Integrator Back-stepping; Linear Quadratic (LQ) Observer

transcript

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

Philosophy of Nonlinear Control Design Using Lyapunov Theory

Philosophy of Feedback Control Design Using Lyapunov Theory

( )( )

Motivation

Goal Design such that

is asymptotically stable

Design Idea:

Choose a pdf

X f X U

X f X X

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

Feedback Control Design Using Lyapunov Theory: An Example

( )( ) ( )

Problem: Design a stabilizing controller for the following system

Solution: Let 2

Let us choose

x ax x u

V x x x ax x u

ax x xu

V X V X

= − +

= = − +

= − +

∴ ≤ −3 4 2 ax x xu x⇒ − + ≤ −

Philosophy of feedback control Design Using Lyapunov Theory

2 3 3 2

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

≤ − + −

= − + −

= − − + −

ed. (which is not desirable)

3 4 2 4

Let us choose:

leads to:

V X x x

ax x xu x x

ax xu x

ax u x u x ax

− + ≤ − −

+ ≤ −

+ = − = − −

Closed Loop system:

i.e. The destabilizing nonlinearity got cancelled,

but the benefitical nonlinearity is retained !

Another Problem: If

x ax x x ax

= − − −

= − −

( ) 22

, only if is accurate

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

,V X x

Can be potentially destabilizing termif high

. . The global stability reduces to local stability.

This excits robustness issues!

However, if is made "Sufficiently powerful",

x x a a x

= − + −

then the destabilizing effect can be minimized.

Hence, Lyapunov based designs can be "very robust"

Control Design UsingIntegrator Back-stepping

Integrator Back-stepping

( ) ( )

Design a state feedback asymptotically stabilizing controller for the following system

where , ,

X f X g X

∈ ∈ ∈

⎡ ⎤⎢⎣

Problem :

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

Assumptions:

, : are smooth

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

nf g D

∗ →

( ) ( )

( ) ( ) ( ) ( ) ( )

of the from Moreover,

a Lyapunov function such that

where, is a pdf functi

, 0 0.

V f X g X X V X X DX

ξ ϕ ϕ

∃ →

∂⎛ ⎞= + ≤ − ∀ ∈⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠

( ) ( )

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

→ →

follows.

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

( ) ( ) ( ) ( )

( ) ( ) ( )

Step - 1:

By this construction, when

which is asymptotically stable (i.e.

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

= + + −

= + + −⎡ ⎤⎣ ⎦

→ = +

( ) ( ) ( ) ( )

This is backstepping, since is

stepped back by differentiation

So, we have

This system is equivalent to the original sys

X f X g X X g X zz v

⎡ ⎤= − ⎢ ⎥

⎣ ⎦

( ) ( )

Note: T T

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

Back-stepping:Conceptual Block Diagram

Back-stepping

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

( ) ( ) ( )

( ) ( )

Step-2: Let

1 , ( )2

V X z V X z

VV f X g X X g X z z vX

VV X g X v zX

≤−

∂⎛ ⎞= + + +⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠

⎡ ⎤∂⎛ ⎞≤ − + +⎢ ⎥⎜ ⎟∂⎝ ⎠⎢ ⎥⎣ ⎦

Then (ndf)

Vv g X k z kX

V V X k z

∂⎛ ⎞= − − >⎜ ⎟∂⎝ ⎠≤ − − <

( ) ( )

Control Solution

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

Vv u g X k zX

Vu g X k XX

f X g X kX

ϕ ξ ϕ

ϕϕ ξ

∂⎛ ⎞= − = − −⎜ ⎟∂⎝ ⎠

∂⎛ ⎞= − − −⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠

∂⎛ ⎞= + >⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠

first⎡ ⎤⎣ ⎦

Integrator Back-stepping:An Example

( ) ( )( )

2 31 1 1 2

2 31 1 1 1 2 1

21 1 1

2 31 1 1 1 1 1 2

Problem

Solution

To find

: , , , 1

x ax x xx u

X x f x ax x x g x

V x x x ax x x

= − +=

= = − = =

= = − + ≤ ( )2 41 1

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

Modified system

x ax x x

ax x x

x ax x x

x ax x x x x

+ ≤ −

+ = −

⇒ = − −

= − + + −⎡ ⎤⎣ ⎦

= ( )( )( ) ( )

21 1 1

2 31 11 1 1 1

Let1 ,2

V x z V x z

V VV V z v ax x x v zx x

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

( ) ( )

( )( )( )( ) ( )

2 3 21 1 2 1 2 1 1

2 3 21 1 1 2 1 2 1 1

2 3 21 1 1 2 1 1 2 1

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

⎛ ⎞∂= − − >⎜ ⎟∂⎝ ⎠

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

⎡ ⎤= − − − + − − + +⎣ ⎦

= − + − + − − + +

where 0k >

221 2 1

22 21 2 1 1

Note: The composite Lyapunov function is:1 2

1 1 2 21 1 2 2

x x x ax

= + −⎡ ⎤⎣ ⎦

= + + +

Integrator Back-stepping:More General Case

( ) ( ) 1

System Dynamics:

Idea : Successive iteration.

Note: The procedure for order system is entirely analogous

X f X g X

⎡ ⎤⎣ ⎦

( ) ( )

( )( ) ( )

Step-1 Consider the subsystem

Assumption:

is a stabiliting feedback law for

and is the coresponding Lyapunov fun

X f X g X

ction.

( ) ( ) ( ) ( ) ( )

12 1 1

By the result obtained before, we have

We also have

, ( 0)

X Vf X g X g X k XX X

ϕξ ξ ξ ϕ

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

= ( ) 21 1

V Xξ ϕ+ −⎡ ⎤⎣ ⎦

( ) ( )

1 11 1

1 21 1 1 1 2

Step - 2:

where,

Using the same idea,

g Xf X

X f X g XX

Vu f X g XX X

⎡ ⎤ ⎡ + ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦

⎡ ⎤= ⎢ ⎥

⎣ ⎦

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

( ) ( )

( ) ( ) ( )

1 1 1 2 1 1 1

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

1 1 12 2 2

g X k X k

ξ ϕ ξ ϕ ξ ϕ= + +

− − >⎡ ⎤⎣ ⎦

− = − + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Integrator Back-stepping for Strict Feedback Systems

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

1 1 1 1 1 2

2 2 1, 2 2 1 2 3

System Dynamics

X f X g X

f X g X

ξ ξ ξ ξ

ξ ξ ξ ξ ξ ξ

= ( ) ( )

( ) ( ) ( )

1 1 2 1 2 1

Strong Assumption:

over the domain of interest

, , , , ,

, , , , , , , , , 0

g X g X g X

ξ ξ ξ ξ

ξ ξ ξ ξ ξ∀

… …

Integrator Back-stepping for Strict Feedback Systems

( ) ( )( ) ( )

( ) ( )

Special Case:

Solution:

Define and carryout the design for as before.

Finally

X f X g X

f X g X u

f X g X u v

ξ ξ ξ

( ) ( )

( ) Note: By assumption,

u v f Xg X

= −⎡ ⎤⎣ ⎦

≠ ∀

Linear Quadratic (LQ) Observer

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.

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)

Observer Design for Linear Systems

State observer( )

(sensor output vector)

ˆLet the observed state be and the be

ˆˆ ˆ ˆ

X AX BUY CX

X AX BU K Y

Plant :

Observer dynamics

Error :

Observer Design for Linear Systems

( ) ( )

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

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

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

ˆand 0

This results in ˆ

A A K C

− − =

Observer dynamics:

Necessary and sufficient condition for the existence of Ke :

The system should be “observable”.

( )ˆ ˆ ˆeX AX BU K Y CX= + + −

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!

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= = −

( ) ( )

T T Te

A K C A K C

⎡ ⎤− = −⎣ ⎦

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= −

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,

X A BK XX as t= −

→ →∞

CL system (control design)

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

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)

X Xii W t V t

iii E W t W t Q Q

E V t V t

τ δ τ

⎡ ⎤+ = ≥⎣ ⎦

+ ) ( ), 0 (pdf)R Rδ τ⎡ ⎤ = >⎣ ⎦

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

X t X t X t⎡ ⎤−⎣ ⎦

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

Objective:

Observer/Estimator/Filter Dynamics

( )( ) ( )( ) ( )

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

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 ?

CXiii K

Problem :

( )ˆ ˆ ˆ eX AX BU K Y Y= + + −

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

AP PA PC R CP GQG

K PC R

+ − + =

( )ˆ ˆ ˆ

where is the measurement vectoreX AX BU K Y CX

= + + −

Solution: Summary

Integrator Back-stepping; Linear Quadratic (LQ) Observer

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