Preconditioned conjugate gradient system

Claim 1

A controller for optimizing the operational state of a system characterized by a set of controllable parameters that interact with each other subject to specified constraint relationships, said controller comprising: means for storing said constraint relationships in the form of a matrix A, where each row in A expresses one of said constraint relationships in terms of the inter-relationship of said controllable parameters, where said set of parameters is expressed as a vector x, and where specified constraint limits are expressed as a vector b; processing means coupled to said means for storing for determining preferred values of said set of controllable parameters, x, such that the matrix relationship Ax=b is substantially satisfied and a preselected cost function is improved, where said processing means includes parameter modification means responsive to a vector u for developing a set of intermediate values of x and setting said set of intermediate values of x to aid preferred values of x when said cost function is sufficiently improved, and matrix inversion means responsive to said parameter modification means for developing said vector u such that AD 2 A T u=AD, where D is a diagonal matrix with the elements of the intermediate values of x on the diagonal; and interface means responsive to said processor means for imparting control signals to said system, to force said controllable parameters of said system to the preferred value of x determined by said processing means; wherein said matrix inversion means further comprises first means for setting an iteration count index i to zero, vector u.sup.(i) to zero, a direction vector d.sup.(i) to AD, and a gradient vector g.sup.(i) to -d.sup.(i) ; second means for computing a vector q.sup.(i) =AD 2 A T d.sup.(i) ; third means for computing a scalar α=(g.sup.(i).spsp.T g.sup.(i))/d.sup.(i).spsp.T q.sup.(i)); fourth means for updating the vector u, the gradient vector g and the direction vector d by evaluating u.sup.(i+1) =u.sup.(i) +αd.sup.(i) g.sup.(i+1) =g.sup.(i) +αq.sup.(i), and; d.sup.(i+1) =-g.sup.(i+1) +((g.sup.(i+1).spsp.T g.sup.(i+1))/(g.sup.(i).spsp.T g.sup.(i)))d.sup.(i) ; fifth means for setting i=i+1 and returning control to said second means when said iteration count index is below a preselected value and the vector u.sup.(i+1) developed in said fourth means is such that AD 2 A T u.sup.(i+1) is within a chosen threshold of AD, and for setting said u.sup.(i+1) to u and applying said vector u to said parameter modifications means, otherwise; where the superscript, T, associated with any matrix or any vector represents the transpose of the matrix of the vector, respectively.

Claim 6

A system including a) a large plurality of controllable parameters that affect the operation of said system and affect a preselected cost measure of said system, expressible as a vector x, b) constraint relationship related to said parameters, expressible in the form of a matrix A associated with said parameters, and with a set of constraint limits, b, such that Ax=b, and a controller for optimizing the operational state of said system by applying control signals to said system that impose on said system values for said controllable parameters x, THE IMPROVEMENT IN SAID CONTROLLER COMPRISING: means for storing said constraint relationships in the form of said matrix A, where each row in A expressed one of said constraint relationships in terms of the inter-relationship of said controllable parameters, where said set of parameters is expressed as said vector x, and where specified constraint limits are expressed as said vector b; processing means coupled to said means for storing for determining preferred values of said set of controllable parameters, x, such that the matrix relationship Ax=b is substantially satisfied and a preselected cost function is improved, where said processing means includes parameter modifications means responsive to a vector u for developing a set of intermediate values of x and setting said set of intermediate values of x to said preferred values of x when said cost function is sufficiently improved, and matrix inversion means responsive to said parameter modification means for developing said vector u such that AD 2 A T u=AD, where D is a diagonal matrix with the elements of the intermediate values of x on the diagonal; and interface means responsive to said processor means for imparting control signals to said system, for force said controllable parameters of said system to the preferred values of x determined by said processing means; wherein said matrix inversion means further comprises first means for setting an iteration count index i to zero, a vector u.sup.(i) to zero, a direction vector d.sup.(i) to AD, and a gradient vector g.sup.(i) to -d.sup.(i) ; second means for computing a vector q.sup.(i) =AD 2 A T d.sup.(i) ; third means for computing a scalar α=(g.sup.(i).spsp.T g.sup.(i))/d.sup.(i) T q.sup.(i)); fourth means for updating the vector u, the gradient vector g and the direction vector d by evaluating u.sup.(i+1) =u.sup.(i) +αd.sup.(i) g.sup.(i+1) =g.sup.(i) +αq.sup.(i), and; d.sup.(i+1) =-g.sup.(i+1) +((g.sup.(i+1).spsp.T g.sup.(i+1))/(g.sup.(i).spsp.T g.sup.(i)))d.sup.(i) ; and fifth means for setting i=i+1 and returning control to said second means when said iteration count index is below a preselected value and the vector u.sup.(i+1) developed in said fourth means is such that AD 2 A T u.sup.(i+1) is within a chosen threshold of AD, and for setting said u.sup.(i+1) to u and applying said vector u to said parameter modification means, otherwise; where the superscript, T, associated with any matrix or any vector represents the transpose of the matrix or the vector, respectively.

Claim 7

A system including a) a large plurality of controllable parameters that affect the operation of said system and affect a preselected cost measure of said system, expressible as a vector x, b) constraint relationship related to said parameters, expressible in the form of a matrix A associated with said parameters, and with a set of constraint limits, b, such that Ax=b, and a controller for optimizing the operational state of said system by applying control signals to said system that impose on said system values for said controllable parameters x, THE IMPROVEMENT IN SAID CONTROLLER COMPRISING: means for storing said constraint relationships in the form of said matrix A, where each row in A expressed one of said constraint relationships in terms of the inter-relationship of said controllable parameters, where said set of parameters is expressed as said vector x, and where specified constraint limits are expressed as said vector b; processing means coupled to said means for storing for determining preferred values of said set of controllable parameters, x, such that the matrix relationship Ax=b is substantially satisfied and a preselected cost function is improved, where said processing means includes parameter modifications means responsive to a vector u for developing a set of intermediate values of x and setting said set of intermediate values of x to said preferred values of x when said cost function is sufficiently improved, and matrix inversion means responsive to said parameter modification means for developing said vector u such that AD 2 A T u=AD, where D is a diagonal matrix with the elements of the intermediate values of x on the diagonal; and interface means responsive to said processor means for imparting control signals to said system, for force said controllable parameters of said system to the preferred values of x determined by said processing means; wherein said matrix inversion means further comprises first means for setting an iteration count index i to zero, a vector u.sup.(i) to zero, a direction vector d.sup.(i) to AD, and a gradient vector g.sup.(i) to -d.sup.(i) ; second means for computing a vector q.sup.(i) =AD 2 A T d.sup.(i) ; third means for computing a scalar α=(g.sup.(i).spsp.T g.sup.(i))/d.sup.(i) T q.sup.(i)); fourth means for updating the vector u, the gradient vector g and the direction vector d by evaluating u.sup.(i+1) =u.sup.(i) +αd.sup.(i) g.sup.(i+1) =g.sup.(i) +αq.sup.(i), and; d.sup.(i+1) =-g.sup.(i+1) +((g.sup.(i+1).spsp.T g.sup.(i+1))/(g.sup.(i).spsp.T g.sup.(i)))d.sup.(i) ; and fifth means for setting i=i+1 and returning control to said second means when said iteration count index is below a preselected value and the vector u.sup.(i+1) developed in said fourth means is such that AD 2 A T u.sup.(i+1) is within a chosen threshold of AD, and for setting said u.sup.(i+1) to u and applying said vector u to said parameter modification means, otherwise; where the superscript, T, associated with any matrix or any vector represents the transpose of the matrix or the vector, respectively.

Claim 8

A system including a) a large plurality of controllable parameters that affect the operation of said system and affect a preselected cost measure of said system, expressible as a vector x, b) constraint relationship related to said parameters, expressible in the form of a matrix A associated with said parameters, and with a set of constraint limits, b, such that Ax=b, and a controller for optimizing the operational state of said system by applying control signals to said system that impose on said system values for said controllable parameters X, THE IMPROVEMENT IN SAID CONTROLLER COMPRISING: means for storing said constraint relationships in the form of a matrix A, where each row in A expressed one of said constraint relationships in terms of the inter-relationship of said controllable parameters, where said set of parameters is expressed as a vector x, and where specified constraint limits are expressed as a vector b; processing means coupled to said means for storing for determining preferred values of said set of controllable parameters, x, such that the matrix relationship Ax=b is substantially satisfied and a preselected cost function is improved, wherein said processing means includes parameter modification means responsive to a vector u for developing a set of intermediate values of x and setting said set of intermediate values of x to said preferred values of x when said cost function is sufficiently improved, and matrix inversion means responsive to said parameter modification means for developing said vector u such that AD 2 A T u=AD, where D is a diagonal matrix with the elements of the intermediate values of x on the diagonal; and interface means responsive to said processor means for imparting control signals to said system, to force said controllable parameters of said system to the preferred values of x determined by said processing means; wherein said matrix inversion means further comprises first means for setting an iteration count index i to zero, a vector v.sup.(i) to zero, a direction vector d.sup.(i) to G -1/2 LAD, and a gradient vector g.sup.(i) to -d.sup.(i) ; second means for computing a vector q.sup.(i) =G -1/2 LAD 2 A T L T G -1/2 d.sup.(i), third means for computing a scalar α=(g.sup.(i).spsp.T g.sup.(i))/d.sup.(i).spsp.T q.sup.(i)); fourth means for updating the vector v, the gradient vector g and the direction vector d by evaluating g.sup.(i+1) =g.sup.(i) +αq.sup.(i), v.sup.(i+1) =v.sup.(i) +αd.sup.(i) and d.sup.(i+1) =g.sup.(i+1) +((g.sup.(i+1).spsp.T g.sup.(i+1))/(g.sup.(i).spsp.T g.sup.(i)))d.sup.(i) ; and fifth means for incrementing said iteration count index and returning control to said second means when said iteration count index is below a preselected value and the vector v.sup.(i+1) developed in said fourth means is such that AD 2 A T v.sup.(i+1) is within a chosen threshold of AD, and for setting said v.sup.(i+1) to us and applying said vector u to said parameter modification means, otherwise where L is a product of a plurality of elementary transformation matrices E j , where j=1,2, . . . N and N is related to the number of non-zero off-diagonal terms in AD 2 A T/ each E j is a matrix with a zero upper triangular half, 1's on the diagonal and one non-zero term in the lower triangular half such that the matrix product E j AD 2 A T has one more zero term than said matrix product AD 2 A T , and L is a product of a plurality of those elementary transformation matrices E j with the element in the lower triangular half being larger than a preselected threshold, G is a matrix that equals LAD 2 A T L T G is the diagonal of LAD 2 A T L T G -1/2 is such that G -1/2 GG -178 =I, where I is the identity matrix; and where the superscript, T, associated with any matrix or any vector represents the transpose of the matrix or the vector, respectively.

Claim 9

A system including a) a large plurality of controllable parameters that affect the operation of said system and affect a preselected cost measure of said system, expressible as a vector x, b) constraint relationship related to said parameters, expressible in the form of a matrix A associated with said parameters, and with a set of constraint limits, b, such that Ax=b, and a controller for optimizing the operational state of said system by applying control signals to said system that impose on said system values for said controllable parameters X, THE IMPROVEMENT IN SAID CONTROLLER COMPRISING: means for storing said constraint relationships in the form of a matrix A, where each row in A expressed one of said constraint relationships in terms of the inter-relationship of said controllable parameters, where said set of parameters is expressed as a vector x, and where specified constraint limits are expressed as a vector b; processing means coupled to said means for storing for determining preferred values of said set of controllable parameters, x, such that the matrix relationship Ax=b is substantially satisfied and a preselected cost function is improved, wherein said processing means includes parameter modification means responsive to a vector u for developing a set of intermediate values of x and setting said set of intermediate values of x to said preferred values of x when said cost function is sufficiently improved, and matrix inversion means responsive to said parameter modification means for developing said vector u such that AD 2 A T u=AD, where D is a diagonal matrix with the elements of the intermediate values of x on the diagonal; and interface means responsive to said processor means for imparting control signals to said system, to force said controllable parameters of said system to the preferred values of x determined by said processing means; wherein said matrix inversion means further comprises first means for setting an iteration count index i to zero, a vector v.sup.(i) to zero, a direction vector d.sup.(i) to G -1/2 LAD, and a gradient vector g.sup.(i) to -d.sup.(i) ; second means for computing a vector q.sup.(i) =G -1/2 LAD 2 A T L T G -1/2 d.sup.(i), third means for computing a scalar α=(g.sup.(i).spsp.T g.sup.(i))/d.sup.(i).spsp.T q.sup.(i)); fourth means for updating the vector v, the gradient vector g and the direction vector d by evaluating g.sup.(i+1) =g.sup.(i) +αq.sup.(i), v.sup.(i+1) =v.sup.(i) +αd.sup.(i) and d.sup.(i+1) =g.sup.(i+1) +((g.sup.(i+1).spsp.T g.sup.(i+1))/(g.sup.(i).spsp.T g.sup.(i)))d.sup.(i) ; and fifth means for incrementing said iteration count index and returning control to said second means when said iteration count index is below a preselected value and the vector v.sup.(i+1) developed in said fourth means is such that AD 2 A T v.sup.(i+1) is within a chosen threshold of AD, and for setting said v.sup.(i+1) to us and applying said vector u to said parameter modification means, otherwise where L is a product of a plurality of elementary transformation matrices E j , where j=1,2, . . . N and N is related to the number of non-zero off-diagonal terms in AD 2 A T/ each E j is a matrix with a zero upper triangular half, 1's on the diagonal and one non-zero term in the lower triangular half such that the matrix product E j AD 2 A T has one more zero term than said matrix product AD 2 A T , and L is a product of a plurality of those elementary transformation matrices E j with the element in the lower triangular half being larger than a preselected threshold, G is a matrix that equals LAD 2 A T L T G is the diagonal of LAD 2 A T L T G -1/2 is such that G -1/2 GG -178 =I, where I is the identity matrix; and where the superscript, T, associated with any matrix or any vector represents the transpose of the matrix or the vector, respectively.

Claim 10

A system including a) a large plurality of controllable parameters that affect the operation of said system and affect a preselected cost measure of said system, expressible as a vector x, b) constraint relationship related to said parameters, expressible in the form of a matrix A associated with said parameters, and with a set of constraint limits, b, such that Ax=b, and a controller for optimizing the operational state of said system by applying control signals to said system that imposed on said system values for said controllable parameters x, THE IMPROVEMENT IN SAID CONTROLLER COMPRISING: means for storing said constraint relationships in the form of a matrix A, where each row in A expressed one of said constraint relationships in term of the inter-relationship of said controllable parameters, where said set of parameters is expressed as a vector x, and where specified constraint limits are expressed as a vector b; processing means coupled to said means for storing for determining preferred values of said set of controllable parameters, x, such that the matrix relationship Ax=b is substantially satisfied and a preselected cost function is improved, where said processing means includes parameter modification means responsive to a vector u for developing a set of intermediate values of x and setting said set of intermediate values of x to said preferred values of x when said cost function is sufficiently improved, and matrix inversion means responsive to said parameter modification means for developing said vector u such that AD 2 A T u=AD, where D is a diagonal matrix with the elements of the intermediate values of x on the diagonal; and interface means responsive to said processor means for imparting control signals to said system, to force said controllable parameters of said system to the preferred values of x determined by said processing means; wherein said matrix inversion means further comprises first means for setting an iteration count index i to zero, a vector v.sup.(i) ' to zero, a direction vector d.sup.(i) ' to L T G -1 LAD, and a gradient vector g.sup.(i) ' to -d.sup.(i) '; second means for computing a vector q.sup.(i) '=L T G -1 LAD 2 A T d.sup.(i) ', third means for computing a scalar α=(g.sup.(i).spsp.T 'g.sup.(i) ')/d.sup.(i).spsp.T 'q.sup.(i) '); fourth means for updating the vector v', the gradient vector g' and the direction vector d' by evaluating g.sup.(i+1) '=g.sup.(i) '+αq.sup.(i) ', v.sup.(i+1) '=v.sup.(i) '+αd.sup.(i) ' and d.sup.(i+1) '=g.sup.(i+1) '+((g.sup.(i+1).spsp.T 'g.sup.(i+1) ')/(g.sup.(i).spsp.T 'g.sup.(i) '))d.sup.(i) '; and fifth means for incrementing said iteration count index and returning control to said second means when said iteration count index is below a preselected value and the vector v.sup.(i+1) ' developed in said fourth means is such that AD 2 A T v.sup.(i+1) ' is within a chosen threshold of AD, and for setting said v.sup.(i+1) ' to u and applying said vector u to said parameter modification means, otherwise where L is a product of a plurality of elementary transformation matrices E j , where j=1,2, . . . N and N is related to the number of off-diagonal terms in AD 2 A T , each E j is a matrix with a zero upper triangular half, 1's on the diagonal and one non-zero term in the lower triangular half such that the matrix product E j AD 2 A T has one more zero term than said matrix product AD 2 A T , and L is a product of a plurality of those transformation matrices E j with the element in the lower triangular half being larger than a preselected threshold, G is a matrix that equals LAD 2 A T L T , G is a diagonal of LAD 2 A T L T G -1/2 is the inverse of matrix G; and where the superscript, T, associated with any matrix or any vector represents the transpose of the matrix or the vector, respectively.

Claim 11

A system including a) a large plurality of controllable parameters that affect the operation of said system and affect a preselected cost measure of said system, expressible as a vector x, b) constraint relationship related to said parameters, expressible in the form of a matrix A associated with said parameters, and with a set of constraint limits, b, such that Ax=b, and a controller for optimizing the operational state of said system by applying control signals to said system that imposed on said system values for said controllable parameters x, THE IMPROVEMENT IN SAID CONTROLLER COMPRISING: means for storing said constraint relationships in the form of a matrix A, where each row in A expressed one of said constraint relationships in term of the inter-relationship of said controllable parameters, where said set of parameters is expressed as a vector x, and where specified constraint limits are expressed as a vector b; processing means coupled to said means for storing for determining preferred values of said set of controllable parameters, x, such that the matrix relationship Ax=b is substantially satisfied and a preselected cost function is improved, where said processing means includes parameter modification means responsive to a vector u for developing a set of intermediate values of x and setting said set of intermediate values of x to said preferred values of x when said cost function is sufficiently improved, and matrix inversion means responsive to said parameter modification means for developing said vector u such that AD 2 A T u=AD, where D is a diagonal matrix with the elements of the intermediate values of x on the diagonal; and interface means responsive to said processor means for imparting control signals to said system, to force said controllable parameters of said system to the preferred values of x determined by said processing means; wherein said matrix inversion means further comprises first means for setting an iteration count index i to zero, a vector v.sup.(i) ' to zero, a direction vector d.sup.(i) ' to L T G -1 LAD, and a gradient vector g.sup.(i) ' to -d.sup.(i) '; second means for computing a vector q.sup.(i) '=L T G -1 LAD 2 A T d.sup.(i) ', third means for computing a scalar α=(g.sup.(i).spsp.T 'g.sup.(i) ')/d.sup.(i).spsp.T 'q.sup.(i) '); fourth means for updating the vector v', the gradient vector g' and the direction vector d' by evaluating g.sup.(i+1) '=g.sup.(i) '+αq.sup.(i) ', v.sup.(i+1) '=v.sup.(i) '+αd.sup.(i) ' and d.sup.(i+1) '=g.sup.(i+1) '+((g.sup.(i+1).spsp.T 'g.sup.(i+1) ')/(g.sup.(i).spsp.T 'g.sup.(i) '))d.sup.(i) '; and fifth means for incrementing said iteration count index and returning control to said second means when said iteration count index is below a preselected value and the vector v.sup.(i+1) ' developed in said fourth means is such that AD 2 A T v.sup.(i+1) ' is within a chosen threshold of AD, and for setting said v.sup.(i+1) ' to u and applying said vector u to said parameter modification means, otherwise where L is a product of a plurality of elementary transformation matrices E j , where j=1,2, . . . N and N is related to the number of off-diagonal terms in AD 2 A T , each E j is a matrix with a zero upper triangular half, 1's on the diagonal and one non-zero term in the lower triangular half such that the matrix product E j AD 2 A T has one more zero term than said matrix product AD 2 A T , and L is a product of a plurality of those transformation matrices E j with the element in the lower triangular half being larger than a preselected threshold, G is a matrix that equals LAD 2 A T L T , G is a diagonal of LAD 2 A T L T G -1/2 is the inverse of matrix G; and where the superscript, T, associated with any matrix or any vector represents the transpose of the matrix or the vector, respectively.