Riassunto teoria Process Control

Assumptions:

() rational

- () asymptotically stable

- () ≠ 0, ∀ . . {} ≥ 0 ( ℎ )

- () ⋯ 0

11

() () ⋮ ⋱ ⋮

= = () ∙ ∆() = [ ]

0 ⋯ ()

()

⋯ 0

1

() ⋮ ⋱ ⋮

= () ∙ ∆() = , ℎ ℎ . ℎ

[ ]

()

0 ⋯

∆()

How to design in the case mxm:

Case 1 – m=1 (SISO process) ()

()

= () ∙ ∆() → ∆() = , ∆() . .

()

Case 2 – Forward decoupling approach

()

= () ∙ ∆()

() ()

∆ = ∆ = 1

11 22 ()

12

1 −

() () ()

12 21 11

() ()

∆ = − & ∆ = − → ∆() = [ ]

12 21 ()

() ()

21

11 22 − 1

()

22

() () ()

= − () ∙ ∆

11 11 12 21 ()

≠ ()

{ , the elemements of from the elements on the diagonal of

() () ()

= ∙ ∆ + ()

22 22 12 22

Case 3 – Backward decoupling approach

̅() ̅ (),

̅() = ℸ() + ℎℎ ℸ() () ()

−1

∆() = ( − ℸ())

0, =

()

()

ℸ = {

− , ≠

()

Thanks to the Decoupling based structure:

() ()

- We can design relying on

1 11

() ()

- We can design relying on

2 22

LEZIONE 16-17

The Relative Gain Array (RGA) is a classical widely-used decentralized method for determining the best input-

output pairings for multivariable process control systems. In a decentralized control scheme any component of

the control vector depend on a single component of the error vector.

=

ℎ =

⋯

11 1

⋮ ⋱ ⋮

→ = = [ ]

ℎ

⋯

1

in the relative gain array the sum of all the elements on a row or a column is always positive and equal to one.

The best pairings are for positive and close to 1.

0,8 0,2

◼ ( )

=[ ] → , ( , )

1 1 2 2

0,2 0,8 0,5 0,5

◼ Limit case: = → ;

[ ]

0,5 0,5 −1

() → = (0)⨀((0) ) , ℎ ℎ det((0)) ≠ 0

LEZIONE 18

Causal PID controller

1

()

= (1 + + ); → ℎ ℎ .

1+

Derivative part is critical because generates impulse signal added to the control input and this can create

damage of the attuator and the system can become non-linear. To solve this problem the derivative part doesn’t

receive the error but the output y.

Parameters tuning using closed loop Ziegler-Nichols method: The Ziegler–Nichols tuning method is a heuristic

method of tuning a PID controller. It’s performed by setting the I (integral) and D (derivative) gains to zero. The

̅̅̅̅

"P" (proportional) gain is then increased (from zero) until it reaches the ultimate gain at which the output

̅̅̅̅ ̅

of the control loop has stable and consistent oscillations. and the oscillation period are then used to set

the P, I, and D gains depending on the type of controller used and behaviour desired.

Wind-up effect: refers to the situation in a PID feedback controller where a large change in setpoint occurs (say

a positive change) and the integral term accumulates a significant error during the rise (windup), thus

overshooting and continuing to increase as this accumulated error is unwound (offset by errors in the other

direction). The specific problem is the excess overshooting.

Integral windup particularly occurs as a limitation of physical systems, compared with ideal systems, due to the

ideal output being physically impossible (process saturation: the output of the process being limited at the top

or bottom of its scale, making the error constant). For example, the position of a valve cannot be any more open

than fully open and also cannot be closed any more than fully closed. In this case, anti-windup can actually

involve the integrator being turned off for periods of time until the response falls back into an acceptable range.

This usually occurs when the controller's output can no longer affect the controlled variable, or if the controller

is part of a selection scheme and it is selected right.

Nowadays it is much easier to prevent integral windup by either limiting the controller output or by using

external reset feedback, which is a means of feeding back the selected output to the integral circuit of all

controllers in the selection scheme so that a closed loop is maintained.

LEZIONE 19

Discrete time-system ( )

̅

0 0

( ) ( ( ), ),

̅ = ̅ ̅ + 1) = (̅ ̅( )

{ ℎ {

0 0 0 0 0 0

( (

̅(), ∀ ≥ ̅ + 2) = (̅ + 1), ̅( + 1), + 1)

0 0 0 0 0

( ()

ℎ − → ̅ + 1) = ̅

∀ > 0, ∃ > 0 . .

( )

||̅ − ̅ || < ℎ ( )

̅ → ; lim ||̅ − ̅ || = 0 → ̅ . .

0

0

→∞

( )

||̅ − ̅ || < , ∀ ≥

{ 0 0

−1

(: )

ℎ → () = ∙ + ∙ + ⋯ + ∙ + ; ∈ ℂ

0 1 −1

: the stability property is a property of the system (not depend on the input).

: A LTI discrete-time system is A.S. if and only if all the eigenvalues of the system lie inside a circle

of radius < 1 centered at the origin of the complex plane.

: It is sufficient that an eigenvalue has a norm > 1 to be able to conclude about the instability of

the LTI discrete-time system.

Useful tools (for a LTI discrete-time system):

→ | | < 1; ? . .

0

→ ∙ (1) > 0; ? . .

0

→ (−1) ∙ ∙ (−1) > 0; ? . .

0 ′

→ > > > ⋯ > > 0; ? . ; ? ℎ

0 1 2

=1 ′

∑ | | | |

→ < ; ? . ; ? ℎ

0

LEZIONE 20

Jury test (necessary and sufficient): LTI discrete-time system is A.S. if and only if the Jury table is well-defined

and all the elements of the first column have the same sign.

…

0 1 ℎ ℎ

1 1 −+1

ℎ ℎ … ℎ ; = ∙ | |

1 2 ℎ ℎ

ℎ

1

…

1 2 −1

3 2

◼ () = + 2 + + 3

1 2 1 3 1 3

1

→ = ∙ | | = −8 → − . . .

−8 1 3 1

ℎ

1

LEZIONE 21

Digital controller ()

→ () = ()

Shannon theorem

→ =

>

- The signal spectrum is pratically zero for

- To avoid the aliasing phenomenon the Nyqwist pulse must be greater than the signal limited bandwidth

>

⁄

> 2 ( = 2 && = 1 )

- To avoid the aliasing phenomenon the sampling pulse

LEZIONE 22

Zero-order hold (ZOH)

The zero-order hold (ZOH) is a mathematical model of the practical signal reconstruction done by a conventional

digital-to-analog converter (DAC). That is, it describes the effect of converting a discrete-time signal to a

continuous