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

• US 6,505,085 B1
• Filed: 03/04/1999
• Issued: 01/07/2003
• Est. Priority Date: 03/04/1999
• Status: Expired due to Fees

• Current Assignee: Massachusetts Institute of Technology
• Original Assignee: Massachusetts Institute of Technology
• Inventors: Tuttle, Timothy D., Seering, Warren P.
• Primary Examiner(s): Grant, William
• Assistant Examiner(s): Garland, Steven R.
• Application Number:
  US09/261,987
• Time in Patent Office: 1,405 Days
• Field of Search: 700/28, 700/31, 700/254, 700/29, 700/32-35, 700/44, 700/45, 700/39, 700/52, 700/54, 700/55, 700/250, 318/561
• US Class Current: 700/28
• CPC Class Codes: B25J 9/1605 Simulation of manipulator lay-out, design, modelling of manipulator
• CPC Class Codes: G05B 13/047 the criterion being a time optimal performance criterion
• CPC Class Codes: G05B 2219/35585 Motion command profile
• CPC Class Codes: G05B 2219/39229 Linear parameterization of robot dynamics
• CPC Class Codes: G05B 2219/39358 Time optimal control along path for singular points, having veloctiy constraints
• CPC Class Codes: G05B 2219/43168 Motion profile planning for point to point control

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.

Claims

1. 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.

2. A method according to claim 1, wherein the command profile is based on the following function:

$u<br><br>\left(t\right)=\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>l<br><br>\left(t-t_j\right)+\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>c_j<br><br>{\mathrm{<br><br>}}^{z_j<br><br>\left(t-t_n\right)}<br><br>l<br><br>\left(t-t_n\right),$

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.

3. 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.

4. 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.

5. A method according to claim 4, wherein the boundary condition constraints are based on the following equations:

6. 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.

7. A method according to claim 1, wherein the determining step comprises determining the switch times based on:

(i) a command profile, u(t), comprising;

$u<br><br>\left(t\right)=\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>l<br><br>\left(t-t_j\right)+\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>c_j<br><br>{\mathrm{<br><br>}}^{z_j<br><br>\left(t-t_n\right)}<br><br>l<br><br>\left(t-t_n\right),$

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;

$\left(\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>t_j^{\left(q_i-1\right)}<br><br>{\mathrm{<br><br>}}^{-{\mathrm{st}}_j}+{\mathrm{<br><br>}}^{-{\mathrm{st}}_n}<br><br>\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>\left[c_j<br><br>t_n^{\left(q_i-1\right)}+\underset{k=1}{\overset{q_i}{\sum <br><br>}}<br><br>\left({\mathrm{\beta <br><br>}}_{q_i}<br><br>\left(k\right)\right)<br><br>\frac{c_j<br><br>z_j}{{\left(s-z_j\right)}^k}<br><br>t_n^{\left(q_i-k\right)}\right]\right)<br><br>{|}_{s=p_i}=0,$

where “

p_i”

represents system poles, “

q_i”

represents a multiplicity of each system pole “

p_i”

, and where β

is defined as follows;

8. A method according to claim 7, wherein the switch times are determined using an optimization algorithm.

9. 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:

$u<br><br>\left(t\right)=\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>l<br><br>\left(t-t_j\right),$

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.

10. A method according to claim 9, wherein (i) the dynamics cancellation constraint comprises:

$\left(\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>t_j^{\left(q_i-1\right)}<br><br>{\mathrm{<br><br>}}^{-{\mathrm{st}}_j}\right)<br><br>{|}_{s=p_i}=0,$

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;

$<br><br>y=\underset{s-><br><br>0}{\lim }<br><br>\left(\frac{N<br><br>\left(s\right)}{D<br><br>\left(s\right)}\right)<br><br>\left({\int <br><br>}^{\left(\mathrm{rbp}\right)}<br><br>u<br><br>\left(t\right)<br><br><br><br>t\right)<br><br>{|}_{t=\mathrm{\infty <br><br>}},$

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.

11. 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.

12. 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.

32. 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:

$u<br><br>\left(t\right)=\underset{j=0}{\overset{u}{\sum <br><br>}}<br><br>a_j<br><br>l<br><br>\left(t-t_j\right),$

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.

33. An apparatus according to claim 32, wherein (i) the dynamics cancellation constraint comprises:

$\left(\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>t_j^{\left(q_i-1\right)}<br><br>{\mathrm{<br><br>}}^{-{\mathrm{st}}_j}\right)<br><br>{|}_{s=p_i}=0,$

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;

13. 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;

$\left(\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>t_j^{\left(q_i-1\right)}<br><br>{\mathrm{<br><br>}}^{-{\mathrm{st}}_j}+{\mathrm{<br><br>}}^{-{\mathrm{st}}_n}<br><br>\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>\left[c_j<br><br>t_n^{\left(q_i-1\right)}+\underset{k=1}{\overset{q_i}{\sum <br><br>}}<br><br>\left({\mathrm{\beta <br><br>}}_{q_i}<br><br>\left(k\right)\right)<br><br>\frac{c_j<br><br>z_j}{{\left(s-z_j\right)}^k}<br><br>t_n^{\left(q_i-k\right)}\right]\right)<br><br>{|}_{s=p_i}=0,$

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;

14. 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;

15. 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.

16. 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.

17. 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.

18. 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.

19. A method according to claim 18, wherein the tail is defined by a function u_tail(t) which comprises:

$u_{\mathrm{tail}}<br><br>\left(t\right)=\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>c_j<br><br>{\mathrm{<br><br>}}^{z_j<br><br>\left(t-t_n\right)}<br><br>l<br><br>\left(t-t_n\right),$

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”

.

20. A method according to claim 19, wherein the values of the coefficients “

c_j”

are limited by $\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>c_j<br><br>{\mathrm{<br><br>}}^{{\mathrm{\zeta <br><br>}}_i<br><br>{\mathrm{\omega <br><br>}}_{\mathrm{nj}}<br><br>t_r}\le <br><br>X<br><br>\text{\hspace{1em}<br><br>}<br><br>u_{\lim },$

where “

u_lim”

defines an actuator limit, “

X”

defines a percentage relative to limit u_lim within which the tail settles, and ζ

_iω

_nj define a real component of zeros “

z”

within the system.

21. 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.

22. 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.

23. 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.

24. An apparatus according to claim 23, wherein the command profile is based on the following function:

$u<br><br>\left(t\right)=\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>l<br><br>\left(t-t_j\right)+\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>c_j<br><br>{\mathrm{<br><br>}}^{z_j<br><br>\left(t-t_n\right)}<br><br>l<br><br>\left(t-t_n\right),$

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.

25. 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.

26. 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;

$\left(\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>t_j^{\left(q_i-1\right)}<br><br>{\mathrm{<br><br>}}^{-{\mathrm{st}}_j}+{\mathrm{<br><br>}}^{-{\mathrm{st}}_n}<br><br>\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>\left[c_j<br><br>t_n^{\left(q_i-1\right)}+\underset{k=1}{\overset{q_i}{\sum <br><br>}}<br><br>\left({\mathrm{\beta <br><br>}}_{q_i}<br><br>\left(k\right)\right)<br><br>\frac{c_j<br><br>z_j}{{\left(s-z_j\right)}^k}<br><br>t_n^{\left(q_i-k\right)}\right]\right)<br><br>{|}_{s=p_i=0,}$

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;

$\begin{array}{c}{\mathrm{\beta <br><br>}}_1<br><br>\left(k\right)=\text{\hspace{1em}<br><br>}<br><br>1,k=1\\ =\text{\hspace{1em}<br><br>}<br><br>0,k\ne <br><br>1\\ {\mathrm{\beta <br><br>}}_{q_i}<br><br>\left(k\right)=\text{\hspace{1em}<br><br>}<br><br>{\mathrm{\beta <br><br>}}_{q_i-1}<br><br>\left(k\right)+\left(k-1\right)<br><br>{\mathrm{\beta <br><br>}}_{q_i-1}<br><br>\left(k-1\right).\end{array}$

27. 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.

28. An apparatus according to claim 27, wherein the boundary condition constraints are based on the following equations:

29. 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.

30. An apparatus according to claim 23, wherein the processor determines the switch times based on:

(i) a command profile, u(t), comprising;

$u<br><br>\left(t\right)=\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>l<br><br>\left(t-t_j\right)+\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>c_j<br><br>{\mathrm{<br><br>}}^{z_j<br><br>\left(t-t_n\right)}<br><br>l<br><br>\left(t-t_n\right),$

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;

$\left(\underset{j=0}{\overset{n}{\sum <br><br>}}<br><br>a_j<br><br>t_j^{\left(q_i-l\right)}<br><br>{\mathrm{<br><br>}}^{-{\mathrm{zs}}_j}+{\mathrm{<br><br>}}^{-s_{\mathrm{tu}}}<br><br>\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>\left[c_j<br><br>t_u^{\left(q_i-l\right)}+\underset{k=1}{\overset{q_1}{\sum <br><br>}}<br><br>\left({\mathrm{\beta <br><br>}}_{q_1}<br><br>\left(k\right)\right)<br><br>\frac{c_j<br><br>z_j}{{\left(s-z_j\right)}^k}<br><br>t_n^{\left(q_1-k\right)}\right]\right)<br><br>{|}_{s=p_1}=0,$

where “

p_i”

represents system poles, “

q_i”

represents a multiplicity of each system pole “

p_i”

, and where β

is defined as follows;

31. An apparatus according to claim 30, wherein the switch times are determined using an optimization algorithm.

34. 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.

35. An apparatus according to claim 34, wherein the tail is defined by a function u_tail(t) which comprises:

$u_{\mathrm{tail}}<br><br>\left(t\right)=\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>c_j<br><br>{\mathrm{<br><br>}}^{z_j<br><br>\left(t-t_n\right)}<br><br>l<br><br>\left(t-t_n\right),$

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”

.

36. An apparatus according to claim 35, wherein the values of the coefficients “

c_j”

are limited by $\underset{j=1}{\overset{r}{\sum <br><br>}}<br><br>c_j<br><br>{\mathrm{<br><br>}}^{{\mathrm{\zeta <br><br>}}_i<br><br>{\mathrm{\omega <br><br>}}_{\mathrm{nj}}<br><br>t_r}\le <br><br>X<br><br>\text{\hspace{1em}<br><br>}<br><br>u_{\lim },$

where “

u_lim”

defines an actuator limit, “

X”

defines a percentage relative to limit u_lim within which the tail settles, and ζ

_iω

_nj define a real component of zeros “

z”

within the system.

37. 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.

38. 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.

39. 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;

40. 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.

41. 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.

42. 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.

43. 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.

44. 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.