Integrator Back-stepping; Linear Quadratic (LQ) Observer

Lecture – 39

Integrator Back-stepping;Linear Quadratic (LQ) Observer

Dr. Radhakant PadhiAsst. Professor

Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

2

Philosophy of Nonlinear Control Design Using Lyapunov Theory




3

Philosophy of Feedback Control Design Using Lyapunov Theory

( )( )

( )( )

( )( )

1

1

Motivation

Goal Design such that

is asymptotically stable

Design Idea:

Choose a pdf

Make

: ,

:

,

U

V X

V X

X f X U

X

X f X X

ϕ

ϕ

=

=

=

∗

∗ ( ) ( )2 2, where (pdf)0V X V X≤ − >


4

Feedback Control Design Using Lyapunov Theory: An Example

( )

( )

( )( ) ( )

2 3

21

2 31

3 4

22

1 2

Problem: Design a stabilizing controller for the following system

1

Solution: Let 2

Let us choose

x ax x u

V X x

V x x x ax x u

ax x xu

V X x

V X V X

= − +

=

= = − +

= − +

=

∴ ≤ −3 4 2 ax x xu x⇒ − + ≤ −


5

Philosophy of feedback control Design Using Lyapunov Theory

2 4 3

3 2

2 3 3 2

i.e

Analysis:

Advantage: The closed loop system is globally asymptotically stable.

Problem: The benefitial nonlinearity got cancell

xu x x ax

u x x ax

x ax x x x ax

x x

≤ − + −

= − + −

= − − + −

= −

ed. (which is not desirable)


6


( )

( )

2 42

1 2

3 4 2 4

3 2

2 2

Let us choose:

Then

leads to:

or

V X x x

V V X

ax x xu x x

ax xu x

ax u x u x ax

= +

−

− + ≤ − −

+ ≤ −

+ = − = − −

≤


7


2 3 2

3

Closed Loop system:

i.e. The destabilizing nonlinearity got cancelled,

but the benefitical nonlinearity is retained !

Another Problem: If

x ax x x ax

x x x

= − − −

= − −

⇒

( ) 22

, only if is accurate

If the actual parameter value is , then the feedback loop operates with

,V X x

x x a

a

=

= −


8


( )

( )

( )

is

Can be potentially destabilizing termif high

2

2

. . The global stability reduces to local stability.

This excits robustness issues!

However, if is made "Sufficiently powerful",

a a

x x a a x

i e

V X

−

= − + −

then the destabilizing effect can be minimized.

Hence, Lyapunov based designs can be "very robust"


9

Control Design UsingIntegrator Back-stepping




10

Integrator Back-stepping

( ) ( )

Design a state feedback asymptotically stabilizing controller for the following system

where , ,

Note:

n

X

X f X g X

u

X u

ξ

ξ

ξ

ξ

= +

=

∈ ∈ ∈

⎡ ⎤⎢⎣

Problem :

R R R

1: State of the system, Control input (single input) : n u+∈⎥⎦

R


11


( )

Assumptions:

, : are smooth

0 0

Considering state as a "control input" of subsystem (1) we assume that a state feedback control law

nf g D

f

ξ

∗ →

∗ =

∗

∃

R

( ) ( )

( ) ( ) ( ) ( ) ( )

( )

1

11

of the from Moreover,

a Lyapunov function such that

where, is a pdf functi

, 0 0.

:

,

:

T

a

a

V D

V X

D

X

V f X g X X V X X DX

V X

ξ ϕ ϕ

ϕ

+

+

∃ →

→

= =

∂⎛ ⎞= + ≤ − ∀ ∈⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠

R

R on.


12


( ) ( )

( )

An important observation:

When 0, = &

(i.e. everything is nice)

However, when 0, and hence 0

in general. That is the core problem!

We need some algebraic manipulation as

0 0 0 0

X

X

X f

X f X

ξ ϕ

ξ

=

→ →

∴

= = =

=

follows.


13


( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

Step - 1:

By this construction, when

which is asymptotically stable (i.e.

0,0)!

z

X f X g X g X X g X X

f X g X X g X X

f X g X X g X z

z X f X g X XX

ξ ϕ ϕ

ϕ ξ ϕ

ϕ

ϕ

= + + −

= + + −⎡ ⎤⎣ ⎦

= + +

→ = +

→


14


( )

( ) ( ) ( ) ( )

This is backstepping, since is

stepped back by differentiation

So, we have

This system is equivalent to the original sys

v

z

Xu

X f X g X X g X zz v

ϕ

ξ ϕ

ϕ

ϕ

= −

⎡ ⎤= − ⎢ ⎥

⎣ ⎦

= + +

=

[ ]

( ) ( )

tem

Note: T T

X f X g XX Xϕ ϕϕ ξ∂ ∂⎛ ⎞ ⎛ ⎞= = +⎡ ⎤⎜ ⎟ ⎜ ⎟ ⎣ ⎦∂ ∂⎝ ⎠ ⎝ ⎠


15

Back-stepping:Conceptual Block Diagram

Back-stepping

Ref: H. J. Marquez, Nonlinear Control Systems: Analysis and Design,Wiley, 2003.


16


( )

( ) ( ) ( )

( )

( )

( ) ( )

21

1

1

Step-2: Let

Then

Let

1 , ( )2

a

T

V X

T

a

V X z V X z

VV f X g X X g X z z vX

VV X g X v zX

ϕ

≤−

= +

∂⎛ ⎞= + + +⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠

⎡ ⎤∂⎛ ⎞≤ − + +⎢ ⎥⎜ ⎟∂⎝ ⎠⎢ ⎥⎣ ⎦

( )

( )

1

2

ndf

Then (ndf)

, 0

0

T

anddf

Vv g X k z kX

V V X k z

∂⎛ ⎞= − − >⎜ ⎟∂⎝ ⎠≤ − − <


17


( )

( ) ( )

( ) ( )

( )

1

1

Control Solution

where

Note: In the design, there is a need to design

:

, , 0

T

T

T

Vv u g X k zX

Vu g X k XX

f X g X kX

Xϕ

ϕ

ϕ ξ ϕ

ϕϕ ξ

∂⎛ ⎞= − = − −⎜ ⎟∂⎝ ⎠

∂⎛ ⎞= − − −⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠

∂⎛ ⎞= + >⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠

first⎡ ⎤⎣ ⎦


18

Integrator Back-stepping:An Example

( ) ( )( )

( )

( )

2 31 1 1 2

2

2 31 1 1 1 2 1

1

21 1 1

2 31 1 1 1 1 1 2

Problem

Solution

To find

:

: , , , 1

:

1 2

x ax x xx u

X x f x ax x x g x

x

V x x

V x x x ax x x

ϕ

ξ

= − +=

= = − = =

=

= = − + ≤ ( )2 41 1

1a

x x

V x+

−


19


3 41 1 ax x− 2 4

1 2 1 1x x x x+ ≤ − −

( )

( ) ( )

( ) ( )

2 21 1 2 1

21 2 1

22 1 1 1

2 31 1 1 1 2 1

2

Let

Modified system

:

z

x ax x x

ax x x

x ax x x

x ax x x x x

z v x

ϕ

ϕ ϕ

+ ≤ −

+ = −

⇒ = − −

= − + + −⎡ ⎤⎣ ⎦

= ( )( )( ) ( )

( )

1

21 1 1

2 31 11 1 1 1

1 1

Let1 ,2

ux

V x z V x z

V VV V z v ax x x v zx x

ϕ

ϕ

−

= +

⎛ ⎞ ⎛ ⎞∂ ∂⎡ ⎤= + = − + + +⎜ ⎟ ⎜ ⎟⎣ ⎦∂ ∂⎝ ⎠ ⎝ ⎠


20


( )

( ) ( )

( )( )( )( ) ( )

1

1

1 2 1

2 3 21 1 2 1 2 1 1

1

2 3 21 1 1 2 1 2 1 1

2 3 21 1 1 2 1 1 2 1

Let

, 0

2 1

1 2

Vv k z kx

u x k x x

u ax x x x k x ax xx

ax ax x x x k x ax x

u ax ax x x x k x x ax

ϕ ϕ

ϕ

⎛ ⎞∂= − − >⎜ ⎟∂⎝ ⎠

− = − − −⎡ ⎤⎣ ⎦∂ ⎡ ⎤= − + − − − − −⎣ ⎦∂

⎡ ⎤= − − − + − − + +⎣ ⎦

= − + − + − − + +

where 0k >


21


( )

( )

21

221 2 1

22 21 2 1 1

Note: The composite Lyapunov function is:1 2

1 1 2 21 1 2 2

V V z

x x x

x x x ax

ϕ

= +

= + −⎡ ⎤⎣ ⎦

= + + +


22

Integrator Back-stepping:More General Case

( ) ( ) 1

1 2

2

System Dynamics:

Idea : Successive iteration.

Note: The procedure for order system is entirely analogous

thn

X f X g X

u

ξ

ξ ξ

ξ

= +

=

=

⎡ ⎤⎣ ⎦


23


( ) ( )

( )( ) ( )

( )

1

1 2

1

1

1

Step-1 Consider the subsystem

Assumption:

is a stabiliting feedback law for

and is the coresponding Lyapunov fun

:

=

X f X g X

X

X f X g X

V X

ξ

ξ ξ

ξ ϕ

ξ

= +

=

= +

ction.


24


( ) ( ) ( ) ( ) ( )

( )

12 1 1

1 1

2

By the result obtained before, we have

We also have

, ( 0)

T T

X

X Vf X g X g X k XX X

X k

V

ϕξ ξ ξ ϕ

ϕ ξ

∂⎛ ⎞ ∂⎛ ⎞= + − − −⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦∂ ∂⎝ ⎠⎝ ⎠

>

= ( ) 21 1

12

V Xξ ϕ+ −⎡ ⎤⎣ ⎦


25


( ) ( )

( ) ( )

( ) ( )

1 11 1

11 2

1

2 11

1 21 1 1 1 2

1 1

Step - 2:

where,

Using the same idea,

0

10

g Xf X

T

X

X f X g XX

Xu

Vu f X g XX X

ξξ

ξ

ξξ

ϕ ξ

∴

⎡ ⎤ ⎡ + ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦

⎡ ⎤= ⎢ ⎥

⎣ ⎦

⎛ ⎞ ⎛ ⎞∂ ∂= + −⎡ ⎤⎜ ⎟ ⎜ ⎟⎣ ⎦∂ ∂⎝ ⎠ ⎝ ⎠

( ) ( )

( ) ( ) ( )

1 1 1 2 1 1 1

2 2 22 2 1 1 1 1 2 1 1and

, 0

1 1 12 2 2

T

V V V

g X k X k

X X X

ξ ϕ

ξ ϕ ξ ϕ ξ ϕ= + +

− − >⎡ ⎤⎣ ⎦

− = − + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦


26

Integrator Back-stepping for Strict Feedback Systems

( ) ( )( ) ( )( ) ( )

1

1 1 1 1 1 2

2 2 1, 2 2 1 2 3

System Dynamics

, ,

, , ,

,k k

X f X g X

f X g X

f X g X

f X

ξ

ξ ξ ξ ξ

ξ ξ ξ ξ ξ ξ

ξ

= +

= +

= +

= ( ) ( )

( ) ( ) ( )

1 1

1 1 2 1 2 1

Strong Assumption:

over the domain of interest

, , , , ,

, , , , , , , , , 0

k k k

k k

t

g X u

g X g X g X

ξ ξ ξ ξ

ξ ξ ξ ξ ξ∀

+

≠

… …

… …


27

Integrator Back-stepping for Strict Feedback Systems

( ) ( )( ) ( )

( ) ( )

Special Case:

Solution:

Define and carryout the design for as before.

Finally

, ,

, ,

. .

a a

a a

v

X f X g X

f X g X u

v

f X g X u v

i e

ξ

ξ ξ ξ

ξ

ξ ξ

= +

= +

=

+ =

( ) ( )

( ) Note: By assumption,

1 ,,

, 0

aa

a

u v f Xg X

g X t

ξξ

ξ

= −⎡ ⎤⎣ ⎦

≠ ∀

Linear Quadratic (LQ) Observer




29

Why Observers?State feedback control designs need the state information for control computationIn practice all the state variables are not available for feedback. Possible reasons are: • Non-availability of sensors

• Expensive sensors

• Quality of some sensors may not acceptable due to noise (its an issue in output feedback control design as well)

A state observer estimates the state variables based on the measurement of some of the output variables as well as the plant information.


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Observer

An observer is a dynamic system whose output is an estimate of the state vector

Full-order Observer

Reduced-order Observer

• Observability condition must be satisfied for designing an observer (this is true for filter design as well)

X


31

Observer Design for Linear Systems

Plant

State observer( )

(sensor output vector)

ˆLet the observed state be and the be

ˆˆ ˆ ˆ

ˆ

e

X AX BUY CX

X

X AX BU K Y

X X X

= +=

= + +

−

Plant :

Observer dynamics

Error :


32

Observer Design for Linear Systems

( ) ( )

( ) ( )

ˆ

ˆ ˆ ˆ

Add and Substract and substitute ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ( ) ( ) ( )ˆ ˆ ˆ

1. Make the err

Error Dynamics:

Goals: o

e

e

e

e

X X X

AX BU AX BU K Y

AX Y CX

X AX AX AX AX BU BU K C X

A A X A X X B B U K C X

AX A A K C X B B U

= −

= + − + +

=

= − + − + − −

= − + − + − −

= + − − + −

r dynamics independent of

( may be large, even though may be small) 2. Eliminate the effect of from eror dynamics

X

X XU

∵


33

Observer Design for Linear SystemsThis can be done by enforcing ˆ 0

ˆand 0

This results in ˆ

ˆ

e

e

A A K C

B B

A A K C

B B

− − =

− =

= −

=

Observer dynamics:

Necessary and sufficient condition for the existence of Ke :

The system should be “observable”.

( )ˆ ˆ ˆeX AX BU K Y CX= + + −


34

Observer Design: Full Order Order of the observer is same as that of the system (i.e. all states are estimated, irrespective of whether they are measured or not).

Goal: Obtain gain Ke such that the error dynamics are asymptotically stable with sufficient speed of response.

This means that Â = A – Ke C is Hurwitz (i.e. it has all eigenvalues strictly in the left half plane.

Note: ÂT =AT – CTKeT and the eigen values of both Â

and ÂT are same!


35

Comparison of Control and Observer Design Philosophies

CL Dynamics

Objective

CL Error Dynamics

Objective

Notice that

Control Design Observer Design

( )X A BK X= −

( ) 0, as X t t→ →∞ ( ) 0, as X t t→ →∞

( )ˆeX AX A K C X= = −

( ) ( )

( )

Te e

T T Te

A K C A K C

A C K

λ λ

λ

⎡ ⎤− = −⎣ ⎦

= −


36

Algebraic Riccati Equation (ARE)Based Observer Design

X AX BU= + T TZ A Z C V= +Y CX=

1nM B AB A B−⎡ ⎤= ⎣ ⎦1nT T T T TN C A C A C−⎡ ⎤= ⎣ ⎦

Tn B Z=

1nT T T T TM C A C A C−⎡ ⎤= ⎣ ⎦

1nN B AB A B−⎡ ⎤= ⎣ ⎦

System Dual System

LQR DesignU K X= −


37

ARE Based Observer Design

1 , 0TK R B P P−= >

1 0T TPA A P PBR B P Q−+ − + =

Error Dynamics

( )eX A K C X= −

( )T T T Te eA K C A C K− = −

Analogous1T

eK R CP−=where,

1 0T TPA AP PC R CP Q−+ − + =Observer Dynamics

ˆ ˆ ( )eX AX BU K Y CX= + + −

Acts like a controller

gainwhere,

( )0

X A BK XX as t= −

→ →∞

CL system (control design)

Continuous-time Kalman Filter Design for Linear Time Invariant (LTI) Systems




39

Problem Statement

( ) ( ) ( )0 0

System Dynamics: ( ) : Process noise vectorMeasured Output: ( ) : Sensor noise vector

Assumptions:

( ) (0) , , ( ) 0, and ( ) 0,

are "mu

X AX BU GW W tY CX V V t

i X X P W t Q V t R

= + += +

∼ ∼ ∼

[ ]tually orthogonal" (0) : initial condition for ( ) ( ) and ( ) are uncorrelated white noise

( ) ( ) ( ) ( ), 0 (psdf)

( ) (

T

T

X Xii W t V t

iii E W t W t Q Q

E V t V t

τ δ τ

τ

⎡ ⎤+ = ≥⎣ ⎦

+ ) ( ), 0 (pdf)R Rδ τ⎡ ⎤ = >⎣ ⎦


40

Problem Statement

( )ˆTo obtain an estimate of the state vector using the state dynamics as well as a "sequence of measurements"as accurate as possible.

ˆi.e., to make sure that the error ( ) ( ) ( ) becomes

v

X t

X t X t X t⎡ ⎤−⎣ ⎦

( )ery small ideally ( ) 0 as .X t t→ →∞

Objective:


41

Observer/Estimator/Filter Dynamics

( )( ) ( )( ) ( )

ˆwhere ( ) ( ) : Estimate of the state ˆ ( ) ( ) : Estimate of the output

( ) 0

i X E X X

ii Y E Y YE CX V

E CX E V

CE X E V

=

=

= +

= +

= =∵ˆ

( ) : Estimator/Filter/Kalman Gain How to design ?

e

e

CXiii K

K

=

Problem :

( )ˆ ˆ ˆ eX AX BU K Y Y= + + −


42

1

1

ˆ (i) Initialize (0) (ii) Solve for Riccati matrix from the Filter ARE: 0 (iii) Compute Kalman Gain: (iv) Propagate the Filter dynamics:

T T T

Te

XP

AP PA PC R CP GQG

K PC R

−

−

+ − + =

=

( )ˆ ˆ ˆ

where is the measurement vectoreX AX BU K Y CX

Y

= + + −

Solution: Summary


43

Integrator Back-stepping; Linear Quadratic (LQ) Observer

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