US6505085B1
Expired / Lapsed
US
⏱️ 0.0 years remaining
Method and apparatus for creating time-optimal commands for linear systems
Assignee: Massachusetts Institute of Technology
•
Filing: 1999-03-04
•
Granted: 2003-01-07
•
Expires: 2019-03-04
Abstract
Abstract The system described herein determines an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one point to another point. The system identifies characteristics of the dynamic system, and then selects a command profile which defines an input to the dynamic system based on the identified characteristics. The command profile comprises one or more pulses which rise and fall at switch times, and the command profile is useable with substantially any dynamic system that can be modeled as a linear system. The system then imposes a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times, and determines the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Inventors
- 👤 Timothy D. Tuttle
- 👤 Warren P. Seering
Bibliographic Details
Application Number
—
Jurisdiction
US
Data Source
google_patents
Last Synchronized
2026-10-31 21:09
CPC Classifications
B; B25; B25J; B25J9/00; B25J9/16; B25J9/1602; B25J9/1605; G; G05; G05B; G05B13/00; G05B13/02; G05B13/04; G05B13/047; G05B2219/00; G05B2219/30; G05B2219/35; G05B2219/35585; G05B2219/39; G05B2219/39229; G05B2219/39358; G05B2219/43; G05B2219/43168
Claims Analysis
Total Claims: 54
Claim 1
Independent Claim
187 words
A method of determining an input command profile for a dynamic system tat can be modeled as a linear system, the input command profile for tansitioning an output of the dynamic system from one state to another state, the method comprising the steps of: identifying characteristics of the dynamic system, wherein the characteristics relate to poles and zeros of the dynamic system; selecting a command profile which defines an input to the dynamic system based on the characteristics identified in the identifying step, the command profile comprising one or more pulses which rise and fall at switch times and the command profile being useable with substantially any dynamic system that can be modeled as a linear system, wherein the command profile further comprises, after a last one of the one or more pulses, a tail which trails off to a constant value; imposing a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times; and determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 2
Independent Claim
187 words
A method of determining an input command profile for a dynamic system tat can be modeled as a linear system, the input command profile for tansitioning an output of the dynamic system from one state to another state, the method comprising the steps of: identifying characteristics of the dynamic system, wherein the characteristics relate to poles and zeros of the dynamic system; selecting a command profile which defines an input to the dynamic system based on the characteristics identified in the identifying step, the command profile comprising one or more pulses which rise and fall at switch times and the command profile being useable with substantially any dynamic system that can be modeled as a linear system, wherein the command profile further comprises, after a last one of the one or more pulses, a tail which trails off to a constant value; imposing a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times; and determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 3
Dependent Claim
132 words
A method according to claim 1 , wherein the command profile is based on the following function: u ( t ) = ∑ j = 0 n a j l ( t - t j ) + ∑ j = 1 r c j z j ( t - t n ) l ( t - t n ) , where “a j ” represents an amplitude of pulses in the command profile, “t j ” represents switch times for the pulses, “c j ” represents coefficients of exponential terms in the tail, “z j ” represents zeros in the system, “n” represents a total number of switches excluding a first switch, and “r” represents a number of zeros in the system.
Claim 4
Dependent Claim
45 words
A method according to claim 1 , wherein the plurality of constraints include a dynamics cancellation constraint which specifies that the input moves the dynamic system from a first state to a second state such that the dynamic system remains substantially at the second state.
Claim 5
Dependent Claim
38 words
A method according to claim 1 , wherein the plurality of constraints include boundary condition constraints, the boundary condition constraints including an initial time of the input and a specified change “dy” in the output of the system.
Claim 6
Dependent Claim
111 words
A method according to claim 4 , wherein the boundary condition constraints are based on the following equations: t 0 =0 y = lim s -> 0 ( N ( s ) D ( s ) ) ( ∫ ( rbp ) u ( t ) t ) | t = ∞ , where u(t) represents the command profile, N(s) represents at least a part of a numerator of a transfer function of the system, D(s) represents at least a part of a denominator of the transfer function of the system, and “rbp” corresponds to rigid body poles in the system.
Claim 7
Dependent Claim
36 words
A method according to claim 1 , wherein the plurality of constraints include an actuator limit constraint, the actuator limit constraint correspond to limits in the system which keep the input within limits of the actuator.
Claim 8
Dependent Claim
447 words
A method according to claim 1 , wherein the determining step comprises determining the switch times based on: (i) a command profile, u(t), comprising: u ( t ) = ∑ j = 0 n a j l ( t - t j ) + ∑ j = 1 r c j z j ( t - t n ) l ( t - t n ) , where “a j ” represents an amplitude of pulses in the function, “t j ” represents switch times for the pulses, “c j ” represents coefficients of exponential terms in a tail of the command profile, “z j ” represents zeros in the system, “n” represents a total number of switches excluding a first switch, and “r” represents a number of zeros in the system, (ii) a dynamics cancellation constraint comprising: ( ∑ j = 0 n a j t j ( q i - 1 ) - st j + - st n ∑ j = 1 r [ c j t n ( q i - 1 ) + ∑ k = 1 q i ( β q i ( k ) ) c j z j ( s - z j ) k t n ( q i - k ) ] ) | s = p i = 0 , where “p i ” represents system poles, “q i ” represents a multiplicity of each system pole “p i ”, and where β is defined as follows: β 1 ( k )=1 , k =1=0 , k ≠1 β q 1 ( k )=β q 1 −1 ( k )+( k −1)β q 1 −1 ( k −1). (iii) boundary condition constraints comprising: t 0 = 0 y = lim s -> 0 ( N ( s ) D ( s ) ) ( ∫ ( rbp ) u ( t ) t ) LINESUBt = ∞ , where N(s) represents at least a part of a numerator of a transfer function of the system, D(s) represents at least a part of a denominator or of the transfer function of the system, and “rbp” corresponds to rigid body poles in the system, and (iv) an actuator limit constraints comprising: u min ≦u tail ( t )≦ u max, where u min represents aminimum value of the input, u tail (t) represents a value of a tail of the input at time “t”, and u max represents a maximum value of the input.
Claim 9
Dependent Claim
17 words
A method according to claim 7 , wherein the switch times are determined using an optimization algorithm.
Claim 10
Dependent Claim
93 words
A method according to claim 1 , wherein, in a case that the identifying step identifies that the dynamic system contains poles only and no zeros, (i) the command profile comprises: u ( t ) = ∑ j = 0 n a j l ( t - t j ) , where “a j ” represents an amplitude of pulses in the command profile, “t j ” represents switch times for the pulses, and (ii) the plurality of constraints include only a dynamics cancellation constraint and boundary condition constraints.
Claim 11
Dependent Claim
158 words
A method according to claim 9 , wherein (i) the dynamics cancellation constraint comprises: ( ∑ j = 0 n a j t j ( q i - 1 ) - st j ) | s = p i = 0 , where “p i ” represents system poles, and “q i ” represents a multiplicity of each system pole “p i ”, and (ii) the boundary condition constraints comprise: y = lim s -> 0 ( N ( s ) D ( s ) ) ( ∫ ( rbp ) u ( t ) t ) | t = ∞ , where N(s) represents at least a part of numerator of a transfer function of the system, D(s) represents at least a part of denominator of the transfer function of the system, and “rbp” corresponds to rigid body poles in the system.
Claim 12
Dependent Claim
27 words
A method according to claim 1 , further comprising the step of verifying optimality of the input which includes the switch times determined in the determining step.
Claim 13
Dependent Claim
30 words
A method according to claim 11 , wherein the optimality of the input which includes the switch times determined in the determining step is verified based on Pontryagin's Minimum Principle.
Claim 14
Independent Claim
421 words
A method of determining an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the method comprising the steps of: identifying characteristics of the dynamic system; selecting a command profile which defines an input to tie dynamic system based on the characteristics identified in the identifying step, the command profile comprising one or more pulses which rise and fall at switch times and a tail following the one or more pulses, the command profile being useable with substantially any dynamic system that can be modeled as a linear system; imposing a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times, wherein the plurality of constraints include a dynamics cancellation constraint which specifies that the input moves the dynamic system from a fist state to a second state such that the dynamic system remains substantially at the second state, and wherein the dynamics cancellation constraint is based on the following equation: ( ∑ j = 0 n a j t j ( q i - 1 ) - st j + - st n ∑ j = 1 r [ c j t n ( q i - 1 ) + ∑ k = 1 q i ( β q i ( k ) ) c j z j ( s - z j ) k t n ( q i - k ) ] ) | s = p i = 0 , where “a j ” represents an amplitude of pulses in the command profile, “t j ” represents switch times for the pulses, “c j ” represents coefficients of exponential terms in the tail, “z j ” represents zeros in the system, “n” represents a total number of switches excluding a first witch, “r” represents a number of zeros in the system, “p i ” represents system poles, “q i ” represents a multiplicity of each system pole “p”, and where β is defined as follows: β 1 ( k )=1 , k =1=0 , k ≠1 β q 1 ( k )=β q 1 −1 ( k )+( k −1)β q 1 −1 ( k −1); and determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 15
Independent Claim
421 words
A method of determining an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the method comprising the steps of: identifying characteristics of the dynamic system; selecting a command profile which defines an input to tie dynamic system based on the characteristics identified in the identifying step, the command profile comprising one or more pulses which rise and fall at switch times and a tail following the one or more pulses, the command profile being useable with substantially any dynamic system that can be modeled as a linear system; imposing a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times, wherein the plurality of constraints include a dynamics cancellation constraint which specifies that the input moves the dynamic system from a fist state to a second state such that the dynamic system remains substantially at the second state, and wherein the dynamics cancellation constraint is based on the following equation: ( ∑ j = 0 n a j t j ( q i - 1 ) - st j + - st n ∑ j = 1 r [ c j t n ( q i - 1 ) + ∑ k = 1 q i ( β q i ( k ) ) c j z j ( s - z j ) k t n ( q i - k ) ] ) | s = p i = 0 , where “a j ” represents an amplitude of pulses in the command profile, “t j ” represents switch times for the pulses, “c j ” represents coefficients of exponential terms in the tail, “z j ” represents zeros in the system, “n” represents a total number of switches excluding a first witch, “r” represents a number of zeros in the system, “p i ” represents system poles, “q i ” represents a multiplicity of each system pole “p”, and where β is defined as follows: β 1 ( k )=1 , k =1=0 , k ≠1 β q 1 ( k )=β q 1 −1 ( k )+( k −1)β q 1 −1 ( k −1); and determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 16
Independent Claim
235 words
A method of determining an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the method comprising the steps of: identifying characteristics of the dynamic system; selecting a command profile, u(t), which defines an input to the dynamic system based on the characteristics identified in the identifying step, the command profile comprising one or more pulses which rise and fall at switch times and a tail which approaches a constant value, the command profile being useable with substantially any dynamic system that can be modeled as a linear system; imposing a plurality of constraints on the dynamic system at least one of the constraints being defined in forms of the switch times, wherein the plurality of constraints include an actuator limit constraint corresponding to limits in the system which keep the input within limits of the actuator, and wherein the actuator limit constraint comprises: u min ≦u tail ( t )≦ u max , where u min represents a minimum value of the input, u tail (t) represents a value of a tail of the input at time “t”, and u max represents a maximum value of the input; and determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 17
Independent Claim
235 words
A method of determining an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the method comprising the steps of: identifying characteristics of the dynamic system; selecting a command profile, u(t), which defines an input to the dynamic system based on the characteristics identified in the identifying step, the command profile comprising one or more pulses which rise and fall at switch times and a tail which approaches a constant value, the command profile being useable with substantially any dynamic system that can be modeled as a linear system; imposing a plurality of constraints on the dynamic system at least one of the constraints being defined in forms of the switch times, wherein the plurality of constraints include an actuator limit constraint corresponding to limits in the system which keep the input within limits of the actuator, and wherein the actuator limit constraint comprises: u min ≦u tail ( t )≦ u max , where u min represents a minimum value of the input, u tail (t) represents a value of a tail of the input at time “t”, and u max represents a maximum value of the input; and determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 18
Independent Claim
168 words
A method of determining an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the method comprising the steps of: identifying characteristics of the dynamic system; selecting a command profile which defines an input to the dynamic system based on the characteristics identified in the identifying step, the command profile comprising one or more pulses which rise and fall at switch times and the command profile being useable with substantially any dynamic system that can be modeled as a linear system; imposing a plurality of constraints on the dynamic system, wherein the plurality of constraints comprise insensitivity constraints which improve system robustness with respect to predetermined system parameters, at least one of the constraints being defined in terms of the switch times; and determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 19
Independent Claim
168 words
A method of determining an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the method comprising the steps of: identifying characteristics of the dynamic system; selecting a command profile which defines an input to the dynamic system based on the characteristics identified in the identifying step, the command profile comprising one or more pulses which rise and fall at switch times and the command profile being useable with substantially any dynamic system that can be modeled as a linear system; imposing a plurality of constraints on the dynamic system, wherein the plurality of constraints comprise insensitivity constraints which improve system robustness with respect to predetermined system parameters, at least one of the constraints being defined in terms of the switch times; and determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 20
Dependent Claim
41 words
A method according to claim 15 , wherein an insensitivity constraint is determined by setting to zero a derivative of a function, y p (t), which defines a residual amplitude response of the system for each pole “p” in the system.
Claim 21
Dependent Claim
44 words
A method according to claim 15 , wherein an insensitivity constraint is determined by setting to zero a derivative of a function y z (t), which defines a contribution of each zero “z” in the system to a residual amplitude response of the system.
Claim 22
Independent Claim
189 words
A method of determining an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the method comprising the steps of: identifying characteristics of the dynamic system; selecting a command profile which defines an input to the dynamic system based on the characteristics identified in the identifying step, the command profile comprising one or more pulses which rise and fall at switch times and, after a last one of the one or more pulses, a tail which trails off to a constant value, the command profile being useable with substantially any dynamic system that can be modeled as a linear system; imposing a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times, and wherein the plurality of constraints comprise insensitivity constraints that include a constraint which limits a length of the tail; and determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 23
Independent Claim
189 words
A method of determining an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the method comprising the steps of: identifying characteristics of the dynamic system; selecting a command profile which defines an input to the dynamic system based on the characteristics identified in the identifying step, the command profile comprising one or more pulses which rise and fall at switch times and, after a last one of the one or more pulses, a tail which trails off to a constant value, the command profile being useable with substantially any dynamic system that can be modeled as a linear system; imposing a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times, and wherein the plurality of constraints comprise insensitivity constraints that include a constraint which limits a length of the tail; and determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 24
Dependent Claim
114 words
A method according to claim 18 , wherein the tail is defined by a function u tail (t) which comprises: u tail ( t ) = ∑ j = 1 r c j z j ( t - t n ) l ( t - t n ) , where “c j ” represents coefficients of exponential terms in the tail, “z j ” represents zeros in the system, “n” represents a total number of switches excluding a first switch, and “r” represents a number of zeros in the system; and wherein the length of the tail is limited by limiting values of coefficients “c j ”.
Claim 25
Dependent Claim
81 words
A method according to claim 19 , wherein the values of the coefficients “c j ” are limited by ∑ j = 1 r c j ζ i ω nj t r ≤ X u lim , where “u lim ” defines an actuator limit, “X” defines a percentage relative to limit u lim within which the tail settles, and ζ i ω nj define a real component of zeros “z” within the system.
Claim 26
Independent Claim
243 words
A method of determining an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the method comprising the steps of: selecting a command profile for the input to the dynamic system, the command profile comprising a pulse train with pulses that rise and fall at switch times and being useable with substantially any dynamic system that can be modeled as a linear system, the selecting step comprising setting pulse train amplitudes based on actuator limits, determining if there are system zeros and including a tail in the command profile in a case that there are system zeros, making an initial guess for the switch times and, in a case that there are system zeros, making an initial guess for coefficients defining the tail; imposing constraints on the system, the constraints comprising dynamics cancellation constraints, boundary condition constraints, and, in a case that there are system zeros, actuator limit constraints; solving for the switch times based on (i) the command profile selected in the selecting step, (ii) the initial guess for the switch times, (iii) in a case that there are zeros in the system, the initial guess for the coefficients defining the tail, and (iv) the problem constraints imposed in the imposing step; and determining whether the switch times define an input which makes the command profile time optimal.
Claim 27
Independent Claim
243 words
A method of determining an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the method comprising the steps of: selecting a command profile for the input to the dynamic system, the command profile comprising a pulse train with pulses that rise and fall at switch times and being useable with substantially any dynamic system that can be modeled as a linear system, the selecting step comprising setting pulse train amplitudes based on actuator limits, determining if there are system zeros and including a tail in the command profile in a case that there are system zeros, making an initial guess for the switch times and, in a case that there are system zeros, making an initial guess for coefficients defining the tail; imposing constraints on the system, the constraints comprising dynamics cancellation constraints, boundary condition constraints, and, in a case that there are system zeros, actuator limit constraints; solving for the switch times based on (i) the command profile selected in the selecting step, (ii) the initial guess for the switch times, (iii) in a case that there are zeros in the system, the initial guess for the coefficients defining the tail, and (iv) the problem constraints imposed in the imposing step; and determining whether the switch times define an input which makes the command profile time optimal.
Claim 28
Dependent Claim
78 words
A method according to claim 21 , wherein, in a case that the determining step determines that the switch times do not define an input which make the command profile time optimal, the method further comprises: repeating the solving step using different values for the switch times and, in a case that there are zeros in the system, different values for the coefficients defining the tail; and repeating the determining step for newly-determined values of the switch times.
Claim 29
Independent Claim
206 words
An apparatus which determines an input command profile for a dynamic system that can be monitored as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the apparatus comprising: a memory which stores computer-executable process steps; and a processor which executes the process steps stored in the memory so as (i) to identify characteristics of the dynamic system, wherein the characteristics identified by the processor relate to poles and zeros of the dynamic system (ii) to select a command profile which defines an input to the dynamic system based on identified characteristics, the command profile comprising one or more pulses which rise and fall at switch times and, after a last one of the one or more pulses, a tail which trails off to a constant value, the command profile being useable with substantially any dynamic system that can be modeled as a linear system, (iii) to impose a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times, and (iv) to determine toe switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 30
Independent Claim
206 words
An apparatus which determines an input command profile for a dynamic system that can be monitored as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the apparatus comprising: a memory which stores computer-executable process steps; and a processor which executes the process steps stored in the memory so as (i) to identify characteristics of the dynamic system, wherein the characteristics identified by the processor relate to poles and zeros of the dynamic system (ii) to select a command profile which defines an input to the dynamic system based on identified characteristics, the command profile comprising one or more pulses which rise and fall at switch times and, after a last one of the one or more pulses, a tail which trails off to a constant value, the command profile being useable with substantially any dynamic system that can be modeled as a linear system, (iii) to impose a plurality of constraints on the dynamic system, at least one of the constraints being defined in terms of the switch times, and (iv) to determine toe switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 31
Dependent Claim
132 words
An apparatus according to claim 23 , wherein the command profile is based on the following function: u ( t ) = ∑ j = 0 n a j l ( t - t j ) + ∑ j = 1 r c j z j ( t - t n ) l ( t - t n ) , where “a j ” represents an amplitude of pulses in the command profile, “t j ” represents switch times for the pulses, “c j ” represents coefficients of exponential terms in the tail, “z j ” represents zeros in the system, “n” represents a total number of switches excluding a first switch, and “r” represents a number of zeros in the system.
Claim 32
Dependent Claim
47 words
An apparatus according to claim 23 , wherein the plurality of constraints include a dynamics cancellation constraint which specifies that the input moves the dynamic system from a first state to a second state such that the dynamic system remains substantially at rest at the second state.
Claim 33
Dependent Claim
273 words
An apparatus according to claim 25 , wherein the command profile includes a tail following the one or more pulses; and wherein the dynamics cancellation constraint is based on the following equation: ( ∑ j = 0 n a j t j ( q i - 1 ) - st j + - st n ∑ j = 1 r [ c j t n ( q i - 1 ) + ∑ k = 1 q i ( β q i ( k ) ) c j z j ( s - z j ) k t n ( q i - k ) ] ) | s = p i = 0 , where “a j ” represents an amplitude of pulses in the command profile, “t j ” represents switch times for the pulses, “c j ” represents coefficients of exponential terms in the tail, “z j ” represents zeros in the system, “n” represents a total number of switches excluding a first switch, “r” represents a number of zeros in the system, “p i ” represents system poles, “q i ” represents a multiplicity of each system pole “p i ”, and where β is defined as follows: β 1 ( k ) = 1 , k = 1 = 0 , k ≠ 1 β q i ( k ) = β q i - 1 ( k ) + ( k - 1 ) β q i - 1 ( k - 1 ) .
Claim 34
Dependent Claim
38 words
An apparatus according to claim 23 , wherein he plurality of constraints include boundary condition constraints, the boundary condition constraints including an initial time of the input and a specified change “dy” in the output of the system.
Claim 35
Dependent Claim
111 words
An apparatus according to claim 27 , wherein the boundary condition constraints are based on the following equations: t 0 =0 y = lim s -> 0 ( N ( s ) D ( s ) ) ( ∫ ( rbp ) u ( t ) t ) | t = ∞ , where u(t) represents the command profile, N(s) represents at least a part of a numerator of a transfer function of the system, D(s) represents at least a part of a denominator of the transfer function of the system, and “rbp” corresponds to rigid body poles in the system.
Claim 36
Dependent Claim
36 words
An apparatus according to claim 23 , wherein the plurality of constraints include an actuator limit constraint, the actuator limit constraint corresponding to limits in the system which keep the input within limits of the actuator.
Claim 37
Dependent Claim
446 words
An apparatus according to claim 23 , wherein the processor determines the switch times based on: (i) a command profile, u(t), comprising: u ( t ) = ∑ j = 0 n a j l ( t - t j ) + ∑ j = 1 r c j z j ( t - t n ) l ( t - t n ) , where “a j ” represents an amplitude of pulses in the function, “t j ” represents switch times for the pulses, “c j ” represents coefficients of exponential terms in a tail of the command profile, “z j ” represents zeros in the system, “n” represents a total number of switches excluding a first switch, and “r” represents a number of zeros in the system, (ii) a dynamics cancellation constraint comprising: ( ∑ j = 0 n a j t j ( q i - l ) - zs j + - s tu ∑ j = 1 r [ c j t u ( q i - l ) + ∑ k = 1 q 1 ( β q 1 ( k ) ) c j z j ( s - z j ) k t n ( q 1 - k ) ] ) | s = p 1 = 0 , where “p i ” represents system poles, “q i ” represents a multiplicity of each system pole “p i ”, and where β is defined as follows: β 1 ( k )=1 , k =1=0 , k ≠1 β q 1 ( k )=β q 1 −1 ( k )+( k −1)β q 1 −1 ( k −1). (iii) boundary condition constraints comprising: t 0 = 0 y = lim s = 0 ( N ( s ) D ( s ) ) ( ∫ ( rbp ) u ( t ) t ) LINESUBt = ∞ , where N(s) represents at least a part of a numerator of a transfer function of the system, D(s) represents at least a part of a denominator of the transfer function of the system, and “rbp” corresponds to rigid body poles in the system; and (iv) an actuator limit constraint comprising: u min ≦ tail ( t )≦ u max , where u min represents a minimum value of the input u tail (t) represents a value of a tail of the input at time “t”, and u max represents a maximum value of the input.
Claim 38
Dependent Claim
17 words
An apparatus according to claim 30 , wherein the switch times are determined using an optimization algorithm.
Claim 39
Dependent Claim
93 words
An apparatus according to claim 3 , wherein, in a case that the identifying step identifies that the dynamic system contains poles only and no zeros, (i) the command profile comprises: u ( t ) = ∑ j = 0 u a j l ( t - t j ) , where “a j ” represents an amplitude of pulses in the command profile, “t j ” represents switch times for the pulses, and (ii) the plurality of constraints include only a dynamics cancellation constraint and boundary condition constraints.
Claim 40
Dependent Claim
163 words
An apparatus according to claim 32 , wherein (i) the dynamics cancellation constraint comprises: ( ∑ j = 0 n a j t j ( q i - 1 ) - st j ) | s = p i = 0 , where “p i ” represents system poles, and “q i ” represents a multiplicity of each system pole “p i ”, and (ii) the boundary condition constraints comprise: t 0 =0 y = lim s -> 0 ( N ( s ) D ( s ) ) ( ∫ ( rbp ) u ( t ) t ) | t = ∞ , where N(s) represents at least a part of a numerator of a transfer function of the system, D(s) represents at least a part of a denominator of the transfer function of the system, and “rbp” corresponds to rigid body poles in the system.
Claim 41
Dependent Claim
52 words
An apparatus according to claim 23 , wherein the command profile further comprises, after a last one of the one or more pulses, a tail which trails off to a constant value; and wherein the plurality of constraints comprise insensitivity constraints that include a constraint which limits a length of the tail.
Claim 42
Dependent Claim
114 words
An apparatus according to claim 34 , wherein the tail is defined by a function u tail (t) which comprises: u tail ( t ) = ∑ j = 1 r c j z j ( t - t n ) l ( t - t n ) , where “c j ” represents coefficients of exponential terms in the tail, “z j ” represents zeros in the system, “n” represents a total number of switches excluding a first switch, and “r” represents a number of zeros in the system; and wherein the length of the tail is limited by limiting values of coefficients “c j ”.
Claim 43
Dependent Claim
81 words
An apparatus according to claim 35 , wherein the values of the coefficients “c j ” are limited by ∑ j = 1 r c j ζ i ω nj t r ≤ X u lim , where “u lim ” defines an actuator limit, “X” defines a percentage relative to limit u lim within which the tail settles, and ζ i ω nj define a real component of zeros “z” within the system.
Claim 44
Dependent Claim
26 words
An apparatus according to claim 23 , further comprising the step of verifying optimality of the input which includes the switch times determined by the processor.
Claim 45
Dependent Claim
30 words
An apparatus according to claim 37 , wherein the optimality of the input which includes the switch times determined in the determining step is verified based on Pontryagin's Minimum Principle.
Claim 46
Independent Claim
258 words
An apparatus which determines an input command profile for a dynamic system that can be monitored as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the apparatus comprising: a memory which stores computer-executable process steps; and a processor which executes the process steps stored in the memory so as (i) to identify characteristics of the dynamic system, (ii) to select a command profile, u(t), which defines an input to the dynamic system based on identified characteristics, the command profile comprising one or more pulses which rise and fall at switch times and a tail which approaches a constant value, the command profile being useable with substantially any dynamic system that can be modeled as a linear system, (iii) to impose a plurality of constraints on the dynamic system, wherein the plurality of constraints include an actuator limit constraint corresponding to limits in the system which keep the input within limits of the actuator, and wherein the actuator limit constraint comprises: u min ≦u tail ( t )≦ u max , where u min represents a minimum value of the input, u tail (t) represents a value of a tail of the input at time “t”, and u max represents a maximum value of the input, and wherein at least one of the constraints being defined in terms of the switch times, and (iv) to determine the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 47
Independent Claim
258 words
An apparatus which determines an input command profile for a dynamic system that can be monitored as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the apparatus comprising: a memory which stores computer-executable process steps; and a processor which executes the process steps stored in the memory so as (i) to identify characteristics of the dynamic system, (ii) to select a command profile, u(t), which defines an input to the dynamic system based on identified characteristics, the command profile comprising one or more pulses which rise and fall at switch times and a tail which approaches a constant value, the command profile being useable with substantially any dynamic system that can be modeled as a linear system, (iii) to impose a plurality of constraints on the dynamic system, wherein the plurality of constraints include an actuator limit constraint corresponding to limits in the system which keep the input within limits of the actuator, and wherein the actuator limit constraint comprises: u min ≦u tail ( t )≦ u max , where u min represents a minimum value of the input, u tail (t) represents a value of a tail of the input at time “t”, and u max represents a maximum value of the input, and wherein at least one of the constraints being defined in terms of the switch times, and (iv) to determine the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 48
Independent Claim
190 words
An apparatus which determines an input command profile for a dynamic system that can be monitored as a linear system, the input command profile for tansitioning an output of the dynamic system from one state to another state, the apparatus comprising: a memory which stores computer, executable process steps; and a processor which executes the process steps stored in the memory so as (i) to identity characteristics of the dynamic system, (ii) to select a command profile which defines an input to the dynamic system based on identified characteristics, the command profile comprising one or more pulses which rise and fall at switch times and the command profile being useable with substantially any dynamic system that can be modeled as a linear system, (iii) to impose a plurality of constraints on the dynamic system, wherein the plurality of constraints comprise insensitivity constraints which improve system robustness with respect to predetermined system parameters, at least one of the constraints being defined in terms of the switch times, and (iv) to determine the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 49
Independent Claim
190 words
An apparatus which determines an input command profile for a dynamic system that can be monitored as a linear system, the input command profile for tansitioning an output of the dynamic system from one state to another state, the apparatus comprising: a memory which stores computer, executable process steps; and a processor which executes the process steps stored in the memory so as (i) to identity characteristics of the dynamic system, (ii) to select a command profile which defines an input to the dynamic system based on identified characteristics, the command profile comprising one or more pulses which rise and fall at switch times and the command profile being useable with substantially any dynamic system that can be modeled as a linear system, (iii) to impose a plurality of constraints on the dynamic system, wherein the plurality of constraints comprise insensitivity constraints which improve system robustness with respect to predetermined system parameters, at least one of the constraints being defined in terms of the switch times, and (iv) to determine the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.
Claim 50
Dependent Claim
41 words
An apparatus according to claim 40 , wherein an insensitivity constraint is determined by setting to zero a derivative of a function, y p (t), which defines a residual amplitude response of the system for each pole “p” in the system.
Claim 51
Dependent Claim
44 words
An apparatus according to claim 40 , wherein an insensitivity constraint is determined by setting to zero a derivative of a function y z (t), which defines a contribution of each zero “z” in the system to a residual amplitude response of the system.
Claim 52
Independent Claim
272 words
An apparatus which determines an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the apparatus comprising: a memory which stores computer-executable process steps; and a processor which executes the process steps stored in the memory so as (i) to select a command profile for the input to the dynamic system, the command profile comprising a pulse train with pulses that rise and fall at switch times and being useable with substantially any dynamic system that can be modeled as a linear system, the processor selecting the command profile by setting pulse train amplitudes based on actuator limits, determining if there are system zeros and including a tail in the command profile in a case that there are system zeros, making an initial guess for the switch times and, in a case that there are system zeros, making an initial guess for coefficients defining the tail, (ii) to impose constraints on the system, the constraints comprising dynamics cancellation constraints, boundary condition constraints, and, in a case that there are system zeros, actuator limit constraints, (iii) to solve for the switch times based on (a) the command profile selected in the selecting step, (b) the initial guess for the switch times, (c) in a case that there are zeros in the system, the initial guess for the coefficients defining the tail, and (d) the problem constraints imposed in the imposing step, and (iv) to determine whether the switch times define an input which makes the command profile time optimal.
Claim 53
Independent Claim
272 words
An apparatus which determines an input command profile for a dynamic system that can be modeled as a linear system, the input command profile for transitioning an output of the dynamic system from one state to another state, the apparatus comprising: a memory which stores computer-executable process steps; and a processor which executes the process steps stored in the memory so as (i) to select a command profile for the input to the dynamic system, the command profile comprising a pulse train with pulses that rise and fall at switch times and being useable with substantially any dynamic system that can be modeled as a linear system, the processor selecting the command profile by setting pulse train amplitudes based on actuator limits, determining if there are system zeros and including a tail in the command profile in a case that there are system zeros, making an initial guess for the switch times and, in a case that there are system zeros, making an initial guess for coefficients defining the tail, (ii) to impose constraints on the system, the constraints comprising dynamics cancellation constraints, boundary condition constraints, and, in a case that there are system zeros, actuator limit constraints, (iii) to solve for the switch times based on (a) the command profile selected in the selecting step, (b) the initial guess for the switch times, (c) in a case that there are zeros in the system, the initial guess for the coefficients defining the tail, and (d) the problem constraints imposed in the imposing step, and (iv) to determine whether the switch times define an input which makes the command profile time optimal.
Claim 54
Dependent Claim
84 words
An apparatus according to claim 43 , wherein, in a case that the processor determines that the switch times do not define an input which make the command profile time optimal, the processor executes process steps so as (i) to repeat the solving step using different values for the switch times and, in a case that there are zeros in the system, different values for the coefficients defining the tail, and (ii) to repeat the determining step for newly-determined values of the switch times.
References Cited (Backward Citations)
USPTO Annuity & Maintenance Windows
| Annuity Stage | Window Opens | Payment Deadline | Surcharge Period Until | Estimated Fee | Status |
|---|---|---|---|---|---|
| 3.5 Years (Year 4) | 2006-01-06 | 2006-07-08 | 2007-01-07 | $2000.00 | Window Closed / Lapsed |
| 7.5 Years (Year 8) | 2010-01-06 | 2010-07-08 | 2011-01-07 | $3760.00 | Window Closed / Lapsed |
| 11.5 Years (Year 12) | 2014-01-06 | 2014-07-08 | 2015-01-07 | $7700.00 | Window Closed / Lapsed |