lyapunov stab closed loop for linear system theory

7/25/2019 Lyapunov Stab Closed Loop for linear system theory

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Copyright F.L. Lewis 2008

All rights reserved

Updated: Tuesday, November 11, 2008

Lyapunov Stabil ity Analysis for Feedback Control Design

Lyapunov Theorems

Lyapunov Analysis allows one to analyze the stability of continuous-time dynamical systems anddiscrete-time dynamical systems of the form:

Continuous-time nonlinear system( ), (0)x f x x=

with state ( ) nx t R and f(0)=0 so the origin is an equilibrium point. We assume that f(.) is

continuous so there exist solutions, and thatf(x)is Lipschitz so there exists a unique solution.

Discrete-time nonlinear system

1 0( ),k kx f x x+ =

with nkx R ,f(.)continuous and Lipschitz, andf(0)=0.

Definition. Lyapunov Function Candidate (LFC).

A scalar function ( ) : nV x R R is said to be a Lyapunov Function Candidate (LFC) if:

1. V(x)is a continuous real-valued function

2. ( ) 0V x > , i.e. V(x) is positive definite

A Lyapunov function is a LFC that is nonincreasing with time and hence bounded.

Definition: Continuous-time (CT) Lyapunov Function

( ) : nV x R R is said to be a CT Lyapunov Function if:

V(x)is a LFC and

3. 0dV

Vdt

= , i.e. ( )V x is negative semi-definite

Definition: Discrete-Time (DT) Lyapunov Function


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( ) : nV x R R is said to be a DT Lyapunov Function if:

V(x)is a LFC and

3. 1( ) ( ) ( ) 0k k kV x V x V x+ = , i.e. the first difference ( )kV x is negative semi-

definite

The following results were proven by A.M. Lyapunov. Stability for CT systems refers to

the j-axis stability boundary in the s-plane. Stability for DT systems refers to the unit circlestability boundary in the s-plane.

Lyapunov Theorem. SISL. Suppose that for a given system there exists a Lyapunov function.Then the system is SISL.

Lyapunov Theorem. AS. Suppose that for a given system there exists a Lyapunov function

which also satisfies the stronger third condition:

For CT systems: 0dV

Vdt

=


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Theorem. SISL for LTI CT Systems. Let Q be a symmetric positive semidefinite matrix.

Then the system x Ax= is SISL (e.g. marginally stable) if and only if the (symmetric) matrix Pwhich solves the CT Lyapunov equation

TA P PA Q+ =

is positive definite.

Theorem. SISL for LTI DT Systems. Let Q be a symmetric positive semidefinite matrix.

Then the system 1k kx Ax+ = is SISL (e.g. marginally stable) if and only if the (symmetric) matrix

Pwhich solves the DT Lyapunov equationTA PA P Q =

is positive definite.

The proof relies on the fact that, if the Lyapunov equations have solutions as specified,

then12

( ) TV x x Px=

serves as a Lyapunov function, with constant kernel matrix Psymmetric and positive definite,i.e. 0TP P= > .

If TQ Q= is in fact positive definite, the theorems yields AS.

Note that the solution properties of the CT Lyapunov equationTA P PA Q+ =

refer to the locations of the poles of system matrixAwith respect to the left half of the complex

plane.

On the other hand, the solution properties of the DT Lyapunov equationTA PA P Q =

refer to the locations of the poles of system matrix A with respect to the unit circle of thecomplex plane.

Lyapunov Analysis for Controlled Systems

We now want to use Lyapunov Analysis to study the stability of systems with control inputs.

We shall see that instead of the Lyapunov equationsTA P PA Q+ = for CT systemsTA PA P Q = for DT systems

we obtain Riccati equations, namely the equations above but with extra terms quadratic in P.


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Control Design for Nonlinear CT Systems

Let us see how direct Lyapunov techniques are by confronting a nonlinear CT system.

THIS SECTION IS NOT ON THE EE 5307 Homework or exams. It is for fun only.

Consider the nonlinear system be given by

)(

)()(

xhy

uxgxfx

=

+=

with state ( ) nx t R and f(0)=0 so the origin is an equilibrium point. We assume that f(.) is

continuous so there exist solutions, and thatf(x)is Lipschitz so there exists a unique solution.

Then for any scalar C1function V(x)one hasT T T

T T

x x

dV V V V V x f gu V f V gu

dt x x x

= = = + +

with x

V

V x

the gradient, which is assumed here to be a column vector.

Completing the squares for any matrix 0TR R= > yields

RuuVggRVuVgRRugRVfVguVfVV T

x

TT

xx

TTT

x

T

x

T

x

T

x 211

2111

21 )()( +++=+= .

Now suppose that V(x)>0, V(0)=0, and satisfies the Hamilton-Jacobi (HJ) inequality

0121

21 + x

TT

x

TT

x VggRVhhfV .

Assume the system is locally input/output detectable in the sense that there exists a neighborhood

such thatttyandtu == 0)(0)( implies that 0)( tx .

This is implied by i/o observability.

Then the closed-loop system is asymptotically stable if one selects the control input as

x

T VxgRu )(1= .

To prove this, note that, according to the HJ equation

RuuhhuVgRRugRVV TTxTTT

x 21

2111

21 )()( ++

and according to the control selectionRuuyyRuuhhV TTTT

21

21

21

21 =

which is negative definite under the i/o detectable assumption. Therefore V(x) is a Lyapunov

function with 0


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Based on this analysis one has the next theorem.

Theorem for Control of Nonlinear CT Systems. Suppose that ( ) : nV x R R is continuous,

V(x)>0, V(0)=0, and V(x) satisfies the Hamilton-Jacobi-Bellman (HJB) equation

11 12 2 0T T T T x x xV f h h V gR g V + = .

Then the closed-loop system is stable using the control input

x

T VxgRu )(1= .

To prove this, note that if V(x)satisfies the HJB equation, it solves the HJ inequality.

THIS MEANS that to design a feedback control for a nonlinear system, one may first

solve the HJB equation for the value V(x), then compute the control using the formula above.

Control Design for Linear Time Invariant CT Systems

Here we specialize the previous development to the case of LTI CT systems of the formx Ax Bu= +

We want to find a SVFBu Kx=

to stabilize the system.

For any scalar C

1

function V(x)one hasT T TT T

x x

dV V V V V x Ax Bu V Ax V Bu

dt x x x

= = = + +

Completing the squares for any matrix 0TR R= > yields1 1 11 1 1

2 2 2( ) ( )T T T T T T T T T

x x x x x x xV V Ax V Bu V Ax V BR u R R B V u V BR B V u Ru = + = + + + .

Now suppose that V(x)>0, V(0)=0, and satisfies the Hamilton-Jacobi (HJ) inequality

11 12 2

0T T T T T x x xV Ax x C Cx V BR B V +

for any matrix Csuch that (A,C)is observable.

Note that the selection of C in the HJ effectively defines an outputy Cx=

Through which the state is observable. This output is NOT used for control purposes, but only to

obtain a suitable value V(x)through solution of the HJ.


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Let us make the HJ symmetric. Note that the first term is a scalar so that it equals its own

transpose, i.e.T T T

x xV Ax x A V = .

Thus, write the HJ inequality equivalently in symmetric form11 1 1

2 2 2( ) 0T T T T T T x x x xx A V V Ax y y V BR B V

+ +

(To se that this is symmetric, transpose it and you will get the same equation.)

Now select the control input as

1 T

xu R B V = .

Then the system is AS. To prove this, note that, according to the HJ equation1 11 1 1

2 2 2( ) ( )T T T T T

x xV V BR u R R B V u y y u Ru + +

and according to the control selection

RuuyyRuuhhV TTTT21

21

21

21 =

which is negative definite under the observabilityassumption. Therefore V(x) is a Lyapunov

function with 0 . Therefore12

( )Tx PxVPx

x x

= =

and the HJ becomes11 1 1

2 2 2

112

( )

( ) 0

T T T T T T T

T T T T

x A Px x PAx x C Cx x PBR B Px

x A P PA C C PBR B P x

+ +

= + +

Since this must hold for every statex(t)it is equivalent to1

0

T T TA P PA C C PBR B P+ +

The stabilizing control is selected according to1 Tu R B Px Kx

=

Theorem. Stabilizing Control Design for CT LTI Systems.


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Let 0, 0T TQ Q R R= > = > be symmetric positive definite matrices. Then the system

x Ax Bu= + is AS if there exists a positive definite solution P to the CT algebraic Riccatiequation (ARE)

1 0T TA P PA Q PBR B P+ + =

Then, the state feedback1 TK R B P=

makes the closed-loop system stable.

To prove this theorem, note that if Qis positive definite then (A,C) is observable for any

square root Cof Q, TQ C C= , for then Cis nonsingular. Moreover, if Pis a solution to the CT

ARE then the HJ inequality holds.

Compare the CT ARE to the CT Lyapunov equation

0T

A P PA Q+ + =

The theorem shows the following design method for CT LTI SVFB controllers

1. Select design matrices 0, 0T TQ Q R R= > = >

2. Solve the CT ARE1 0T TA P PA Q PBR B P+ + =

3. Compute the SVFB1 TK R B P

=

The ARE is easily solved using many routines, among them the MATLAB routine[ , ] ( , , , )K P lqr A B Q R=

Optimal Control for CT LTI Systems

Now we see that the above constructions actually yield the OPTIMAL CT controller.

Consider the LTI CT systemx Ax Bu= +

We want to find a SVFBu Kx=

to minimize the performance index

12

0

( (0), ) ( )T T

J x u x Qx u Ru dt

= +

Theorem. Optimal Control Design for CT LTI Systems.


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Let 0, 0T T

Q Q R R= > = > be symmetric positive definite matrices. Suppose there exists a

positive definite solution Pto the CT algebraic Riccati equation (ARE)1 0

T TA P PA Q PBR B P+ + =

Then, the state feedback1 TK R B P=

minimizesJ(x(0),u)and makes the closed-loop system stable.

Proof:

Select the Lyapunov function12

( ) TV x x Px=

Then it was shown above that if Psatisfies the ARE and one selects the given SVFB, one has12

( )T TV x Qx u Ru= +

Integrating both sides yields

12( )

lim ( ( )) ( ( )) ( ( ), )

T T

t t

t

V dt x Qx u Ru dt

V x t V x t J x t u

= +

=

However, the system has already been shown AS, so that

lim ( ) 0t

x t

=

which implies

lim ( ( )) 0t

V x t

=

or

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

T

V x t x t Px t J x t u= = with 1 Tu Kx R B Px= = .

It remains to show that V(x(t))is the optimal value of J(x(0),u). Can you do it?

This theorem shows how to select design matrices Q, R. namely, as discussed in the notes

on LQR.

Control Design for Linear Time Invariant DT SystemsGiven the LTI DT system

1k k kx Ax Bu+ = +

it is desired to find a stabilizing SVFB control

k ku Kx=


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Note that for any scalar quadratic form V(x)one has the first forward difference evaluated

along the system trajectories given as

1 1 1( ) ( ) ( )

2

T T T T

k k k k k k k k k k k k k

T T T T T T T

V x V V x Px x Px Ax Bu P Ax Bu x Px

x A PAx x Px x A PBu u B PBu

+ + + = = = + +

= + +

where lack of a subscript on time functions indicates their values at time k.

Note that the first difference of V(.)is QUADRATIC in the state dynamics Ax+Bu. Bycontrast, in the CT LTI case the differential of V(.) is LINEAR in the state dynamics Ax+Bu.

This makes expressions for DT systems more complex than for CT systems.

Complete the square to see that, for any nonsingular matrix one has1 1 1( ) ( ) 2

T T T T T T T T T T x A PB u B PAx u x A PB B PAx x A PBu u u + + = + +

So that one writes ( )V x as

1 1 1

( )

( ) ( )

T T T T T

k

T T T T T T T T

V x x A PAx x Px u B PBu

x A PB u B PAx u x A PB B PAx u u

= +

+ + +

Now, to get rid of the third term on the right-hand side and introduce a positive definite quadraticterm in u, define

TB PB R = + .

Then,

1 1 1

( )

( ) ( )

T T T T

k

T T T T T T T

V x x A PAx x Px u Ru

x A PB u B PAx u x A PB B PAx

=

+ + +

Now suppose ( ) 0, (0) 0V x V> = and it satisfies the DT HJ inequality1( ) 0T T T T A PA P Q A PB B PB R B PA + +

Then1 1( ) ( ) ( )T T T T T T

kV x x Qx u Ru x A PB u B PAx u + + + .

Selecting the control as1( )T T

k ku B PB R B PAx= +

yields

( ) T TkV x x Qx u Ru .

so that V(x)is a Lyapunov function. Now assume i/o detectability with the outputk k

y Cx=

withT

TQ Q Q C C = , and the system is AS.

It is not hard to show that in fact this choice of control also minimizes the performance

measure


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12

( , ) ( )T Tk k k k k i k

J x u x Qx u Ru

=

= +

where u denotes the control sequence 1{ , }k ku u + . That is, for the above ( )kV x one has

12

( ) min ( , ) min ( )T Tk k k k k k u u

i k

V x J x u x Qx u Ru

=

= = +

One calls ( )k

V x the VALUE FUNCTION.

Summary of Design procedure for DT Systems:

1. Select design matrices 0, 0T TQ Q R R= > = >

2. Solve the DT ARE1( ) 0T T T T A PA P Q A PB B PB R B PA

+ + =

3.

Compute the SVFB 1( )T TK B PB R B PA= +

The ARE is easily solved using many routines, among them the MATLAB routine

[ , ] ( , , , )K P dlqr A B Q R=

lyapunov stab closed loop for linear system theory

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