Optimal distributed control system for a linear distributed parameter system

• US 5,329,442 A
• Filed: 08/29/1991
• Issued: 07/12/1994
• Est. Priority Date: 08/29/1991
• Status: Expired due to Fees

• Current Assignee: the united states of america as represented by the secretary of the navy
• Original Assignee: the united states of america as represented by the secretary of the navy
• Inventors: Moshfegh, Allen
• Primary Examiner(s): Smith, Jerry
• Assistant Examiner(s): OAKES, BRIAN
• Application Number:
  US07/751,939
• Time in Patent Office: 1,048 Days
• Field of Search: 364/148, 364/149, 364/164, 364/150, 364/153, 364/180, 364/508
• US Class Current: 700/29
• CPC Class Codes: G05B 13/02 electric

Abstract

An optimal control system for canceling the undesired responses of a linearistributed-parameter system in response to an input stimulus. The control system is optimally distributed and its design is based on the extension of the Pontryagin'"'"'s maximum principle for a lumped-parameter system to a distributed-parameter system. The technique is to transform the stimulus-to-undesired-output transfer function of the distributed-parameter system into a multidimensional discrete state-space model, and to require a performance criterion to be defined. Theorems have been established to give conditions for asymptotic stability of the closed-loop distributed-parameter system, and to set up weighting factors for the performance criterion. An optimal distributed-control system can force the states of the distributed system to behave according to the prescribed performance criterion regardless of the boundary and initial conditions. The optimal control law can be implemented either as a state-feedback or an output-feedback controller.

Claims

1. 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, which has a continuous distributed transfer function describable by a partial differential equation with respect to at least two independent variables, at least one of which is limited in length,

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 said continuous distributed transfer function of said system, said function describing the transfer between input signal and at least one undesired output response, into a digital distributed transfer function in a discrete format and for obtaining from the digital distributed transfer function a digital distributed state-space model of said system capable of use with a digital computer, said model being in the form of state-space equations,

D. means for using said model and responsive to the input signal for deriving an optimal 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

E. transducer means responsive to the computed output signal for applying the derived output response to said body to cancel the effects of the desired output response.

2. A control system in accordance with claim 1, wherein the means for converting further comprises:

A. means for converting the continuous distributed transfer function into Chester transform equations to obtain said digital distributed transfer function, and

B. means for converting the Chester transform equations into said state-space model of said system.

3. A control system in accordance with claim 2, wherein the means for deriving an optimal computed output signal further comprises:

A. means for deriving a plurality of possible computed output signals, and

B. optimizing means for selecting the best one of the plurality of possible computed output signals to obtain the actual computed output signal.

4. 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.

5. A control system in accordance with claim 4, wherein

A. the continuous distributed transfer function is derived from equations having the form ##EQU37## and B. said state-space model has the form ##EQU38##

6. A control system in accordance with claim 4, wherein the means for deriving a computed output signal further comprises:

A. means for deriving a plurality of possible computed output signals, and

B. optimizing means for selecting the best one of the plurality of possible computed output signals to obtain the actual computed output signal.

7. A control system in accordance with claim 6, 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 ##EQU39## 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^o (i,j), which is an optimum quartic controller in accordance with the control equation ##EQU40##

8. 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.

9. 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^o (i,j), which is an optimum quartic controller in accordance with the control equation ##EQU42##