Nonlinear Predictive Controlfor Fast Constrained Systems

By

Ahmed Youssef

What’s MBPC

N=N2-N1

Nu

CV

MV

N1 N2t

Introduction

CV: controlled variable MV: manipulated variable

IntroductionShortcomings of current industrial nonlinear MBPC•Computing the MBPC control law demands significant on-line computation effort •Inability to deal explicitly with the plant model uncertainty.

Objective of research workReducing the computational complexity of nonlinear MBPC & adding the robustness property whilst preserving its good attributes to make it more effective practical tool for controlling systems of fast-constrained dynamics.

Given the nonlinear dynamic model

Reformulate into nonlinear state-dependent form

This is not a linearisation

Trivial example

tttttttt xxCyuxBxxAx )()()(1

)(),(1 ttkkk xgyuxfx

)cos()(;)sin(

)(

)cos()sin(1

kkkk

kk

kkkkk

xxxBx

xxA

uxxxx

State Dependent State-Space Models

Hamilton-Jacobi-Bellman

uxgxfx )()(

0

)()()(min)( dtuxRuxxQxxV TT

tu

0)()(4

1)( 1

xQx

x

VxgRxg

x

Vxf

x

V

t

V T

T

T

T

The NLQGPC quadratic infinite horizon cost function:

The optimal control vector in terms of the states of the system and reference model:

11

, ( , ) ( , ) ( ) ( ) ( , )

1 2( , ) ( ) ( ) ( , ) ,ˆ

T T Tt N u t t N e t t N t t t t N e

TT Tt t N t t t N t t N e t N

U Q S Q S H S Q

A H A x H S Q R

N

jjtu

Tjtjtjte

Tjtjtt

T

tt

T

uQuryQryJ

JT

J

01111

01

1lim

NLQGPC Control Law

The Coupled Algebraic Riccati Equations

1 1 11 1

11 1

1 1

T TT Tj N e N j N e N j

T TT Tu N e N j N e N j

H A Q H A A Q S H

Q S Q S H S Q H A

2 2 11 1

11 2

1 1

TT T T Tj N e j N e N j

T TT Tu N e N j j N N e

H A Q A H A Q S H

Q S Q S H H S Q

Control Lyapunov Function

A C1 function V(x): n is said to be a discrete CLF for the system:

if V(x) is positive definite, unbounded, and if

for all x 0

Dealing with Stability Issue

Satisficing is based on a point-wise cost / benefit comparison of an action.

The benefits are given by the “Selectability” function Ps(u,x), while the costs are given by the

“Rejectability” function Pr(u,x).

The “satisficing” set is those options for which selectability exceeds rejectability:

Stability via Satisficing

Satisficing generates the state dependent set of controls that render the closed-loop system stable with respect to a known CLF.

uaug = uNLQGPC - ()(BTPB)-1BTPf

Start

uS

f, B, P,

Calculate

uNLQGPC

No

uS

implies

impl

ies

fPBBPBBPf

fPBuTTT

TTNLQGPC

1)(

fPBBPBBPf

xPxfPfTTT

TT

1)(

CLF-Based Satisficing Technique

• Magnitude Saturation

• Rate-Limited Actuators

• Actuator Dead-Zone

Therefore the common term is the Saturation function

,uu,u

,uuu,u

,uu,u

)u(satu

mininmin

maxinminin

maxinmax

inout

,u,

,u,u

,u,

)u(satu

mininmin

maxinminin

maxinmax

inout

)()( usatuuD dd

Dealing with Input Constraints

Examples of Actuator Constraints

u0 u

the actuator range of operation is limited

Limiting functions that map the interval (-,) onto (0, 1)

Limiting functions that map the interval (-,) onto (-1, 1)

Approximation of Magnitude Saturation

Error function (Blue)

Tanh function (Green)

Sigmoid function (Red)11S

10SSigmoid function (Black)

Approximation of Magnitude Saturation

Case Studies

F-8 fighter aircraft

F-16 fighter aircraft Caltech Ducted Fan

Controlling of F-8 Fighter

0 2 4 6 8 10 12 14-0.1

0

0.1

0.2

0.3

0.4

0.5

Time [s]

Ang

le o

f A

ttac

k [r

ad]

Unconstrained NLQGPC Constrained NLQGPC CNLQGPC-Satisficing (Nominal) CNLQGPC-Satisficing (Wind Gust)

0 2 4 6 8 10 12 14-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Time [s]

Pitc

h R

ate

[rad

/s]

0 2 4 6 8 10 12 14-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

Time [s]

u [r

ad]

Elevator Deflection

radu 05236.0

-0.500.5

-1

0

1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

x2

x1

Trajectories of Controlled System

x3

CONCLUSIONS

Properties of NLQGPC controller:

1. High performance

2. Less computational burden

3. Dealing with input constraints

4. Guaranteeing asymptotic stability to the closed-loop system.

5. Possesses both performance robustness & stability robustness