Optimal distributed control system for a linear distributed parameter system

A control system comprising:

• A. a linear continuous distributed parameter system, which, when excited by an input signal, produces both desired output responses and at least one undesired output response,
• B. a physical body which is so placed that it is subject to the undesired output response,
• C. means using a Moshfegh transform format for converting a continuous distributed model of the linear distributed system with at least two independent variables, at least one of which is limited in length, into a continuous distributed transfer function,
• D. means using a Chester transform format for converting the continuous distributed transfer function into a discrete distributed transfer function capable of use with a digital computer,
• E. means for converting the discrete distributed transfer function into a distributed state-space model of the linear distributed parameter system,
• F. means for using the distributed state-space model of the transfer function and responsive to the input signal for deriving a computed output signal representing a derived output response which is coincident in time and space with the undesired output response, equal in magnitude and opposite in phase, and
• G. transducer means responsive to the computed output signal for applying the derived output response to the physical body to cancel the effects of the undesired output response.

A control system comprising:

• A. a linear continuous distributed parameter system, which, when excited by an input signal, produces both desired output responses and at least one undesired output response,
• B. a physical body which is so placed that it is subject to the undesired output response,
• C. means using a first transform format operating with both space and time as independent variables, wherein the space variable is limited in length, for converting a continuous distributed model of the linear distributed system into a continuous distributed transfer function,
• D. means using a second transform format for converting the continuous distributed transfer function into a discrete distributed transfer function capable of use with a digital computer,
• E. means for converting the discrete distributed transfer function into a distributed state-space model of the linear distributed parameter system,
• F. means for using the distributed state-space model of the transfer function and responsive to the input signal for deriving a computed output signal representing a derived output response which is coincident in time and space with the undesired output response, equal in magnitude and opposite in phase, and
• G. transducer means responsive to the computed output signal for applying the derived output response to the physical body to cancel the effects of the undesired output response,
• wherein the first transform format is mathematically in the nature of a Laplace transform and the second transform format is mathematically in the nature of a Z-transform format except that at least one of the variable dimensions each of the first transform format and of the second transform format are length-limited spatial dimensions in addition to the strictly time dimensions usually used in Laplace transforms and in z-transforms.

A control system comprising:

• A. a linear continuous distributed parameter system, which, when excited by an input signal, produces both desired output responses and at least one undesired output response,
• B. a physical body which is so placed that it is subject to the undesired output response,
• C. means using a Moshfegh transform format for converting a continuous distributed model of the linear distributed system with at least two independent variables, at least one of which is limited in length, into a continuous distributed transfer function,
• D. means using a Chester transform format for converting the continuous distributed transfer function into a discrete distributed transfer function capable of use with a digital computer,
• E. means for converting the discrete distributed transfer function into a distributed state-space model of the linear distributed parameter system,
• F. means for using the distributed state-space model of the transfer function and responsive to the input signal for deriving a computed output signal representing a derived output response which is coincident in time and space with the undesired output response, equal in magnitude and opposite in phase, and
• G. transducer means responsive to the computed output signal for applying the derived output response to the physical body to cancel the effects of the undesired output response.
• and wherein the means for deriving a computed output signal further comprises:
• H. means for deriving a plurality of possible computed output signals, and
• I. optimizing means for selecting the best one of the plurality of possible computed output signals to obtain the actual computed output signal,
• and wherein the plurality of possible computed output signals is represented by the expression y(i,j) for the derived output in the digital distributed state-space model equations ##EQU41## and wherein the equations for the best one of the plurality of possible output signals has been selected when the value of the expression u(i,j) is chosen as the optimum value u.sup.o (i,j), which is an optimum quartic controller in accordance with the control equation ##EQU42##