PID Controller Design for Nonlinear Systems Using Discrete ...

Motivation PID Controller Design Feedforward Control Conclusion & Outlook

PID Controller Design for Nonlinear Systems UsingDiscrete-Time Local Model Networks

4. Workshop fur Modellbasierte Kalibriermethoden

Nikolaus Euler-Rolle, Christoph Hametner, Stefan Jakubek

Christian Mayr (AVL List GmbH)

08.11.2013

4. Workshop fur Modellbasierte Kalibriermethoden: Euler-Rolle - PID Controller Design for Nonlinear Systems 1/26


Feedback Control of Nonlinear Systems

Motivation

Implementation of Two-Degrees-of-Freedom control using local model networks

Feedforward part improves the dynamic performance- Reference tracking- Deadtime- Input saturation

Controller design on (semi)-physical process models instead of testbed runs

Opportunity of inexpensive feasibility studies and rapid prototyping

PID Plant

ww*

u*

uy



Feedback Control of Nonlinear Systems

Motivation

Implementation of Two-Degrees-of-Freedom control using local model networks

Feedforward part improves the dynamic performance- Reference tracking- Deadtime- Input saturation

Controller design on (semi)-physical process models instead of testbed runs

Opportunity of inexpensive feasibility studies and rapid prototyping

Approach

Globally nonlinear process model (based on input/output measurements)

Design of nonlinear PID controllers with guaranteed global stability

Fully automated generation of a dynamic feedforward control



Controller Design Workflow

TestbedMaps

Signals

DoE

[n, q, u]

LMNSS-ModelLocal PIDs

DoE

Optimisation

y

SimulationParameter

ControllerMaps

Performance

Stability

Ide

ntifica

tio

n



Controller Design Workflow

TestbedMaps

Signals

DoE

[n, q, u]

LMNSS-ModelLocal PIDs

DoE

Optimisation

y

SimulationParameter

ControllerMaps

Performance

Stability

Ide

ntifica

tio

n

DynamicFF-Control



2 PID Controller Design



Local Model NetworkOverview

1000 1500 2000 2500

5

10

15

20

25

30

InjectionMass,mg/stroke

Engine Speed, rpm

local

global

Local Model Network

Globally nonlinear dynamical systemrepresented by local linear models

Found by system identification

Local stability proof & controllerdesign using linear methods

⇒ Global approach necessary (due totransition, model interpolation...)

o for nonlinear systemso based on Lyapunov stability theory



Typical PID Controller StructureExample: Engine Control Unit

-

min

max

P-Part

I-Part

DT1-Part

anti windup

Map

Map

Feedforward-

Feedback-Control

n

n

n

q

q

q

u

w

y

e

ufb

uff



Feedback Controlled Local Model Network

Concept

One local controller (LC) per local model(LM)

Scheduling of parameters according to thevalidity functions of local models (ParallelDistributed Compensator)

KPID(Φ) =∑

ΦiK(i)PID

Formal split into inputs used for control uand disturbances z

!"#$%"&&'%

(')*+#

,-!.

,- /

,!!.

,! /+

01'%2$*#+!

1"*#$!

('1'#('#$

3,-!/

,--

,!!/

,! -,-!- ,!!-

Nonlinear process is approximated by a local model networkTrade-Off: model fit ↔ simple controller design

Closed-loop state-space representation necessary (to prove Lyapunov stability)



Closed-Loop State-Space RepresentationIncluding Error Signal Adaptation

InputScheduler

q−1I

v(k)B(Φ)

B(Φ)

E(Φ) A(Φ)

KPID(Φ, e)

we(Φ, e)

cTx(k + 1)

x(k)

y(k)w(k)

z(k)z(k)

f(Φ)

-

SystemPre-Filter

Figure: Local model network with PID controller in state-space representation

State Equation

x(k + 1) = [A(Φ)−B(Φ)KPID(Φ, e)]x(k) +B(Φ)G(Φ, e)w(k) +E(Φ)z(k)

+ f(Φ) +B(Φ)we(Φ, e)

y(k) =cTx(k)



Overview of the Design Procedure

Controller Design

Basic calibration (linear design methods per local model)

Generation of a suitable performance sequence (DoE)- Operating range (e.g.: 1000–4000 rpm, 0–70 mg/stroke)- Holding time- Gradients (e.g.: engine speed)- Filtering

Nonlinear, multi-objective optimisation of controller parameters considering- Performance- Stability

Multi-objective optimisation of the parameters of the error signal adaptation(optional)



Multi-Objective Genetic Algorithm

Objective Function

min fm(xopt)subject to gj(xopt) ≥ 0

hk(xopt) = 0

x(lb)i ≤ xi ≤ x

(ub)i

fS Stability (by Lyapunov’s direct method)

fP Performance (by a closed-loop simulation)

0

fP

fS

Paretofrontier

GA Population

1 n

· · ·

· · ·Individuals

Gen

ome

Gen

ome

Fitness

Fitness

Stability

Stability

Perform

ance

Perform

ance



Fitness Function: StabilityLyapunov’s Direct Method for Discrete-Time Systems

Stability of Dynamic Systems

A positive definite, scalar Lyapunov functionV (k) = V (x(k)) with state vector x(k)proves global asymptotic stability if:

o V (x(k) = 0) = 0

o V (k) > 0 for x(k) 6= 0

o V (k) → ∞ as ‖x(k)‖ → ∞

o V (k + 1) < V (k) ∀k ∈ N+

or global exponential stability if:o V (k + 1) ≤ α2V (k) ∀k ∈ N

+

with decay rate 0 < α < 1

Results in Linear Matrix Inequalities (LMIs),which are solved by optimisation

Sufficient but not necessary condition

Common Quadratic Lyapunov Function

V (k) = xT(k)Px(k)

LMI Problem

P ≻ 0

inf{

0 < α < 1 :

ΛTijPΛij +Xij � α2P

}



Fitness Function: Performance

Requirements

Assessment of the closed-loopperformance for a given set ofparameters

Representative synthetic reference isgenerated by DoE

Desired trajectory is PT1-filtered

Fitness Function

Closed-loop simulation of the referencecycle for each genome

Sum of squared errorsfP =

∑

∀k(y(k)− ydmd(k))2

Time

0 5 10 150

0.5

1

1.5

2



Pareto-Optimal Solutions

0.995 1 1.005 1.01 1.015 1.02 1.025 1.03

1

1.5

2

2.5

3

3.5

4x 10

7

A

B

Pe

rfo

rma

nce

Stability

fS

fP



3 Feedforward Control



Feedforward Control

State of the Art: Static Model Inversion

Steady state input is found by static model inversion

u(Φ) = [cT(I − A(Φ))−1B(Φ)]−1(w(Φ) − c

T(I − A(Φ))−1(E(Φ)z(Φ) + f(Φ)))

Stored in a map

Dynamic Feedforward Control

PID Plant

ww*

u*

uy

Dynamic feedforward control improves the closed-loop performance.



Dynamic Feedforward ControlGeneration using Local Model Networks

Benefits

Automatic generation of a dynamic feedforward control law for nonlineardynamic systems

Exploits the generic model structure of local model networks

Model complexity may be arbitrarily high

Applicable online for any reference trajectory without pre-planning

Properties

Based on an open-loop state-space model

Realised by a feedback linearizing input transformation

Restricted to globally minimum-phase local model networks



Feedback LinearizationUndamped Nonlinear Oscillation

Consider an undamped oscillator with anonlinear spring force characteristicf(y), which is to be stabilized usingconstant c and input u

y + f(y) = cu

Figure: Air suspension

Exact Linearization

For this second order system, the state variables are chosen as

y = x1

y = x1 = x2

y = x1 = x2 = cu− f(y)



Feedback LinearizationUndamped Nonlinear Oscillation

Consider an undamped oscillator with anonlinear spring force characteristicf(y), which is to be stabilized usingconstant c and input u

y + f(y) = cu

Figure: Air suspension

Exact Linearization

For this second order system, the state variables are chosen as

y = x1

y = x1 = x2

y = x1 = x2 = cu− f(y) = v

1

s

1

s

v y



Feedforward ControlUndamped Nonlinear Oscillation

Exact Linearization

For a two times differentiable desired trajectory w, the nonlinear feedforward controlinput u∗ can be found from

v!= w = cu∗

− f(w) → u∗ =w + f(w)

c

1

s

1

s

yu

u∗

w

w

C

u∗ =w + f(w)

c



Demonstration ExampleAutomatic Feedforward Control Design

Wiener Model

G(z) =P (z)

U(z)=

0.6z−3

1− 1.3z−1 + 0.8825z−2 − 0.1325z−3

y(k) = f(p(k)) = arctan(p(k))

Figure: Wiener Model approximated by an LMN:

y(k

−1)

u(k − 3)

6

5

43

2

1

−3 −2 −1 0 1 2 3

−1

−0.5

0

0.5

1



Feedforward Controlled SimulationWiener Model

Samples

yW

iener

uw,y

40 60 80 100 120 140 160 180 200 220

40 60 80 100 120 140 160 180 200 220

40 60 80 100 120 140 160 180 200 220

−1

0

1

−3

0

3

−1

0

1



Feedforward Controlled SimulationWiener Model

y FFCw

y

Samples0 50 100 150 200 250 300 350 400 450 500

−1.5

−1

−0.5

0

0.5

1

1.5

PID Plant

ww*

u*

uy



Two-Degrees-of-Freedom ControlWiener Model

y 2DoFy FFCw

Samples

y

0 50 100 150 200 250 300 350 400 450 500−1.5

−1

−0.5

0

0.5

1

1.5

PID Plant

ww*

u*

uy



Two-Degrees-of-Freedom ControlWiener Model

y 2DoFyPIDw

Samples

y

0 50 100 150 200 250 300 350 400 450 500−1.5

−1

−0.5

0

0.5

1

1.5

PID Plant

ww*

u*

uy



4 Conclusion & Outlook



Conclusion & Outlook

Two-Degrees-of-Freedom Control

Nonlinear PID controller design using local model networks

Multi-objective optimisation of controller parameters consideringStabilityPerformance

Automatic feedforward control law generation for minimum-phase local modelnetworks

Outlook

Application of a Lyapunov function to check internal stability

Considering input constraints

Assessment of Two-Degrees-of-Freedom control on a physical process



Thank you for your attention!



Fitness Function: Stability

Common Quadratic Lyapunov Function for Closed-Loop Systems

Exponential stability with decay rate α of the closed-loop feedback system is shown, ifsymmetric matrices P and Xij exist and the following conditions are fulfilled:

P ≻ 0

inf{

0 < α < 1 : ΛTijPΛij + Xij � α

2P}

X =

X11 X12 · · · X1I

X12 X22 · · · X2I

.

.

.. . .

.

.

.X1I X2I · · · XII

≻ 0

∀i ∈ I, ∀i ≤ j ≤ I

using

Λij =Gij + Gji

2,

Gij = Ai − BikTPID,jC.



Fitness Function: Stability

Common Quadratic Lyapunov Function for Closed-Loop Systems

Exponential stability with decay rate α of the closed-loop feedback system is shown, ifsymmetric matrices P and Xij exist and the following conditions are fulfilled:

P ≻ 0

inf{

0 < α < 1 : ΛTijPΛij + Xij � α

2P}

X =

X11 X12 · · · X1I

X12 X22 · · · X2I

.

.

.. . .

.

.

.X1I X2I · · · XII

≻ 0

∀i ∈ I, ∀i ≤ j ≤ I

using

Λij =Gij + Gji

2,

Gij = Ai − BikTPID,jC.

Simultaneous solving for P and kTPID,j is not possible! → fS = α


PID Controller Design for Nonlinear Systems Using Discrete ...

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