PID Controller Design for Nonlinear Systems Using Discrete-Time Local Model Networks

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PID Controller Design for Nonlinear Systems Using

Discrete-Time Local Model Networks

4. Workshop f¨ur Modellbasierte Kalibriermethoden

Nikolaus Euler-Rolle, Christoph Hametner, Stefan Jakubek

Christian Mayr(AVL List GmbH)

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Feedback Control of Nonlinear Systems

Motivation

Implementation ofTwo-Degrees-of-Freedom controlusing local model networks Feedforward part improves the dynamic performance

- Reference tracking - Deadtime - Input saturation

Controller design on (semi)-physical process modelsinstead of testbed runs Opportunity of inexpensivefeasibility studies and rapid prototyping

PID Plant

w

*

u

*

u

y

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Feedback Control of Nonlinear Systems

Motivation

Implementation ofTwo-Degrees-of-Freedom controlusing local model networks Feedforward part improves the dynamic performance

- Reference tracking - Deadtime - Input saturation

Controller design on (semi)-physical process modelsinstead of testbed runs Opportunity of inexpensivefeasibility 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

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Controller Design Workflow

Testbed

Maps Signals DoE [n, q, u]

LMN

SS-Model

Local PIDs

DoE Optimisation y Simulation Parameter Controller Maps Performance Stability Identif ication

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Controller Design Workflow

Testbed

Maps Signals DoE [n, q, u]

LMN

SS-Model

Local PIDs

DoE Optimisation y Simulation Parameter Controller Maps Performance Stability Identif ication Dynamic FF-Control

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Local Model Network

Overview 1000 1500 2000 2500 5 10 15 20 25 30 In je ct io n M a ss , m g / st ro ke Engine Speed, rpm local global

Local Model Network

Globally nonlinear dynamical system represented by local linear models Found by system identification Local stability proof & controller design using linear methods

⇒ Global approach necessary (due to transition, model interpolation...)

o for nonlinear systems

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Typical PID Controller Structure

Example: 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

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Feedback Controlled Local Model Network

Concept

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

Scheduling of parameters according to the validity functions of local models (Parallel

DistributedCompensator)

KP ID(Φ) =PΦiK (i) P ID

Formal split into inputs used for controlu

and disturbancesz !"#$%"&&'% (')*+# ,-!. ,- / ,!!. ,! / + 01'%2$*#+! 1"*#$! ('1'#('#$

3

,-!/ , -,!!/ ,! -,-!- ,!!

-Nonlinear process is approximated by a local model network

Trade-Off: model fit↔simple controller design

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Closed-Loop State-Space Representation

Including Error Signal Adaptation

Input Scheduler q−1_I v(k) B(Φ) B(Φ) E(Φ) _A₍_Φ₎ KP ID(Φ, e) we(Φ, e) cT x(k+ 1) x(k) ˆ y(k) w(k) ˆ z(k) z(k) f(Φ) -System P re -F il te r

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

State Equation

x(k+ 1) = [A(Φ)−B(Φ)KP ID(Φ, e)]x(k) +B(Φ)G(Φ, e)w(k) +E(Φ)ˆz(k)

+f(Φ) +B(Φ)we(Φ, e)

ˆ

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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)

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Multi-Objective Genetic Algorithm

Objective Function min fm(xopt) subject to gj(xopt)≥0 hk(xopt) = 0 x(_ilb)≤xi≤x(_iub) fSStability(by Lyapunov’s direct method) fP Performance(by a closed-loop simulation)

0 fP fS Paretofrontier

GA Population

1 n · · · · · · Individuals G en o m e G en o m e F it n es s F it n es s S ta b il it y S ta b il it y P er fo rm a n ce P er fo rm a n ce

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Fitness Function: Stability

Lyapunov’s Direct Method for Discrete-Time Systems

Stability of Dynamic Systems

A positive definite, scalar Lyapunov function

V(k) =V(x(k))with state vectorx(k)

proves globalasymptoticstability if:

o V(x(k) =0_{) = 0} o V(k)>0 forx(k)6= 0 o V(k)→ ∞ askx(k)k → ∞

o V(k+ 1)< V(k) ∀k∈N+

or globalexponentialstability if:

o V(k+ 1)≤α2V(k) ∀k∈N+ with decay rate0< α <1

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

Sufficient but not necessary condition Common Quadratic Lyapunov Function

LMI Problem

P ≻0

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Fitness Function: Performance

Requirements

Assessment of the closed-loop performance for a given set of parameters

Representative synthetic reference is generated by DoE

Desired trajectory is PT1-filtered

Fitness Function

Closed-loop simulation of the reference cycle for each genome

Sum of squared errors

fP =P∀k(ˆy(k)−ydmd(k))2 Time 0 5 10 15 0 0.5 1 1.5 2

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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 Performance Stability fP

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Feedforward Control

State of the Art: Static Model Inversion

Steady state input is found by static model inversion

u(Φ) = [cT(I−A(Φ))−1_B₍Φ)]−1₍_w₍Φ)−cT(I−A(Φ))−1₍_E₍Φ)ˆz(Φ) +f(Φ)))

Stored in a map

Dynamic Feedforward Control

PID Plant

w

*

u

*

u

y

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Dynamic Feedforward Control

Generation using Local Model Networks

Benefits

Automatic generationof a dynamic feedforward control law for nonlinear dynamic systems

Exploits thegeneric model structureof local model networks

Model complexitymay be arbitrarily high

Applicableonlinefor any reference trajectory without pre-planning

Properties

Based on an open-loop state-space model

Realised by afeedback linearizinginput transformation Restricted to globally minimum-phase local model networks

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Feedback Linearization

Undamped Nonlinear Oscillation

Consider an undamped oscillator with a nonlinear spring force characteristic

f(y), which is to be stabilized using constantcand inputu

¨

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 ¨

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Feedback Linearization

Consider an undamped oscillator with a nonlinear spring force characteristic

f(y), which is to be stabilized using constantcand inputu

¨

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

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Feedforward Control

Exact Linearization

For a two times differentiable desired trajectoryw, the nonlinear feedforward control inputu∗can be found from

v= ¨! w=cu∗−f(w) → u∗=w¨+f(w) c 1 s 1 s y u u∗ w ¨ w C u∗ =w¨+f(w) c

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Demonstration Example

Automatic 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:

ˆ y ( k − 1 ) u(k−3) 6 5 4 3 2 1 −3 −2 −1 0 1 2 3 −1 −0.5 0 0.5 1

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Feedforward Controlled Simulation

Wiener Model yW ie n e r u w , ˆy 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

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Feedforward Controlled Simulation

Wiener Model yFFC w ˆy Samples 0 50 100 150 200 250 300 350 400 450 500 −1.5 −1 −0.5 0 0.5 1 1.5 PID Plant w w* u* u y

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Two-Degrees-of-Freedom Control

Wiener Model y2DoF yFFC w Samples ˆy 0 50 100 150 200 250 300 350 400 450 500 −1.5 −1 −0.5 0 0.5 1 1.5 PID Plant w w* u* u y

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Two-Degrees-of-Freedom Control

Wiener Model y2DoF yPID w Samples ˆy 0 50 100 150 200 250 300 350 400 450 500 −1.5 −1 −0.5 0 0.5 1 1.5 PID Plant w w* u* u y

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Conclusion & Outlook

Two-Degrees-of-Freedom Control

Nonlinear PID controller design using local model networks Multi-objective optimisation of controller parameters considering

Stability Performance

Automatic feedforward control law generation for minimum-phase local model networks

Outlook

Application of a Lyapunov function to check internal stability Considering input constraints

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Fitness Function: Stability

Common Quadratic Lyapunov Function for Closed-Loop Systems

Exponential stability with decay rateαof the closed-loop feedback system is shown, if symmetric matricesP andXijexist and the following conditions are fulfilled:

P ≻0 infn0< α <1 :ΛT ijPΛij+Xijα 2 Po ˜ X=      X11 X12 · · · X1I X12 X22 · · · X2I . . . . .. . . . X1I X2I · · · XII      ≻0 ∀i∈ I,∀i≤j≤I using Λ_ij=Gij+Gji 2 , Gij=Ai−Bik T P ID,jC.

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Fitness Function: Stability

Common Quadratic Lyapunov Function for Closed-Loop Systems

Exponential stability with decay rateαof the closed-loop feedback system is shown, if symmetric matricesP andXijexist and the following conditions are fulfilled:

P ≻0 infn0< α <1 :ΛT ijPΛij+Xijα2P o ˜ X=      X11 X12 · · · X1I X12 X22 · · · X2I . . . . .. . . . X1I X2I · · · XII      ≻0 ∀i∈ I,∀i≤j≤I using Λ_ij=Gij+Gji 2 , Gij=Ai−Bik T P ID,jC.