sup. T.sub.4.
14. Apparatus for determining open-loop parameters for a Second
Configuration, multiple-input, multiple output, closed-loop control
system, the apparatus comprising:
a feedback loop having a difference module, a controller module, a sum
module and a plant module arranged in that order in a feedback loop that
operates in its linear operating region, the controller module having
controller dynamics represented by a matrix C of dimensions n.sub.2
.times.n.sub.1, and the plant module having physical system dynamics
represented by a matrix P of dimensions n.sub.1 .times.n.sub.2, where
n.sub.1 and n.sub.2 are predetermined positive integers;
the difference module having a first input terminal that receives an array
R.sub.1 of n.sub.1 predetermined input signals and having a second input
terminal that receives an array Y.sub.2 of n.sub.1 signals, with the
difference module forming and issuing at an output terminal thereof a
difference array of signals E.sub.1 =R.sub.1 -Y.sub.2 ;
the controller module receiving the array E.sub.1 at an input terminal
thereof and, in response thereto, forming and issuing an array Y.sub.1
=CE.sub.1 of n.sub.2 intermediate output signals at an output terminal
thereof;
the sum module having a first input terminal to receive an array R.sub.2 of
n.sub.2 predetermined input signals and having a second input terminal to
receive the array Y.sub.1 of intermediate output signals, with the sum
module forming and issuing a sum array of signals E.sub.2 =R.sub.2
+Y.sub.1 ;
the plant module receiving the array of signals E.sub.2 at an input
terminal thereof and, in response thereto, forming and issuing an array
Y.sub.2 =PE.sub.2 of n.sub.1 output signals at an output terminal thereof;
measurement means connected to the feedback loop for measuring the arrays
E.sub.l and E.sub.2 of signals received in response to provision of the
arrays R.sub.1 and R.sub.2 of signals; and
computation means for receiving measurements of the arrays E.sub.1 and
E.sub.2 of signals from the measurement means, for determining the entries
of at least one of the matrices S=S.sub.1, S.sub.2, S.sub.3 and S.sub.4 of
dimensions n.sub.1 .times.n.sub.1, n.sub.1 .times.n.sub.2, n.sub.2
.times.n.sub.1 and n.sub.2 .times.n.sub.2, respectively, by solving the
matrix identity
##EQU9##
for these entries, and for determining the entries of P or CP or PC from
the entries of the matrix S.
15. The apparatus of claim 14, wherein S=S.sub.1 and said measurement means
and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, E.sub.1 and E.sub.2 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, E.sub.1 and
E.sub.2 as measured by the signal analyzer, and that determines the
entries of said matrices S.sub.1 and S.sub.1.sup. , the entries of said
matrix C.sup. , and the entries of said matrix P by the relation
P=(S.sub.1.sup. -I)C.sup. .
16. The apparatus of claim 14, wherein S=S.sub.2 and said measurement means
and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, E.sub.1 and E.sub.2 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, E.sub.1 and
E.sub.2 as measured by the signal analyzer, and that determines the
entries of said matrices S.sub.2 and (I+CS.sub.2).sup. , the entries of
said matrix C.sup. , and the entries of said matrix P by the relation
P=-S.sub.2 (I+CS.sub.2).sup. =-(I+S.sub.2 C).sup. S.sub.2.
17. The apparatus of claim 14, wherein S=S.sub.3 and said measurement means
and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, E.sub.1 and E.sub.2 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, E.sub.1 and
E.sub.2 as measured by the signal analyzer, and that determines the
entries of said matrices S.sub.3 and S.sub.3.sup. , the entries of said
matrix C.sup. , and the entries of said matrix P by the relation
P=S.sub.3.sup. -C.sup. .
18. The apparatus of claim 14, wherein S=S.sub.4 and said measurement means
and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, E.sub.1 and E.sub.2 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, E.sub.1 and
E.sub.2 as measured by the signal analyzer, and that determines the
entries of said matrices S.sub.4 and S.sub.4.sup. , the entries of said
matrix C.sup. , and the entries of said matrix P by the relation P=C.sup.
(S.sub.4.sup. -I).
19. Apparatus for determining open-loop parameters for a Third
Configuration, multiple-input, multiple output, closed-loop control
system, the apparatus comprising:
a feedback loop having a difference module, a controller module, a first
sum module a plant module, a second sum module and a sensor module
arranged in that order in a feedback loop that operates in its linear
operating region, the controller module having controller dynamics
represented by a matrix C of dimensions n.sub.2 .times.n.sub.1, the plant
module having physical system dynamics represented by a matrix P of
dimensions n.sub.3 .times.n.sub.2, and the sensor module having sensor
dynamics represented by a matrix H of dimensions n.sub.1 .times.n.sub.3,
where n.sub.1, n.sub.2 and n.sub.3 are predetermined positive integers;
the difference module having a first input terminal that receives an array
R.sub.1 of n.sub.1 predetermined input signals and having a second input
terminal that receives an array Y.sub.3 of n.sub.1 signals, with the
difference module forming and issuing at an output terminal thereof a
difference array of signals E.sub.1 =R.sub.1 -Y.sub.3 ;
the controller module receiving the array E.sub.1 at an input terminal
thereof and, in response thereto, forming and issuing an array Y.sub.1
=CE.sub.1 of n.sub.2 intermediate output signals at an output terminal
thereof;
the first sum module having a first input terminal to receive an array
R.sub.2 of n.sub.2 predetermined input signals and having a second input
terminal to receive the array Y.sub.1 of intermediate output signals, with
the sum module forming and issuing a sum array of signals E.sub.2 =R.sub.2
+Y.sub.1 ;
the plant module receiving the array of signals E.sub.2 at an input
terminal thereof and, in response thereto, forming and issuing an array
Y.sub.2 =PE.sub.2 of n.sub.3 output signals at an output terminal thereof;
the second sum module having a first input terminal to receive an array
R.sub.3 of n.sub.3 predetermined input signals and having a second input
terminal to receive the array Y.sub.2 of intermediate output signals, with
the sum module forming and issuing a sum array of signals E.sub.3 =R.sub.3
+Y.sub.2 ;
the sensor module receiving the array of signals E.sub.3 at an input
terminal thereof and, in response thereto, forming and issuing an array
Y.sub.3 =HE.sub.3 of n.sub.1 output signals at an output terminal thereof;
measurement means connected to the feedback loop for measuring the arrays
E.sub.1, E.sub.2 and E.sub.3 of signals received in response to provision
of signal arrays R.sub.1, R.sub.2 and R.sub.3 ; and
computation means for receiving measurements of the arrays E.sub.1, E.sub.2
and E.sub.3 of signals from the measurement means, for determining the
entries of at least one of the matrices S=S.sub.11, S.sub.12, S.sub.13,
S.sub.21, S.sub.22, S.sub.23, S.sub.31, S.sub. 32 and S.sub.33 of
dimensions n.sub.1 .times.n.sub.1, n.sub.1 .times.n.sub.2, n.sub.1
.times.n.sub.3, n.sub.2 .times.n.sub.1, n.sub.2 .times.n.sub.2, n.sub.2
.times.n.sub.3, n.sub.3 .times.n.sub.1, n.sub.3 .times.n.sub.2 and n.sub.3
.times.n.sub.3, respectively, by solving the matrix identity
##EQU10##
for these entries, and for determining the entries of P or HP or CHP or
PCH or HCP from the entries of the matrix S.
20. The apparatus of claim 19, wherein S=S.sub.11 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, E.sub.1, E.sub.2 and E.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
E.sub.1, E.sub.2 and E.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices S.sub.11 and S.sub.11.sup. , the
entries of said matrix C.sup. , and the entries of said matrix HP by the
relation HP=(S.sub.11.sup. -I)C.sup. .
21. The apparatus of claim 19, wherein S=S.sub.12 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, E.sub.1, E.sub.2 and E.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
E.sub.1, E.sub.2 and E.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices S.sub.12 and (I+CS.sub.12).sup. ,
the entries of said matrix C.sup. , and the entries of said matrix HP by
the relation HP=-S.sub.12 (I+CS.sub.12).sup. =-(I+S.sub.12 C).sup.
S.sub.12.
22. The apparatus of claim 19, wherein S=S.sub.13 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, E.sub.1, E.sub.2 and E.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
E.sub.1, E.sub.2 and E.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices S.sub.13 and HS.sub.13.sup. , the
entries of said matric C.sup. , and the entries of said matrix HP by the
relation HP=-(I+HS.sub.13.sup. )C.sup. .
23. The apparatus of claim 19, wherein S=S.sub.21 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, E.sub.1, E.sub.2 and E.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
E.sub.1, E.sub.2 and E.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices S.sub.21 and S.sub.21.sup. , the
entries of said matrix C.sup. , and the entries of said matrix HP by the
relation HP=S.sub.21.sup. -C.sup. .
24. The apparatus of claim 19, wherein S=S.sub.22 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, E.sub.1, E.sub.2 and E.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
E.sub.1, E.sub.2 and E.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices S.sub.22 and S.sub.22.sup. , the
entries of said matrix C.sup. , and the entries of said matrix HP by the
relation HP=C.sup. (S.sub.22.sup. -I).
25. The apparatus of claim 19, wherein S=S.sub.23 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, E.sub.1, E.sub.2 and E.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
E.sub.1, E.sub.2 and E.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices S.sub.23 and S.sub.23.sup. , the
entries of said matrix (CH).sup. , and the entries of said matrix P by the
relation P=-S.sub.23.sup. -(CH).sup. .
26. The apparatus of claim 19, wherein S=S.sub.31 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, E.sub.1, E.sub.2 and E.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
E.sub.1, E.sub.2 and E.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices S.sub.31 and (I-HS.sub.31).sup.
or (I-S.sub.31 H).sup. , the entries of said matrix C.sup. , and the
entries of said matrix P by the relation P=(I-S.sub.31 H).sup. S.sub.31
C.sup. =S.sub.31 (I-HS.sub.31).sup. C.sup. .
27. The apparatus of clain 19, wherein S=S.sub.32 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, E.sub.1, E.sub.2 and E.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
E.sub.1, E.sub.2 and E.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices S.sub.32 and (I-S.sub.32 CH).sup.
or (I-CHS.sub.32).sup. , the entries of said matrix C.sup. , and the
entries of said matrix P by the relation P=(I-S.sub.32 CH).sup. S.sub.32
=S.sub.32 (I-CHS.sub.32).sup. .
28. The apparatus of claim 19, wherein S=S.sub.33 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, E.sub.1, E.sub.2 and E.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
E.sub.1, E.sub.2 and E.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices S.sub.33 and S.sub.33.sup. , the
entries of said matrix (CH).sup. , and the entries of said matrix P by the
relation P=(S.sub.33.sup. -I)(CH).sup. .
29. Apparatus for determining open-loop parameters for a Third
Configuration, multiple-input, multiple output, closed-loop control
system, the apparatus comprising:
a feedback loop having a difference module, a controller module, a first
sum module a plant module, a second sum module and a sensor module
arranged in that order in a feedback loop that operates in its linear
operating region, the controller module having controller dynamics
represented by a matrix C of dimensions n.sub.2 .times.n.sub.1, the plant
module having physical system dynamics represented by a matrix P of
dimensions n.sub.3 .times.n.sub.2, and the sensor module having sensor
dynamics represented by a matrix H of dimensions n.sub.1 .times.n.sub.3,
where n.sub.1, n.sub.2 and n.sub.3 are predetermined positive integers;
the difference module having a first input terminal that receives an array
R.sub.1 of n.sub.1 predetermined input signals and having a second input
terminal that receives an array Y.sub.3 of n.sub.1 signals, with the
difference module forming and issuing at an output terminal thereof a
difference array of signals E.sub.1 =R.sub.1 -Y.sub.3 ;
the controller module receiving the array E.sub.1 at an input terminal
thereof and, in response thereto, forming and issuing an array Y.sub.1
=CE.sub.1 of n.sub.2 intermediate output signals at an output terminal
thereof;
the first sum module having a first input terminal to receive an array
R.sub.2 of n.sub.2 predetermined input signals and having a second input
terminal to receive the array Y.sub.1 of intermediate output signals, with
the sum module forming and issuing a sum array of signals E.sub.2 =R.sub.2
+Y.sub.1 ;
the plant module receiving the array of signals E.sub.2 at an input
terminal thereof and, in response thereto, forming and issuing an array
Y.sub.2 =PE.sub.2 of n.sub.3 output signals at an output terminal thereof;
the second sum module having a first input terminal to receive an array
R.sub.3 of n.sub.3 predetermined input signals and having a second input
terminal to receive the array Y.sub.2 of intermediate output signals, with
the sum module forming and issuing a sum array of signals E.sub.3 =R.sub.3
+Y.sub.2 ;
the sensor module receiving the array of signals E.sub.3 at an input
terminal thereof and, in response thereto, forming and issuing an array
Y.sub.3 =HE.sub.3 of n.sub.1 output signals at an output terminal thereof;
measurement means connected to the feedback loop for measuring the arrays
E.sub.1, E.sub.2 and E.sub.3 of signals received in reponse to provision
of signal arrays R.sub.1, R.sub.2 and R.sub.3 ; and
computation means for receiving measurements of the arrays Y.sub.1, Y.sub.2
and Y.sub.3 of signals from the measurement means, for determining the
entries of at least one of the matrices T=T.sub.11, T.sub.12, T.sub.13,
T.sub.21, T.sub.22, T.sub.23, T.sub.31, T.sub.32 and T.sub.33 of
dimensions n.sub.2 .times.n.sub.1, n.sub.2 .times.n.sub.2, n.sub.2
.times.n.sub.3, n.sub.3 .times.n.sub.1, n.sub.3 .times.n.sub.2, n.sub.3
.times.n.sub.3, n.sub.1 .times.n.sub.1, n.sub.1 .times.n.sub.2 and n.sub.1
.times.n.sub.3, respectively, by solving the matrix identity
##EQU11##
for these entries, and for determining the entries of P or HP or or HPC or
CHP or PCH from the entries of the matrix T.
30. The apparatus of claim 29, wherein T=T.sub.11 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, Y.sub.1, Y.sub.2 and Y.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
Y.sub.1 Y.sub.2 and Y.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices T.sub.11 and T.sub.11.sup. , the
entries of said matrix C.sup. , and the entries of said matrix HP by the
relation HP=T.sub.11.sup. -C.sup. .
31. The apparatus of claim 29, wherein T=T.sub.12 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, Y.sub.1, Y.sub.2 and Y.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
Y.sub.1, Y.sub.2 and Y.sub.3 as measured by the signal analyzer, and that
that determines the entries of said matrices T.sub.12 and
(I+T.sub.12).sup. , the entries of said matrix C.sup. , and the entries of
said matrix HP by the relation HP=-C.sup. (I+T.sub.12).sup. T.sub.12
=-C.sup. T.sub.12 (I+T.sub.12).sup. .
32. The apparatus of claim 29, wherein T=T.sub.13 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, Y.sub.1, Y.sub.2 and Y.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
Y.sub.1, Y.sub.2 and Y.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices T.sub.13 and T.sub.13.sup. , the
entries of said matrix (CH).sup. , and the entries of said matrix P by the
relation P=-T.sub.13.sup. -(CH).sup. .
33. The apparatus of claim 29, wherein T=T.sub.21 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, Y.sub.1, Y.sub.2 and Y.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
Y.sub.1, Y.sub.2 and Y.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices T.sub.21 and (I-T.sub.21 H).sup. ,
the entries of said matrix C.sup. , and the entries of said matrix P by
the relation P=(I-T.sub.21 H).sup. T.sub.21 C.sup. =T.sub.21
(I-HT.sub.21).sup. C.sup. .
34. The apparatus of claim 29, wherein T=T.sub.22 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, Y.sub.1, Y.sub.2 and Y.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
Y.sub.1, Y.sub.2 and Y.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices T.sub.22 and (I-T.sub.22 CH).sup.
, the entries of said matrix C.sup. , and the entries of said matrix P by
the relation P=(I-T.sub.22 CH).sup. T.sub.22.
35. The apparatus of claim 29, wherein T=T.sub.23 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, Y.sub.1, Y.sub.2 and Y.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
Y.sub.1, Y.sub.2 and Y.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices T.sub.23 and (I+T.sub.23).sup. ,
the entries of said matrix (CH).sup. , and the entries of said matrix P by
the relation P=-(I+T.sub.23).sup. T.sub.23 (CH).sup. .
36. The apparatus of claim 29, wherein T=T.sub.31 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, Y.sub.1, Y.sub.2 and Y.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
Y.sub.1, Y.sub.2 and Y.sub.3 as measured by the signal analyzer, and that
detemines the entries of said matrices T.sub.31 and (I-T.sub.31).sup. ,
the entries of said matrix C.sup. , and the entries of said matrix HP by
the relation HP=(I-T.sub.31).sup. T.sub.31 C.sup. .
37. The apparatus of claim 29, wherein T=T.sub.32 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, Y.sub.1, Y.sub.2 and Y.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
Y.sub.1, Y.sub.2 and Y.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices T.sub.32 and (I-CT.sub.32).sup.
and the entries of said matrix HP by the relation HP=T.sub.32
(I-CT.sub.32).sup. =(I-T.sub.32 C).sup. T.sub.32.
38. The apparatus of claim 29, wherein T=T.sub.33 and said measurement
means and said computation means comprise:
a signal analyzer connected to said feedback loop that measures said arrays
of signals R.sub.1, R.sub.2, R.sub.3, Y.sub.1, Y.sub.2 and Y.sub.3 ; and
a multivariable work station, including a digital computer, a digital
signal processor and associated mathematics module, that receives and
temporarily stores said arrays of signals R.sub.1, R.sub.2, R.sub.3,
Y.sub.1, Y.sub.2 and Y.sub.3 as measured by the signal analyzer, and that
determines the entries of said matrices T.sub.33 and T.sub.33.sup. , the
entries of said matrices C.sup. and H.sup. , and the entries of said
matrix P by the relation P=(T.sub.33.sup. -H.sup. )C.sup. .
39. A method for determining open-loop parameters for a First
Configuration, multiple-input, multiple output, closed-loop control
system, having a difference module, a controller module, a plant module
and a sensor module arranged in that order in a feedback loop that
operates in its linear operating region, the sensor module having sensor
dynamics represented by a matrix H of dimensions n.sub.1 .times.n.sub.2,
the controller module having controller dynamics represented by a matrix C
of dimensions n.sub.3 .times.n.sub.1, and the plant module having physical
system dynamics represented by a matrix P of dimensions n.sub.2
.times.n.sub.3, where n.sub.1, n.sub.2 and n.sub.3 are selected positive
integers, the method comprising the steps of:
providing an array R of n.sub.1 predetermined input signals that are
received at a first input terminal of the difference module;
receiving an array Z of n.sub.1 signals at a second input terminal of the
difference module;
forming and issuing a difference array E=R-Z of n.sub.1 signals at an
output terminal of the difference module;
causing the controller module to receive the array E at an input terminal
thereof and, in response thereto, to issue an array U=CE of n.sub.3
intermediate output signals at an output terminal thereof;
causing the plant module to receive the array U at an input terminal
thereof and, in response thereto, to form and issue an array Y=PU of
n.sub.2 output signals at an output terminal thereof;
causing the sensor module to receive the array Y at an input terminal
thereof and, in response thereto, to form and issue the array Z=HY of
n.sub.1 signals at an output terminal thereof;
providing a control system signal analyzer, connected to the difference
module of the feedback loop, that injects the array of signals R, measures
the array of signals Z and the array of signals R, and determines the
entries of the matrix T of dimensions n.sub.1 .times.n.sub.1 by solving
the matrix identity Z=TR;
measur | | |
