Discrete and continuous dynamic systems

Bounded input bounded output (BIBO) and asymptotic stability

Continuous and discrete time linear time-invariant systems

Katalin Hangos

University of Pannonia
Faculty of Information Technology
Department of Electrical Engineering and Information Systems
hangos.katalin@virt.uni-pannon.hu

Feb 2018
 Lecture overview
1 Previous notions
CT-LTI state space models
DT-LTI state space models
Poles
2 The notion of stability
Signal norms
3 Bounded input-bounded output (BIBO) stability - continuous time
systems
BIBO stability for SISO CT-LTI systems
4 Asymptotic stability - continuous time systems
The notion of asymptotic stability
Motivating example
Asymptotic stability of CT-LTI systems
5 Discrete time stability
Stability of DT systems
Stability of DT-LTI systems
K. Hangos (University of Pannonia) PE Feb 2018 2 / 31
 Previous notions

Overview

1 Previous notions
CT-LTI state space models
DT-LTI state space models
Poles

2 The notion of stability

3 Bounded input-bounded output (BIBO) stability - continuous time

systems

4 Asymptotic stability - continuous time systems

5 Discrete time stability

K. Hangos (University of Pannonia) PE Feb 2018 3 / 31

 Previous notions

Systems

System (S): acts on signals

y = S[u]

inputs (u) and outputs (y )

K. Hangos (University of Pannonia) PE Feb 2018 4 / 31

 Previous notions

CT-LTI I/O system models

Time domain:Impulse response function

is the response of a SISO LTI system to a Dirac-delta input function
with zero initial condition.
The output of S can be written as
Z ∞ Z ∞
y (t) = h(t − τ )u(τ )dτ = h(τ )u(t − τ )dτ
−∞ −∞

K. Hangos (University of Pannonia) PE Feb 2018 5 / 31

 Previous notions CT-LTI state space models

CT-LTI state-space models

General form - revisited

ẋ(t) = Ax(t) + Bu(t) , x(t0 ) = x(0)
y (t) = Cx(t)

with
signals: x(t) ∈ Rn , y (t) ∈ Rp , u(t) ∈ Rr
system parameters: A ∈ Rn×n , B ∈ Rn×r , C ∈ Rp×n (D = 0 by
using centering the inputs and outputs)
Dynamic system properties:
observability
controllability
stability

K. Hangos (University of Pannonia) PE Feb 2018 6 / 31

 Previous notions DT-LTI state space models

DT-LTI state space models

State space model

x(k + 1) = Φx(k) + Γu(k) (state equation)

y (k) = Cx(k) + Du(k) (output equation)

with given initial condition x(0) and

x(k) ∈ Rn , y (k) ∈ Rp , u(k) ∈ Rr

being vectors of finite dimensional spaces and

Φ ∈ Rn×n , Γ ∈ Rn×r , C ∈ Rp×n , D ∈ Rp×r

being matrices

K. Hangos (University of Pannonia) PE Feb 2018 7 / 31

 Previous notions Poles

Poles of CT-LTI and DT-LTI systems

continuous time system discrete time system

state eq. ẋ(t) = Ax(t) + Bu(t) x(kh + h) = Φx(kh) + Γu(kh)

Φ = e Ah

output eq. y (t) = Cx(t) y (kh) = Cx(kh)

poles λi (A) λi (Φ)

λi (Φ) = e λi (A)h

K. Hangos (University of Pannonia) PE Feb 2018 8 / 31

 The notion of stability

Overview

1 Previous notions

2 The notion of stability

Signal norms

3 Bounded input-bounded output (BIBO) stability - continuous time

systems

4 Asymptotic stability - continuous time systems

5 Discrete time stability

K. Hangos (University of Pannonia) PE Feb 2018 9 / 31

 The notion of stability

Stability
Stability expresses the resistance of a system against disturbances.

System response to two kinds of disturbances

Small persistent disturbance on input(s) (d1 ): external or bounded
input bounded output (BIBO) stability
Impulse type effect on the state moving it out of steady state (d2 ):
internal or asymptotic stability
K. Hangos (University of Pannonia) PE Feb 2018 10 / 31
 The notion of stability Signal norms

Scalar valued signals

vector norms: v ∈ Rn
v
u n n
uX X
||v ||2 = t vi2 , ||v ||1 = |vi | , ||v ||∞ = max|vi |
i=1 i=1

discrete time signal: f (k) ∈ R, ∀k ≥ 0

∞
!1
X q

norm: ||f ||q = |f (k)|qν

0

continuous time signal f (t) ∈ R, ∀t ≥ 0

Z ∞ 1
q
norm: ||f ||q = |f (t)|qν
0

K. Hangos (University of Pannonia) PE Feb 2018 11 / 31

 The notion of stability Signal norms

Vector valued signals

continuous time signal: f (t) ∈ Rn , ∀t ≥ 0

|| · ||n is a norm in Rn (e.g. Euclidean)
 Z ∞ 
+ n
Lq (ν) = f : R0 7→ R | f is measurable and ||f (t)||qν <∞
0

Z ∞ 1
q
norm: ||f ||q = ||f (t)||qν
0
Remark: The case L2 is special, because the norm comes from an
inner product (L2 is a Hilbert-space)

K. Hangos (University of Pannonia) PE Feb 2018 12 / 31

Bounded input-bounded output (BIBO) stability - continuous
time systems

Overview

1 Previous notions

2 The notion of stability

3 Bounded input-bounded output (BIBO) stability - continuous time

systems
BIBO stability for SISO CT-LTI systems

4 Asymptotic stability - continuous time systems

5 Discrete time stability

K. Hangos (University of Pannonia) PE Feb 2018 13 / 31

Bounded input-bounded output (BIBO) stability - continuous
time systems

BIBO stability – general

Definition (BIBO stability)

A system is externally or BIBO stable if for any bounded input it responds
with a bounded output

||u|| ≤ M1 < ∞ ⇒ ||y || ≤ M2 < ∞

where ||.|| is a signal norm.

This applies to any type of systems.

Stability is a system property, i.e. it is realization-independent.

K. Hangos (University of Pannonia) PE Feb 2018 14 / 31

Bounded input-bounded output (BIBO) stability - continuous
time systems BIBO stability for SISO CT-LTI systems

BIBO stability – 1

Bounded input-bounded output (BIBO) stability for SISO systems

|u(t)| ≤ M1 < ∞, ∀t ∈ [0, ∞[ ⇒ |y (t)| ≤ M2 < ∞, ∀t ∈ [0, ∞[

Theorem (BIBO stability)

A SISO LTI system is BIBO stable if and only if
Z ∞
|h(t)|dt ≤ M < ∞
0

where M ∈ R+ and h is the impulse response function.

K. Hangos (University of Pannonia) PE Feb 2018 15 / 31

Bounded input-bounded output (BIBO) stability - continuous
time systems BIBO stability for SISO CT-LTI systems

BIBO stability – 2

Proof: R∞
⇐ Assume 0 |h(t)|dt ≤ M < ∞ and u is bounded, i.e.
|u(t)| ≤ M1 < ∞, ∀t ∈ R+ 0 . Then
Z ∞ Z ∞
|y (t)| ≤ | h(τ )u(t − τ )dτ | ≤ M1 |h(τ )|dτ ≤ M1 · M = M2
0 0
R∞
⇒ (indirect) Assume 0 |h(τ )|dτ = ∞, but the system is BIBO stable.
Consider the bounded input:

 1 if h(τ ) > 0
u(t − τ ) = sign h(τ ) = 0 if h(τ ) = 0
−1 if h(τ ) < 0


K. Hangos (University of Pannonia) PE Feb 2018 16 / 31

 Asymptotic stability - continuous time systems

Overview

1 Previous notions

2 The notion of stability

3 Bounded input-bounded output (BIBO) stability - continuous time

systems

4 Asymptotic stability - continuous time systems

The notion of asymptotic stability
Motivating example
Asymptotic stability of CT-LTI systems

5 Discrete time stability

K. Hangos (University of Pannonia) PE Feb 2018 17 / 31

 Asymptotic stability - continuous time systems The notion of asymptotic stability

Asymptotic stability – general

Definition ((local) asymptotic stability)

An equilibrium/steady-state point x ∗ of truncated/autonomous system
with state equation

ẋ(t) = F (x(t)) , x(0) = x0 (6= x ∗ ) , F (x ∗ ) = 0

is internally or asymptotically stable if for any initial state x0 6= x ∗ (from a

neighbourhood of Gx ∗ of x ∗ )

lim x(t) = x ∗
t→∞

This applies to any type of continuous time systems.

For discrete time systems a similar definition is applicable with
x(k + 1) = F (x(k).

K. Hangos (University of Pannonia) PE Feb 2018 18 / 31

 Asymptotic stability - continuous time systems Motivating example

Example: asymptotic stability

RLC circuit, parameters: R = 1 Ω, L = 10−1 H, C = 10−1 F .

uC (0) = 1 V, i(0) = 1 A, ube (t) = 0 V

K. Hangos (University of Pannonia) PE Feb 2018 19 / 31

 Asymptotic stability - continuous time systems Motivating example

Non-asymptotic stability

(R)LC circuit, parameters: R = 0 Ω(!), L = 10−1 H, C = 10−1 F .

uC (0) = 1 V, i(0) = 1 A, ube (t) = 0 V

K. Hangos (University of Pannonia) PE Feb 2018 20 / 31

 Asymptotic stability - continuous time systems Motivating example

Example: instability

ẋ1 = x1 + 0.1x2
, x(0) = [1 2]T
ẋ2 = −0.2x1 + 2x2

K. Hangos (University of Pannonia) PE Feb 2018 21 / 31

 Asymptotic stability - continuous time systems Asymptotic stability of CT-LTI systems

Stability of CT-LTI systems

(Truncated) LTI state equation with (u ≡ 0):

ẋ = A · x, x ∈ Rn , A ∈ Rn×n , x(0) = x0

Equilibrium pont: x ∗ = 0
Solution:
x(t) = e At · x0
Recall: A diagonalizable (there exists invertible T , such that

T · A · T −1

is diagonal) if and only if, A has n linearly independent eigenvectors.

K. Hangos (University of Pannonia) PE Feb 2018 22 / 31

 Asymptotic stability - continuous time systems Asymptotic stability of CT-LTI systems

Asymptotic stability of LTI systems – 1

Stability types:
the real part of every eigenvalue of A is negative (A is a stability
matrix): asymptotic stability
A has eigenvalues with zero and negative real parts
the eigenvectors related to the zero real part eigenvalues are linearly
independent: (non-asymptotic) stability
the eigenvectors related to the zero real part eigenvalues are not
linearly independent: (polynomial) instability
A has (at least) an eigenvalue with positive real part: (exponential)
instability

K. Hangos (University of Pannonia) PE Feb 2018 23 / 31

 Asymptotic stability - continuous time systems Asymptotic stability of CT-LTI systems

Asymptotic stability of LTI systems – 2

Theorem
The eigenvalues of a square A ∈ Rnxn matrix remain unchanged after a
similarity transformation on A by a transformation matrix T :

A0 = TAT −1

Proof:
Let us start with the eigenvalue equation for matrix A

Aξ = λξ , ξ ∈ Rn , λ ∈ C

If we transform it using ξ 0 = T ξ then we obtain

TAT −1 T ξ = λT ξ

A0 ξ 0 = λξ 0
K. Hangos (University of Pannonia) PE Feb 2018 24 / 31
 Asymptotic stability - continuous time systems Asymptotic stability of CT-LTI systems

Asymptotic stability of LTI systems – 3

Theorem
A CT-LTI system is asymptotically stable iff A is a stability matrix.

Sketch of Proof: Assume A is diagonalizable

 
λ1 0 . . . 0
 0 λ2 . . . 0 
Ā = TAT −1 = 
 
.. 
 . 0 
0 ... 0 λn

e λ1 t
 
0 ... 0
Āt Āt
 0 e λ2 t ... 0 
x̄(t) = e · x̄0 , e =
 
.. 
 . 0 
0 ... 0 e λn t

K. Hangos (University of Pannonia) PE Feb 2018 25 / 31

 Asymptotic stability - continuous time systems Asymptotic stability of CT-LTI systems

BIBO and asymptotic stability

Theorem
Asymptotic stability implies BIBO stability for LTI systems.

Proof:
Z t
At
x(t) = e x(0) + e A(t−τ ) Bu(τ )dτ, y (t) = Cx(t)
0
Rt
||x(t)|| ≤ ||e At x(t0 ) + M 0 e A(t−τ ) Bdτ || =
Rt
= ||e At (x(t0 ) + M 0 e −Aτ Bdτ )|| =
= ||e At (x(t0 ) + M[−A−1 e −Aτ B]t0 )|| =
= ||e At [x(t0 ) − MA−1 e −At B + MA−1 B]||
||x(t)|| ≤ ||e At (x(t0 ) + MA−1 B) − MA−1 B||

BIBO stability does not necessarily imply asymptotic stability.

K. Hangos (University of Pannonia) PE Feb 2018 26 / 31
 Discrete time stability

Overview

1 Previous notions

2 The notion of stability

3 Bounded input-bounded output (BIBO) stability - continuous time

systems

4 Asymptotic stability - continuous time systems

5 Discrete time stability

Stability of DT systems
Stability of DT-LTI systems

K. Hangos (University of Pannonia) PE Feb 2018 27 / 31

 Discrete time stability Stability of DT systems

Stability of discrete time systems – 1

Truncated state equation

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

with a ordinary solution x 0 (k) for x 0 (k0 ) and a perturbed solution

x(k) for x(k0 ).

Stability of a solution x 0 (k) is stable if for a given  > 0 there exists

a δ(, k0 ) such that all solutions with ||x(k0 ) − x 0 (k0 )|| < δ fulfill
||x(k) − x 0 (k)|| <  for all k ≥ k0 .

Asymptotic stability x 0 (k) is asymptotically stable if it is stable and

||x(k) − x 0 (k)|| → 0 when k → ∞ provided that ||x(k0 ) − x 0 (k0 )|| is
small enough.

K. Hangos (University of Pannonia) PE Feb 2018 28 / 31

 Discrete time stability Stability of DT systems

Stability of discrete time systems – 2

BIBO stability
A discrete time system is externally or BIBO stable if for any

||u|| ≤ M1 < ∞ ⇒ ||y || ≤ M2 < ∞

where ||.|| is a suitable signal norm.

K. Hangos (University of Pannonia) PE Feb 2018 29 / 31

 Discrete time stability Stability of DT-LTI systems

Stability of DT-LTI systems – 1

Consider a truncated state equation with u(k) = 0, k = 0, 1, 2, ...

x(k + 1) = Φx(k)

x 0 (k) for x 0 (0) = a0 as the ordinary solution and

x(k) for x(0) = a as a "perturbed solution".
The difference x = x − x 0 satisfies

x(k + 1) = Φx(k) , x(0) = a − a0

⇒ Stability is a system property for LTI systems

K. Hangos (University of Pannonia) PE Feb 2018 30 / 31

 Discrete time stability Stability of DT-LTI systems

Stability of DT-LTI systems – 2

Solution of the truncated state equation x(k + 1) = Φx(k), x(0) = x0

x(k) = Φk x(0)

Bring the matrix Φk into diagonal form and use that its eigenvalues
λi (Φk ) = λi (Φ)k thus

x(k) −→ 0 ⇐⇒ |λi (Φ)| < 1

Theorem
A DT-LTI system is asymptotically stable if and only if λi (Φ) are strictly
inside the unit disc.

Theorem
Asymptotic stability implies BIBO stability.
K. Hangos (University of Pannonia) PE Feb 2018 31 / 31