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
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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
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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