학습기반 공정 동적최적화

M0000.026500

학습기반 공정 동적최적화

J^ONG M^IN L^EE

School of Chemical & Biological Engineering

Lecture 3: Dynamic Matrix Control (DMC)

In this lecture, we will discuss

• Process representation: step response model

• Prediction (perfect model)

• Incorporation of “feedback”

• Optimization: unconstrained and constrained QP

• Implementation

The Genesis of MPC

- From Prof. Manfred Morari’s Slides

Who invented predictive control?

God ...

Predictive control is a discovery, not an invention, ...

But God need prophets.

IFAC Congress

Munich, 1987

Dynamic Matrix Control

First appeared in the open literature in 1979 (Cutler ad Ramaker; Prett and Gillette)

Dynamic Matrix Control

• Reformulation as a quadratic program (QP) by Garcia and Morshedi in 1986

“Quadratic Dynamic Matrix Control”

• AspenTech: DMCplus

• Prototype of commercial algorithms presently used

in the process industry

Process Representation

Stable, Single Input Single Output (SISO):

Dynamic System

unit step-response function:

u 1

Ts

y

Ts

S₀ S₁ S₂

S_n S_n+1 S_n = S_n+1= S_n+2 = ... = S_∞ S₀ = 0

u 1

Ts

y

Ts

S₀ S₁ S₂

S_n S_n+1 S_n = S_n+1 = S_n+2 = ... = S_∞ S₀ = 0

S = [S

, S

, S

, . . . , S

]

Complete description of the process requires n step response coefficients 0

6

Principle of Superposition

Linearity

1 2

1/2

1 1

=

-3/2

u(0) u(1)

u(2)

+ +

u u u u

?

y

y(0)

y(1) y(2)

y(3) y(4)

y(5)

u(k i+ 1) =u(k i+ 1) u(k i)

y(1) = y(0) +S₁ u(0)

y(2) = y(0) +S1 u(1) +S2 u(0) ... = ...

y(k+ 1) = y(0) +

nX1

i=1

Si u(k i+ 1) +Sn{ u(k n+ 1) + u(k n) +· · · u(0)}

= y(0) +

nX1

i=1

S_i u(k i+ 1) +S_nu(k n+ 1)

7

Elements of DMC

target

Future Past

k+1 k+m-1 k+p

k input

Horizon

∆u(k)

∆u(k+1)

~y(k) ~y(k+1) y(k+2)

~

1. Prediction (Stable, SISO)

At time k: we know y(k) and need to compute ∆u(k), which we don’t know yet.

i.e., y is measured slightly before time k and u is adjusted slightly after time k.

ˆ

y(k + 1) : prediction of y(k+1) made at time k. Assume y(0) = 0 ˆ

y(k + 1) =

nX1

i=1

S_i u(k i + 1) + S_nu(k n + 1)

Effect of current

control action Effect of past control actions

ˆ

y(k + 1) = S₁ u(k) +

nX1

i=2

S_i u(k i + 1) + S_nu(k n + 1)

Substitute k = k+1, and

Effect of future control action

Effect of current

control action Effect of past control actions

ˆ

y(k + 2) = S₁ u(k + 1) + S₂ u(k) +

nX1

i=3

S_i u(k i + 2) + S_nu(k n + 2)

j-step ahead prediction

y(k + j) = Xj

i=1

S_i u(k + j i) +

nX1

i=j+1

S_i u(k + j i) + S_nu(k + j n)

Effect of current and future control actions Effect of past control actions

Let

ˆ

y⁰(k + j) =

nX1

i=j+1

S_i u(k + j i) + S_nu(k + j n)

This is referred to as “predicted unforced response” with past inputs only.

U = [. . . , u(k 2), u(k 1), 0, 0, 0, . . .]^T for j = 1

ˆ

y(k + j ) =

X

i=1

S

u(k + j i) + ˆ y

(k + j )

Multiple Predictions

Yˆ ⁰(k + 1) = ⇥ ˆ

y⁰(k + 1), yˆ⁰(k + 2), · · · , yˆ⁰(k + p)⇤^T U(k) = [ u(k), u(k + 1), · · · , u(k + m 1)]^T

Y(kˆ + 1) = [ˆy(k + 1), y(kˆ + 1), · · · , y(kˆ + p)]^T p: prediction horizon

m: control horizon m  p  n + m

Y(k ˆ + 1) = S U(k) + ˆ Y

(k + 1) matrix form

S =

2 66 66 66 66 66 4

S₁ 0 0 · · · 0

S₂ S₁ 0 · · · 0

S₃ S₂ S₁ · · · 0

· · · ·

S_m S_{m 1} S_{m 2} · · · S₁

S_m+1 S_m S_{m 1} · · · S₂

· · · ·

S_p S_{p 1} S_{p 2} · · · S_{p m+1}

3 77 77 77 77 77 5

Output Feedback and Bias Correction

So far, we have not utilized the latest observation, y(k)

The fact is that there is no perfect model. Corrected prediction by adding a constant bias term.

˜

y(k + j) = ˆy(k + j) + b(k + j)

b(k + j) = y(k) y(k) ˆ ˆ

y(k)

˜

y(k + j ) = ˆ y(k + j ) + [y(k) y(k)] ˆ

Y(k˜ + 1) = S U(k) + ˆY⁰(k + 1) + [y(k) y(kˆ )]1

Y(k˜ + 1) = [˜y(k + 1), y˜(k + 2), · · · , y(k˜ + p)]^T 1 = [1, 1, ,· · · , 1]^T

Recursive Update of Unforced Response

For stable models, one can update the predicted unforced response after u(k) is computed.

Yˆ_n⁰(k + 1) = MYˆ_n⁰(k) +S^⇤ u(k)

works like a state; hence you need “n” not “p”

or

2 66 64

ˆ

y⁰(k + 1) ˆ

y⁰(k + 2) ...

ˆ

y⁰(k + p) 3 77 75 =

2 66 66 66 4

0 1 0 · · · 0

0 0 1 . .. 0 ... ... ... . .. 0

0 0 · · · 0 1

0 0 · · · 0 1

3 77 77 77 5

2 66 64

ˆ y⁰(k) ˆ

y⁰(k + 1) ...

ˆ

y⁰(k + p 1) 3 77 75 +

2 66 64

S₁ S₂ ... S_p

3 77

75 u(k)

n n n

Why?

To achieve good control performance

• Y˜(k+1) should be close to the true open-loop output

• This requires that n, the number of coefficient matrices in S* is chosen that S

= S

(i.e., plant should be stable), otherwise MY

will be in error.

• It also requires the feedback term stays approximately constant (Step

disturbance).

1. Prediction

Stable, MIMO

2-by-2 System

ˆ

y1(k + 1) =

nX1

i=1

S11,i u1(k i + 1) + S11,nu1(k n + 1)

+

nX1

i=1

S_12,i u₂(k i + 1) + S_12,nu₂(k n + 1)

ˆ

y₂(k + 1) =

nX1

i=1

S_21,i u₁(k i + 1) + S_21,nu₁(k n + 1)

+

nX1

i=1

S_22,i u₂(k i + 1) + S_22,nu₂(k n + 1)

Vector Notation

y = 2 66 64

y1

y2

... ym

3 77

75 u = 2 66 64

u1

u2

... ur

3 77

75 I_p = 2 64

I ... I

3 75 pny-by-ny

Y(k˜ + 1) = S U(k) + ˆY⁰(k + 1) + I_p [y(k) y(k)]ˆ

Y(k˜ + 1) = 2 66 64

˜

y(k + 1)

˜

y(k + 2) ... y(k˜ +p)

3 77

75 U(k) =

2 66 64

u(k) u(k + 1)

...

u(k + m 1) 3 77 75 Yˆ⁰(k + 1) =

2 66 64

yˆ⁰(k + 1) yˆ⁰(k + 2)

... ˆ

y⁰(k +p) 3 77 75

Use up to p only out of n

S = 2 66 66 66 66 66 64

S₁ 0 · · · 0

S₂ S₁ 0 ...

... ... . .. 0

S_m S_{m 1} · · · S₁

S_m+1 S_m · · · S₂

... ... . .. ...

S_p S_{p 1} · · · S_{p m+1}

3 77 77 77 77 77 75

S_i = 2 66 64

S_11,i S_12,i · · · S_1r,i

S_21,i · · · · · · S_2r,i

... ... ... ...

S_m1,i · · · · · · S_mr,i

3 77 75

17

Recursive Update of Unforced Response

Y ˆ

(k + 1) = M Y ˆ

(k) + S

u(k)

or

2 66 64

ˆ

y⁰(k + 1) ˆ

y⁰(k + 2) ... ˆ

y⁰(k +p) 3 77 75 =

2 66 66 66 4

0 I_m 0 · · · 0

0 0 I_m . .. 0 ... ... ... . .. 0

0 0 · · · 0 I_m

0 0 · · · 0 I_m

3 77 77 77 5

2 66 64

ˆ y⁰(k) ˆ

y⁰(k + 1) ... ˆ

y⁰(k + p 1) 3 77 75 +

2 66 64

S₁ S₂ ... S_p

3 77

75 u(k) n

n n

2. Control Calculations

Objective functions

At time k, minimize the predicted deviation of the output from the setpoint with some penalty on the input movement size measured in terms of the quadratic norm.

minU(k)

( _p X

i=1

(y_r(k + i) y(k˜ + i))^T Q(y_r(k + i) y(k˜ + i))

+

mX1

`=0

u^T(k + `)R u(k + `) )

Q, R: weighting matrices (diagonal)

Constraints

ymax target

Future Past

umax future input

t+1 t+m-1 t+p

t input

measurements

Horizon

projected output

Input magnitude

u_min  u(k + `)  u_max

Input rate

| u(k + `)|  u_max

Output magnitude

ymin  y(k˜ + i)  ymin

Solve Quadratic Program

A U(k)  b min

⇢ 1

2 U

(k) H U(k) + f

U(k)

H f

A b

U(k)

: Hessian matrix : gradient vector : constraint matrix : constraint vector

: decision variable

We need to convert the MPC objective and constraints to the standard QP form.

Unconstrained QP

min

⇢ 1

2 U

(k) H U(k) + f

U(k)

Take the gradient w.r.t. the input:

H U(k) + f = 0 U(k) = H ¹f

Objective Function in Quadratic Form

minU(k)

( _p X

i=1

(yr(k + i) y(k˜ + i))^T Q(yr(k + i) y(k˜ + i)) +

mX1

`=0

u^T(k + `)R u(k + `) )

2 66 64

y_r(k + 1) y(k˜ + 1) y_r(k + 2) y(k˜ + 2)

...

yr(k + p) y(k˜ + p) 3 77 75

T 2 66 64

Q Q

. ..

Q 3 77 75

2 66 64

y_r(k + 1) y(k˜ + 1) y_r(k + 2) y(k˜ + 2)

...

yr(k + p) y(k˜ + p) 3 77 75

+ 2 66 64

u(k) u(k + 1)

...

u(k + m 1) 3 77 75

T 2 66 64

R R

. ..

R 3 77 75

2 66 64

u(k) u(k + 1)

...

u(k + m 1) 3 77 75

⇣Y_r(k + 1) Y(k˜ + 1)⌘T

Q¯ ⇣

Y_r(k + 1) Y(k˜ + 1)⌘

+ U^T(k) ¯R U(k)

Not Done Yet!

⇣Y_r(k + 1) Y(k˜ + 1)⌘T

Q¯ ⇣

Y_r(k + 1) Y(k˜ + 1)⌘

+ U^T(k) ¯R U(k)

Y(k˜ + 1) = S U(k) + ˆY⁰(k + 1) + I_p[y(k) y(k)]ˆ This yields

"(k + 1) = Y_r(k + 1) Yˆ ⁰(k + 1) I_p[y(k) y(k)]ˆ

where

is a known term.

"^T(k + 1) ¯Q"(k + 1) 2"^T(k + 1) ¯QS U(k) + U^T(k) S^TQS¯ + ¯R U(k)

Hessian (a constant matrix): H = S^TQS¯ + ¯R

Gradient vector (must be updated at each time):

f

= "

(k + 1) ¯ QS

Unconstrained Solution with a Gain Form

MPC toolbox in Matlab uses a gain form for the unconstrained solution:

u(k ) = ⇥

I 0 · · · 0 ⇤

U(k )

= K

⇥ "(k + 1)

, where U(k) = H

f and K

is a constant gain matrix

Constraints in Linear Inequality Form

u

 u(k + `)  u

| u(k + `) |  u

y

 y(k ˜ + i)  y

A U(k )  b

i = 1, · · · , p

` = 0, · · · , m 1

Input Magnitude Constraint

u

 u(k + `)  u

, ` = 0, · · · , m 1

u(k 1)

X`

i=0

u(k + i)  u_min

u(k 1) + X`

i=0

u(k + i)  u_max

2 66 66 66 66 66 66 66 4

2 66 66 4

I 0 · · · 0

I I 0 ... ... ... . .. 0

I I · · · I

3 77 77 5 2

66 66 4

I 0 · · · 0

I I 0 ... ... ... . .. 0

I I · · · I

3 77 77 5

3 77 77 77 77 77 77 77 5

2 66 64

u(k) u(k + 1)

...

u(k +m 1) 3 77 75 

2 66 66 66 66 66 66 4

2 66 64

u_min u(k 1) u_min u(k 1)

...

u_min u(k 1) 3 77 75

2 66 64

u_max u(k 1) u_max u(k 1)

...

u_max u(k 1) 3 77 75

3 77 77 77 77 77 77 5

I L

27

Input Rate Constraints

| u(k + `) |  u

` = 0, · · · , m 1

u

 u(k + `)  u

u(k + `)  u

u(k + `)  u

2 66 66 66 66 66 66 66 4

2 66 66 4

I 0 · · · 0

0 I 0 ... ... ... . .. 0

0 0 · · · I

3 77 77 5 2

66 66 4

I 0 · · · 0

0 I 0 ... ... ... . .. 0

0 0 · · · I

3 77 77 5

3 77 77 77 77 77 77 77 5

2 66 64

u(k) u(k + 1)

...

u(k +m 1) 3 77 75 

2 66 66 66 66 66 66 4

2 66 64

u_max u_max

... u_max

3 77 75

2 66 64

u_max u_max

... u_max

3 77 75

3 77 77 77 77 77 77 5

I

Output Magnitude Constraints

y

 y(k ˜ + i)  y

, i = 1, · · · , p

y(k ˜ + i)  y

˜

y(k + i)  y

 S U(k) + ˆY⁰(k + 1) + Ip(y(k) y(kˆ ))

S U(k) Yˆ⁰(k + 1) I_p(y(k) y(kˆ )) 

 Ymax

Ymin Y_max = 2 66 64

y_max y_max

... y_max

3 77

75 Y_min = 2 66 64

y_min y_min

... y_min

3 77 75

 S

S U(k) 

 Y_max Yˆ ⁰(k + 1) I_p (y(k) y(k))ˆ Y_min + ˆY⁰(k + 1) + I_p (y(k) y(k))ˆ

In Summary

2 66 66 66 4

I_L I_L

I I

S S

3 77 77 77 5

U(k)  2 66 66 66 66 66 66 66 66 66 66 66 66 66 4

2 64

u_min u(k 1) ...

u_min u(k 1) 3 75

2 64

u_max u(k 1) ...

u_max u(k 1) 3 75

2 64

u_max ... u_max

3 75

2 64

u_max ... u_max

3 75

 Y_max Yˆ⁰(k + 1) I_p(y(k) y(k))ˆ Y_min + ˆY⁰(k + 1) +I_p(y(k) y(k))ˆ

3 77 77 77 77 77 77 77 77 77 77 77 77 77 5

A U(k)  b

2 66 66 66 4

I_L I_L I I S S

3 77 77 77 5

Solving QP

• Quadratic program: minimization of quadratic function subject to linear inequality constraints.

• QPs are convex and therefore fundamentally tractable.

• Off-the-shelf solvers (e.g., QPSOL, QUADPROG) are

available but further customization is desirable (to exploit the structure in the Hessian and constraint matrices)

• Complexity of a QP is a complex function of the dimension/

structure of Hessian, as well as the number of constraints.

Active Set Method

Interior Point Method

Real-Time Implementation

1. Initialization: Initialize the memory vector and the reference vector and the reference vector . Set k = 0.

2. Memory update:

3. Reference vector update

4. Measurement intake: Take in new measurement y(k) and ▵d(k) 5. Calculation of the gradient vector and constraint vector

6. Solve QP

7. Implementation if input

8. Go back to Step 2 after setting and the reference vector Y(0)ˆ

Yˆ⁰(k + 1) = MYˆ⁰(k) + S^⇤ u(k)

u(k) = u(k 1) + u(k)