Method and Apparatus for Adaptive Parallel Proportional-Integral-Derivative Controller

• US 20090248177A1
• Filed: 03/28/2008
• Published: 10/01/2009
• Est. Priority Date: 03/28/2008
• Status: Active Grant

• Current Assignee: Mitsubishi Electric Research Laboratories, Inc. (Mitsubishi Electric Corporation)
• Original Assignee: Mitsubishi Electric Research Laboratories, Inc. (Mitsubishi Electric Corporation)
• Inventors: Rifai, Khalid EL
• Granted Patent: US 7,805,207 B2
• Time in Patent Office: Days
• US Class Current: 700/45
• CPC Class Codes: G05B 11/36 with provision for obtaining particular characteristics, e.g. proportional, integral, differential
• CPC Class Codes: G05B 13/02 electric

Abstract

An adaptive parallel proportional-integral-derivative controller produces a fixed controller output including a fixed proportional-integral-derivative and a fixed feedforward controller command, and an adaptive controller output including an adaptive parallel PID and an adaptive feedforward command all from a reference command. The fixed controller output and the adaptive controller output are added to produce a control command for a controlled system, which provides a measure of an output and a rate of change of the output as feedback for the controller.

Claims

1. An apparatus for controlling a system, comprising:

a fixed controller comprising;

a fixed proportional-integral-derivative (PID) controller; and

a fixed feedforward controller, in which the fixed PID controller and the fixed feedforward controller are configured to receive a reference command and to produce a fixed controller output comprising a fixed PID and a fixed feedforward controller command;

an adaptive parallel controller comprising;

an adaptive parallel PID controller; and

an adaptive feedforward controller, in which the adaptive parallel PID controller and the adaptive feedforward controller are configured to receive the reference command and to produce adaptive controller output comprising an adaptive parallel PID and an adaptive feedforward; and

means for adding the fixed controller output and the adaptive parallel controller output to produce a control command for a controlled system, the controlled system providing a measure of an output and a rate of change of the output as feedback to the fixed controller and the adaptive controller.

2. The apparatus of claim 1, in which gains of the adaptive parallel controller and the adaptive feedforward controller are adjusted dynamically.

3. The apparatus of claim 1, in which the controlled system has a dominant order n, with n=1 for first order dominant processes, such as temperature or velocity controlled systems.

4. The apparatus of claim 1, in which the controlled system has a dominant order n, with n=2 for second order dominant processes, such as position controlled systems.

5. The apparatus of claim 1, in which

z=−

(d/dt+K_pp)<sup>n−

1</sup>e

z_I=∫

zdt

where t is time, K_pp>

0 is a selected scalar gain, e=r−

y is a tracking error for the reference command r, and y is the output.

6. The apparatus of claim 5, in which the control command is

u=K_puz+K_iuz_I+K_ffw_ff+u_adapt

u_adapt={circumflex over (K)}_ffw_ff+u_a

where K_pv>

0 is a fixed proportional gain, K_iv>

0 is a fixed integral gain, K_ff is a fixed feedforward gain, and an adaptive parallel control command u_adapt includes an adaptive parallel PID command u_a and an adaptive feedforward command {circumflex over (K)}_ffw_ff, and the fixed feedforward gain K_ff and the adaptive feedforward gain {circumflex over (K)}_ff are multiplied by a combined feedforward and feedback signal w_ff=y⁽ⁿ⁾−

ż

.

7. The apparatus of claim 6, in which the controlled system has a dominant order n=1 for first order dominant processes, and the fixed controller is a PI controller, and w_ff={dot over (r)}, where a superscript (.) on the variable r denote a first derivative with respect to time t.

8. The apparatus of claim 6, in which the controlled system has a dominant order n=2 for second order dominant processes, and the fixed controller is a PID controller, and w_ff={umlaut over (r)}+K_ppė

, where a superscript (..) on the variable r denote a second derivative with respect to time t.

9. The apparatus of claim 6, in which the adaptive feedforward gain {circumflex over (K)}_ff is updated according to:

{circumflex over ({dot over (K)}_ff=−

γ

_ffw_ffz−

L_ff{circumflex over (K)}_ff,

where γ

_ff>

0 is an adaptation gain for the adaptive feedforward gain, and L_ff≧

0 is a filter gain.

10. The apparatus of claim 6, in which the adaptive parallel PID control command u_a is

u_a={circumflex over (K)}_Pe+{circumflex over (K)}_Dė

+{circumflex over (K)}_I∫

edt,

where {circumflex over (K)}_P is an adaptive proportional gain, {circumflex over (K)}_D is an adaptive derivative gain, {circumflex over (K)}_I is an adaptive integral gain. The module 250 is a differentiator, in which s is a Laplace variable.

11. The apparatus of claim 10, in which adaptive gains of the adaptive parallel PID control command are updated according to

{circumflex over ({dot over (K)}_P=−

γ

_Pez−

L_P{circumflex over (K)}_P

{circumflex over ({dot over (K)}_D=−

γ

_Dė

z−

L_D{circumflex over (K)}_D

{circumflex over ({dot over (K)}_I=−

γ

_I∫

edtz−

L_I{circumflex over (K)}_I,

where γ

_P, γ

_D, γ

_I>

0 are adaptation gains for proportional, derivative, and integral gains, respectively and L_P, L_D, L_I≧

0 are filter gains used to adjust an adaptation response for adaptive PID gains.

12. The apparatus of claim 6, in which the adaptive parallel PID control command is

u_a={circumflex over (K)}(e+{circumflex over (K)}_Dpė

+{circumflex over (K)}_IP∫

edt),

where {circumflex over (K)}_P is an adaptive proportional gain, {circumflex over (K)}_Dp is an adaptive derivative gain scaled by the adaptive proportional gain, K_Ip is an adaptive integral gain scaled by the adaptive proportional gain.

13. The apparatus of claim 12, in which adaptive gains of the adaptive parallel PID control command are updated according to

$\begin{array}{c}{\stackrel{.}{\hat{K}}}_P=-{\gamma }_{}\end{array}$

P 

( e + K ^ D p 2 

e . + K ^ I p 2 

∫

e 



t ) 

z - L P 

K ^ P K ^ . D p = - γ

D 

K ^ P 2 

e . 

z - L D p 

K ^ D p K ^ . I p = - γ

I 

K ^ P 2 

∫

e 



tz - L I p 

K ^ I p ,

where γ

_P, γ

_Dp, γ

_Ip>

0 are adaptation gains for proportional, derivative, and integral gains, respectively and L_P, L_Dp, L_Ip≧

are the filter gains used to adjust an adaptation response for the adaptive PID gains.

14. The apparatus of claim 6, in which the adaptive parallel PID control command is

u_a={circumflex over (K)}_D(ė

+{circumflex over (K)}_Pde+{circumflex over (K)}_Id∫

edt),

where {circumflex over (K)}_D is an adaptive derivative gain, {circumflex over (K)}_Pd is an adaptive proportional gain scaled by the adaptive the derivative gain, and {circumflex over (K)}_Id 420 is an adaptive integral gain scaled by the adaptive derivative gain.

15. The apparatus of claim 14, in which the adaptive gains are updated according to

$\begin{array}{c}{\stackrel{.}{\hat{K}}}_{\mathrm{Pd}}=-{\gamma }_{}\end{array}$

Pd 

K ^ D 2 

ez - L Pd 

K ^ Pd K ^ . D = - γ

D 

( e . + K ^ Pd 2 

e + K ^ Id 2 

∫

e 



t ) 

z - L D 

K ^ D K ^ . Id = - γ

I 

K ^ D 2 

∫

e 



tz - L Id 

K ^ Id ,

where γ

_Pd, γ

_D, γ

_Id>

0 are adaptation gains for proportional, derivative, and integral gains, respectively and L_Pd, L_D, L_Id≧

are filter gains used to adjust an adaptation response for the adaptive gains.

16. The apparatus of claim 6, in which the adaptive parallel PID control command is

u_a={circumflex over (K)}_I(∫

edt+{circumflex over (K)}_Pie+{circumflex over (K)}_Diė

),

where {circumflex over (K)}_I is an adaptive integral gain, {circumflex over (K)}_Di is an adaptive derivative gain scaled by the adaptive integral gain, {circumflex over (K)}_Pi is an adaptive proportional gain scaled by the adaptive integral gain.

17. The apparatus of claim 16, in which the adaptive gains are updated according to

$\begin{array}{c}{\stackrel{.}{\hat{K}}}_{\mathrm{Pi}}=-{\gamma }_{}\end{array}$

Pi 

K ^ I 2 

ez - L Pi 

K ^ Pi K ^ . Di = - γ

Di 

K ^ I 2 

e . 

z - L Di 

K ^ Di K ^ . I = - γ

I 

( ∫

e 



t + K ^ Pi 2 

e + K ^ Di 2 

e . ) 

z - L I 

K ^ I ,

where γ

_P, γ

_D, γ

_I>

0 are adaptation gains for proportional, derivative, and integral gains, respectively and L_Pi, L_Di, L_I≧

are filter gains used to adjust an adaptation response for the adaptive gains.

18. The apparatus of claim 1, in which the adaptive parallel controller compensates for unknown and varying system parameters such as stiffness and inertia of the controlled system.

19. A method for controlling a system, comprising the steps of:

producing a fixed controller output comprising a fixed proportional-integral-derivative (PID) a fixed feedforward controller command from a reference command;

producing an adaptive controller output comprising an adaptive parallel PID and an adaptive feedforward command from the reference command; and

adding the fixed controller output and the adaptive controller output to produce a control command for a controlled system providing a measure of an output and a rate of change of the output as feedback for the producing steps.

20. The method of claim 19, in which the adaptive controller output is produced in parallel.