Method and apparatus for creating time-optimal commands for linear systems

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 ) ⁢ ⅇ - st j + ⅇ - 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 “aj” represents an amplitude of pulses in the command profile, “tj” represents switch times for the pulses, “cj” represents coefficients of exponential terms in the tail, “zj” 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, “pi” represents system poles, “qi” represents a multiplicity of each system pole “p”, and where β is defined as follows:
• β1(k)=1, k=1=0, k≠1
• βq1(k)=βq1−1(k)+(k−1)βq1−1(k−1);
• and
• determining the switch times for the input to the dynamic system based on the command profile and the plurality of constraints.

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:
• umin≦utail(t)≦umax,
• where umin represents a minimum value of the input, utail(t) represents a value of a tail of the input at time “t”, and umax 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.

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.

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.

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.

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:
• umin≦utail(t)≦umax,
• where umin represents a minimum value of the input, utail(t) represents a value of a tail of the input at time “t”, and umax 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.

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.