Stability of Dynamic Systems

Robert Stengel Optimal Control and Estimation, MAE 546

Princeton University, 2018

•  Stability about an equilibrium point

•  Bounds on the system norm•  Lyapunov criteria for stability•  Eigenvalues•  Transfer functions•  Continuous- and discrete-

time systems

Copyright 2018 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE546.html

http://www.princeton.edu/~stengel/OptConEst.html1

Dynamic System Stability

•  Well over 100 definitions of stability•  Common thread: Response, x(t), is bounded as t –> ∞•  Our principal focus: Initial-condition response of NTI

and LTI dynamic systems

2

Vector Norms for Real Variables

L2 norm = x 2 = xTx( )1/2 = x1

2 + x22 ++ xn

2( )1/2

•  �Norm� = Measure of length or magnitude of a vector, x

•  Euclidean or Quadratic Norm

•  Weighted Euclidean Norm

y 2 = yTy( )1/2 = y12 + y2

2 +!+ ym2( )1/2

= xTDTDx( )1/2 = Dx 2

xTDTDx xTQxQ DTD = Defining matrix

3

Uniform Stability§  Autonomous dynamic system

§  Time-invariant§  No forcing input

§  Uniform stability about x = 0

x(t) = f[x(t)]

x t0( ) ≤ δ , δ > 0

§  If system response is bounded, then the system possesses uniform stability

Let δ = δ ε( )If, for every ε ≥ 0,x t( ) ≤ ε, ε ≥ δ > 0, t ≥ t0

Then the system is uniformly stable

4

Local and Global Asymptotic Stability

•  Local asymptotic stability–  Uniform stability plus

x t( ) t→∞⎯ →⎯⎯ 0

•  Global asymptotic stability

•  If a linear system has uniform asymptotic stability, it also is globally stable x(t) = F x(t)

System is asymptotically stable for any ε

5

Exponential Asymptotic Stability

§  Uniform stability about x = 0 plus

x t( ) ≤ ke−α t x 0( ) ; k,α ≥ 0

§  –α = Lyapunov exponent§  If norm of x(t) is contained within an

exponentially decaying envelope with convergence, system is exponentially asymptotically stable (EAS)

§  Linear, time-invariant system that is asymptotically stable is EAS

6

k e−α t dt0

∞

∫ = −kα

⎛⎝⎜

⎞⎠⎟e−α t

0

∞=kα

x t( ) dt0

∞

∫ = xT t( )x t( )⎡⎣ ⎤⎦1/2dt

0

∞

∫ ≤ kα

⎛⎝⎜

⎞⎠⎟ x 0( )

and

x t( ) 2 dt0

∞

∫ is bounded

Exponential Asymptotic Stability

Therefore, time integrals of the norm of an EAS system are

bounded

7

Exponential Asymptotic Stability

Weighted Euclidean norm and its square are bounded if system is EAS

Dx t( ) dt0

∞

∫ = xT t( )DTDx t( )⎡⎣ ⎤⎦1/2dt

0

∞

∫ is bounded

with 0 < DTD ! Q < ∞

xT t( )Qx t( )⎡⎣ ⎤⎦dt0

∞

∫ is bounded

Conversely, if the weighted Euclidean norm is bounded, the LTI system is EAS

8

Initial-Condition Response of an EAS Linear System

•  To be shown–  Continuous-time LTI system is stable if all

eigenvalues of F have negative real parts–  Discrete-time LTI system is stable if all

eigenvalues of Φ lie within the unit circle

x(t) = Φ t,0( )x(0) = eF t( )x(0)

x(t) 2 = xT (0)ΦT t,0( )Φ t,0( )x(0) is bounded

9

Lyapunov�s First Theorem

x(t) = f[x(t)] is stable at xo = 0 if

Δx(t) = ∂f[x(t)]∂x xo =0

Δx(t) is stable

•  A nonlinear system is asymptotically stable at the origin if its linear approximation is stable at the origin, i.e., –  for all trajectories that start �close enough� (in the

neighborhood)–  within a stable manifold (closed boundary)

�At the origin� is a fuzzy concept10

Lyapunov Function*Define a scalar Lyapunov function, a positive definite

function of the state in the region of interest

V x t( )⎡⎣ ⎤⎦ ≥ 0

* Who was Lyapunov? see http://en.wikipedia.org/wiki/Aleksandr_Lyapunov

V = E = mV2

2+mgh; E

mg= Eweight

= V2

2g+ h

V = 12xTx; V = 1

2xTPx

Examples

11

V = xT t( )Px t( )⎡⎣ ⎤⎦2

Lyapunov�s Second Theorem

Evaluate the time derivative of the Lyapunov function

V x t( )⎡⎣ ⎤⎦ ≥ 0V[0]= 0 for t ≥ 0

dVdt

= ∂V∂t

+ ∂V∂x!x = ∂V

∂t+ ∂V∂x

f x t( )⎡⎣ ⎤⎦

= ∂V∂x

f x t( )⎡⎣ ⎤⎦ for autonomous systems

•  If in the neighborhood of the origin, the origin is asymptotically stabledVdt

< 0

12

•  Sufficient but not necessary condition for stability

•  Lyapunov function is not unique

Quadratic Lyapunov Function for LTI System

Lyapunov function

dVdt

= ∂V∂x!x = xT t( )P!x t( ) + !xT t( )Px t( )

= xT t( ) PF + FTP( )x t( ) " −xT t( )Qx t( )

V x t( )⎡⎣ ⎤⎦ = xT t( )Px t( )

Rate of change for quadratic Lyapunov function

x(t) = F x(t)Linear, Time-Invariant System

13

Lyapunov Equation

PF + FTP = −Qwith

P > 0, Q > 0

The LTI system is stable if the Lyapunov equation is satisfied with positive-definite P and Q

14

Lyapunov Stability: 1st-Order Example

PF + FTP = −Qwith p > 0, a < 02pa < 0 and q > 0∴ system is stable

Δx(t) = aΔx(t) , Δx(0) given1st-order initial-

condition response

F = a, P = p,Q = q

Δx(t) = Δ!x(t)dt0

t

∫ = aΔx(t)dt0

t

∫= eatΔx(0)

Unstable, a > 0

Stable, a < 0

PF + FTP = −Qwith p > 0, a > 02pa < 0 and q < 0∴ system is unstable

15

Lyapunov Stability and the HJB Equation

∂V *∂t

= −minu(t )

H

V x t( )⎡⎣ ⎤⎦ = xT t( )Px t( )

dVdt

< 0

Lyapunov stability Dynamic programming optimality

16Rantzer, Sys.Con. Let., 2001

Lyapunov Stability for Nonautonomous (Time-Varying) Systems

For asymptotic stability of time-varying system

Time-derivative of Lyapunov function must be negative-definite

17

!x(t) = f[x(t),t]; f[0,t]= 0 for t ≥ 0

V x t( ),t⎡⎣ ⎤⎦ ≥Va x t( )⎡⎣ ⎤⎦ > 0 in neighborhood of x 0( ) = 0

dV x t( ),t⎡⎣ ⎤⎦dt

=∂V x t( ),t⎡⎣ ⎤⎦

∂t⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪+

∂V x t( ),t⎡⎣ ⎤⎦∂x

f x t( )⎡⎣ ⎤⎦⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪< 0 in neighborhood of x = 0

V x t( ),t⎡⎣ ⎤⎦ ≤Vb x t( )⎡⎣ ⎤⎦ > 0 for large values of x t( )

Time-varying Lyapunov function bounded above and below by time-invariant Lyapunov functions

Laplace Transforms and Linear System Stability

18

Fourier Transform of a Scalar Variable

x(t)

x( jω ) = a(ω ) + jb(ω )

x(t) : real variablex( jω ) : complex variable

= a(ω )+ jb(ω )= A(ω )e jϕ (ω )

�

A : amplitudeϕ : phase angle

19

F x(t)[ ] = x( jω ) = x(t)e− jωt

−∞

∞

∫ dt

ω = frequency, rad / s

j ! i ! −1

Laplace Transforms of Scalar Variables

Laplace transform of a scalar variable is a complex numbers is the Laplace operator, a complex variable

Laplace transformation is a linear operation

x(t) : real variablex(s) : complex variable

= a(ω )+ jb(ω )= A(ω )e jϕ (ω )

Multiplication by a constant

Sum of Laplace transforms

20

L x(t)[ ] = x(s) = x(t)e− st dt

0

∞

∫ , s =σ + jω

L a x(t)[ ] = a x(s)

L x1(t)+ x2 (t)[ ] = x1(s)+ x2 (s)

Laplace Transforms of Vectors and Matrices

Laplace transform of a vector variable

Laplace transform of a matrix variable

21

L x(t)[ ] = x(s) =x1(s)x2 (s)...

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

L A(t)[ ] = A(s) =a11(s) a12 (s) ...a21(s) a22 (s) ...... ... ...

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Laplace transform of a derivative w.r.t. time

L

dx(t)dt

⎡⎣⎢

⎤⎦⎥= sx(s)− x(0)

Laplace transform of an integral over time

L x(τ )dτ∫⎡

⎣⎢

⎤

⎦⎥ =x(s)s

!x(t) = Fx(t)+Gu(t)y(t) = Hxx(t)+Huu(t)

Time-Domain System Equations

Laplace Transforms of System Equations

sx(s)− x(0) = Fx(s)+Gu(s)y(s) = Hxx(s)+Huu(s)

Transformation of the LTI System Equations

Dynamic Equation

Output Equation

Dynamic Equation

Output Equation

22

Laplace Transform of State Vector Response to Initial Condition and

Control Rearrange Laplace Transform of Dynamic Equation

sx(s)− Fx(s) = x(0)+Gu(s)sI− F[ ]x(s) = x(0)+Gu(s)

x(s) = sI− F[ ]−1 x(0)+Gu(s)[ ]

sI − F[ ]−1 = Adj sI − F( )sI − F

(n x n)

The matrix inverse is

Adj sI− F( ) : Adjoint matrix (n × n) Transpose of matrix of cofactorssI− F = det sI− F( ) : Determinant 1×1( )

23

Characteristic Polynomial of a Dynamic System

sI − F[ ]−1 = Adj sI − F( )sI − F

(n x n)

Characteristic polynomial of the system

sI− F = det sI− F( )≡ Δ(s) = sn + an−1s

n−1 + ...+ a1s + a0

Matrix Inverse

sI − F( ) =

s − f11( ) − f12 ... − f1n− f21 s − f22( ) ... − f2n... ... ... ...− fn1 − fn2 ... s − fnn( )

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

(n x n)

Characteristic matrix of the system

24

Eigenvalues

25

Eigenvalues of the LTI System

Δ(s) = sn + an−1sn−1 + ...+ a1s + a0 = 0

= s − λ1( ) s − λ2( ) ...( ) s − λn( ) = 0

Characteristic equation of the system

Eigenvalues, λi , are solutions (roots) of the polynomial, Δ s( ) = 0

�

λi = σ i + jω i

�

λ*i = σ i − jω is Plane

26

s Plane

�

δ = cos−1ζ

λ1 =σ 1 + jω1

λ2 =σ 1 − jω1

λ1 = σ1, λ2 = σ 2

Factors of a 2nd-Degree Characteristic Equation

ω n : natural frequency, rad/sζ : damping ratio, dimensionless

sI− F =s − f11( ) − f12

− f21 s − f22( )! Δ s( )

= s2 − f11 + f22( )s + f11 f22 + f12 f21( )= s − λ1( ) s − λ2( ) = 0 [real or complex roots]

= s2 + 2ζω ns +ω n2 with complex-conjugate roots

27

z Transforms and Discrete-Time Systems

28

Application of Dirac Delta Function to Sampling Process

Δxk = Δx tk( ) = Δx kΔt( )§  Periodic sequence of numbers

Δx kΔt( )δ t0 − kΔt( )

δ t0 − kΔt( )=∞, t0 − kΔt( ) = 00, t0 − kΔt( ) ≠ 0

⎧⎨⎪

⎩⎪

δ t0 − kΔt( )dt = 1t0 − kΔt( )−ε

t0 − kΔt( )+ε∫

§  Periodic sequence of scaled delta functions

§  Dirac delta function

29

Laplace Transform of a Periodic Scalar Sequence

§  Laplace transform of the delta function sequence

L Δx kΔt( )δ t − kΔt( )[ ] = Δx(z) = Δx kΔt( )δ t − kΔt( )e−sΔt0

∞

∫ dt

= Δx kΔt( )e−skΔt dt0

∞

∫ ! Δx kΔt( )z−kk=0

∞

∑

Δxk = Δx tk( ) = Δx kΔt( )§  Periodic sequence of numbers

Δx kΔt( )δ t − kΔt( )§  Periodic sequence of scaled delta functions

30

z Transform of the Periodic Sequence

L Δx kΔt( )δ t − kΔt( )[ ] = Δx kΔt( )k=0

∞

∑ e−skΔt ! Δx kΔt( )k=0

∞

∑ z−k

z ! esΔt advance by one sampling interval[ ]z−1 ! e− sΔt delay by one sampling interval[ ]

z Transform (time-shift) Operator

z transform is the Laplace transform of the delta function sequence

31

z Transform of a Discrete-Time Dynamic System

Δxk+1 = ΦΔxk + Γ Δuk + ΛΔwk

System equation in sampled time domain

Laplace transform of sampled-data system equation (�z Transform�)

zΔx(z) − Δx(0) = ΦΔx(z) + ΓΔu(z) + ΛΔw(z)

32

z Transform of a Discrete-Time Dynamic System

zΔx(z) − ΦΔx(z) = Δx(0) + ΓΔu(z) + ΛΔw(z)

zI − Φ( )Δx(z) = Δx(0) + ΓΔu(z) + ΛΔw(z)

�

Δx(z) = zI−Φ( ) −1 Δx(0) + ΓΔu(z) + ΛΔw(z)[ ]

Rearrange

Collect terms

Pre-multiply by inverse

33

Characteristic Matrix and Determinant of Discrete-Time System

�

zI−Φ( ) −1=Adj zI−Φ( )zI−Φ

(n x n)

Characteristic polynomial of the discrete-time model

zI− Φ = det zI− Φ( ) ≡ Δ(z)

= zn + an−1zn−1 + ...+ a1z + a0

�

Δx(z) = zI−Φ( ) −1 Δx(0) + ΓΔu(z) + ΛΔw(z)[ ]

Inverse matrix

34

Eigenvalues (or Roots) of the LTI Discrete-Time System

Characteristic equation of the system

Eigenvalues are complex numbers that can be plotted in the z plane

λi = σ i + jω i λ*i = σ i − jω i

Δ(z) = zn + an−1zn−1 + ...+ a1z + a0

= z − λ1( ) z − λ2( ) ...( ) z − λn( ) = 0

z Plane

Eigenvalues, λi , of the state transition matrix, Φ, are solutions (roots) of the polynomial, Δ z( ) = 0

35

Laplace Transforms of Continuous- and Discrete-Time State-Space

Models

Δx(z) = zI− Φ( ) −1ΓΔu(z)Δy(z) = H zI− Φ( ) −1ΓΔu(z)

Δx(s) = sI− F( ) −1GΔu(s)Δy(s) = H sI− F( ) −1GΔu(s)

Initial condition and disturbance effect neglected

Equivalent discrete-time model

36

Scalar Transfer Functions of Continuous- and Discrete-Time

Systems

Δy(s)Δu(s)

= H sI − F( ) −1G =HAdj sI − F( )G

sI − F= Y s( )

Δy(z)Δu(z)

= H zI − Φ( ) −1Γ =HAdj sI − Φ( )Γ

sI − Φ= Y z( )

dim(H) = 1× ndim(G) = n ×1

37

Comparison of s-Plane and z-Plane Plots of Poles and Zeros

§  s-Plane Plot of Poles and Zeros§  Poles in left-half-plane are stable§  Zeros in left-half-plane are

minimum phase

§  z-Plane Plot of Poles and Zeros§  Poles within unit circle are stable§  Zeros within unit circle are

minimum phase

Increasing sampling rate moves poles and zeros

toward the (1,0) point

Note correspondence of configurations as sampling rate

increases 38

Δy(s)Δu(s)

= H sI − F( ) −1G =HAdj sI − F( )G

sI − F= Y s( ) Δy(z)

Δu(z)= H zI − Φ( ) −1Γ =

HAdj sI − Φ( )ΓsI − Φ

= Y z( )

Next Time:Time-Invariant Linear-Quadratic Regulators

39

Supplementary Material

40

Differential Equations for 2nd-Order System

Laplace Transforms of 2nd-Order System

Second-Order Oscillator

Dynamic Equation

Output Equation

Δ!x1(t)Δ!x2 (t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

0 1−ω n

2 −2ζω n

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δx1(t)Δx2 (t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥+

0ω n

2

⎡

⎣⎢⎢

⎤

⎦⎥⎥Δu(t)

Δy1(t)Δy2 (t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 1 0

0 1⎡

⎣⎢

⎤

⎦⎥

Δx1(t)Δx2 (t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥+ 0

0⎡

⎣⎢

⎤

⎦⎥Δu(t) =

Δx1(t)Δx2 (t)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

sΔx1(s)− Δx1(0)sΔx2 (s)− Δx2 (0)

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

0 1−ω n

2 −2ζω n

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δx1(s)Δx2 (s)

⎡

⎣⎢⎢

⎤

⎦⎥⎥+

0ω n

2

⎡

⎣⎢⎢

⎤

⎦⎥⎥Δu(s)

Δy1(s)Δy2 (s)

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 1 0

0 1⎡

⎣⎢

⎤

⎦⎥

Δx1(s)Δx2 (s)

⎡

⎣⎢⎢

⎤

⎦⎥⎥+ 0

0⎡

⎣⎢

⎤

⎦⎥Δu(s) =

Δx1(s)Δx2 (s)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Dynamic Equation

Output Equation

41

Small Perturbations from Steady, Level Flight

�

Δ˙ x (t) = FΔx(t) + GΔu(t) + LΔw(t)

Δx(t) =

Δx1

Δx2

Δx3

Δx4

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

ΔVΔγΔqΔα

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

velocity, m/sflight path angle, rad

pitch rate, rad/sangle of attack, rad

Δu(t) =Δu1

Δu2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= ΔδE

ΔδT⎡

⎣⎢

⎤

⎦⎥

elevator angle, radthrottle setting, %

Δw(t) =Δw1

Δw2

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

ΔVwΔαw

⎡

⎣⎢⎢

⎤

⎦⎥⎥

~horizontal wind, m/s~vertical wind/Vnom, rad

42

Eigenvalues of Aircraft Longitudinal Modes of

MotionsI− F = det sI− F( ) ≡ Δ(s) = s − λ1( ) s − λ2( ) s − λ3( ) s − λ4( )

= s − λP( ) s − λ*P( ) s − λSP( ) s − λ*SP( )= s2 + 2ζ Pω nP

s +ω nP2( ) s2 + 2ζ SPω nSP

s +ω nSP2( ) = 0

Eigenvalues determine the damping and natural frequencies of the linear system�s modes of motion

ζ P ,ω nP( ) :phugoid (long-period) mode

ζ SP ,ω nSP( ) :short-period mode43

•  0 - 100 sec•  Reveals Long-Period Mode

Initial-Condition Response of Business Jet at Two Time Scales

•  0 - 6 sec•  Reveals Short-Period Mode

Same 4th-order responses viewed over different periods of time

Δx(t) =

Δx1

Δx2

Δx3

Δx4

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

ΔVΔγΔqΔα

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

velocity, m/sflight path angle, rad

pitch rate, rad/sangle of attack, rad

44