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Lecture 7: Stabilityy

OutlineOutlineBounded-input bounded-output (BIBO) stabilityBounded input bounded output (BIBO) stabilityStability of solutionsStability for linear time invariant systemsStability for linear time invariant systemsThe Nyquist stability criterionStability for equilibria via linearizationStability for equilibria via linearization

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BIBO stabilityBIBO stabilityMany different stability notions exist. y yA general system is BIBO stable, if a bounded input leads to a bounded output.A system is input-output stable if it has finite gain (!).

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Stability of solutionsStability of solutionsHere stability is understood as stability of solutions of system y y yequations with respect to initial conditions. For a general system, stability depends on where in the state space the state vector of the system is.

Ý x (t) = f (x(t)) x(t +1) = f (x(t))( ) f ( ( ))y(t) = h(x(t))x(0) = x0

y(t) = h(x(t))x(0) = x0

The solution x*(t), for the initial state x*(0), is said to be stable if for each ε>0 there is a δ such that i li th t f ll t 0

x*(0) − x(0) < δ

implies that for all t>0.The solution x*(t) is said to be asymptotically if it is stable and there exist a δ such that implies that

x*(t) − x(t) < ε

*(0) (0) < δ

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and there exist a δ such that implies that as t>∞.

x (0) − x(0) < δ

x*(t) − x(t) →0

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Stability for LTI systemsStability for LTI systemsLTI systems: stability is a system property (defined by y y y p p y ( ysystems parameters) and applies notwithstanding the initial conditions

All parameters are in

Initial conditions response

All parameters are inthe matrix A

The difference between two solutions with different initial values (and the same input) is given by

x*(t) − x(t) = eA( t− t0 ) x*(t0) − x(t0)( )It is determined by the properties of A (here we refer to unstable, stable or asymptotically stable systems)Th b h i f At d At i l d h i l f

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The behaviour of eAt and At is related to the eigenvalues of matrix A.

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Stability for LTI systemsStability for LTI systemsInitial conditions responsep

Eigenvalues and eigenvectors of A:g gConsider the case of single and distinct eigenvalues:

Introduce the transformation matrix:For normalized eigenvectors T is unitary, i.e.For normalized eigenvectors T is unitary, i.e.

Let x=Tz z –new state vector x0=Tz0Let x Tz, z new state vector, x0 Tz0

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Stability for LTI systemsStability for LTI systemsContinuous system: Discrete system:

An LTI system is asymptotically stable if and only if all eigenvalues of the matrix A are inside the stability regionmatrix A are inside the stability region.

01Im Ims-plane z-plane

0ReRe

If an eigenvalue λi is outside of the stability region then the system is unstableunstableIf all eigenvalues are inside the stability region or on the stability border and those that are on the stability border are single, then the system is marginall stable The s stem o tp t can tho gh be nbo nded despite

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marginally stable. The system output can though be unbounded despite bounded input.

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Stability for continuous LTI systemStability for continuous LTI systemLTI – linear time-invariant system (model)y ( )• Input-output form

• State space form

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the characteristic polynomial of A, the denominator of W(s)

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Nyquist stability criterionNyquist stability criterionNyquist stability criterion provides information about stability yq y p yof the closed-loop system

W(s)-R(s) Y(s)

C(s)

Y(s) C(s)W (s)Open-loop transfer function: C(s)W(s)

Y(s)R(s)

=C(s)W (s)

1+C(s)W (s)

Stability: The zeros of [1+C(s)W(s)] must be into the stability region

The poles of C(s)W(s) are the same as 1+C(s)W(s) =1+D(s)

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p ( ) ( ) ( ) ( ) ( )

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Cauchy’s principle of argumentCauchy s principle of argumentThe Nyquist stability criterion can be understood by using theyq y y gargument principle (Cauchy’s principle of argument).

Cauchy’s principle of argument: For a function F(s), analytical except in a finite number of poles. If the functionF(s) has Z zeros and P poles in an area Γs then Z-P= number of times the curve F(s) encircles the origin when sfollows the boundary Γ in positive direction (counterfollows the boundary Γs in positive direction (counterclockwise)

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Nyquist contourNyquist contourThe Nyquist plot is a polar plot of the function D(s) yq p p p ( )(modified!) when s travels around the contour given below.

ω= jagIm s planep

R

c

σ=alRe σ=alRe∞→R

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Right half planeLeft half plane

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Nyquist stability criterionNyquist stability criterionThe number of unstable closed-loop poles is equal to the p p qnumber of unstable open loop poles (P) plus the number of encirclements of the origen (N) of the Nyquist plot of the

l f ti D( )complex function D(s)Z=N+P

•N is positive for encirclements in directionclockwise•N is negative for encirclements in opposite

The Nyquist criterion uses this transformation:

N is negative for encirclements in oppositedirection clockwise

D’(s)=D(s)-1=C(s)W(s)

Then, the function C(s)W(s) is plotted and the encirclements of the Nyquist plot around the point [-1, j0] are counted.

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Nyquist stability criterionNyquist stability criterionIf the system C(s)W(s) has no poles in the right half plane, y ( ) ( ) p g p ,the closed loop system Y(s)

R(s)=

C(s)W (s)1+C(s)W (s)

is stable if and only if the Nyquist plot does not encircle the point (-1,0).

If the system C(s)W(s) has poles in the right half plane, the closed loop systemclosed loop system

Y(s)R(s)

=C(s)W (s)

1+C(s)W (s)

is stable if and only if the number of encirclements of the point (-1,0) in opposite direction clockwise of the Nyquist plot

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is equal to the number of poles in the right half plane.

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Nyquist plot (Z=N+P)Nyquist plot (Z=N+P)Nyquist diagrams are always symmetrical withyq g y yrespect to the real axisTwo stable closed loop systems:

4

50 dB

Nyquist DiagramD’(s)=10/(s-1)

4

50 dB

Nyquist DiagramD’(s)=10/(s+1)

2

3

4

-2 dB4dB

2 dB2

3

4

-4dB

-2 dB

4dB

2 dB

-1

0

1-10 dB-6 dB-4 dB

10 dB6 dB

4 dB

Imag

inar

y Ax

is-1

0

1-10 dB

-6 dB-4 dB

10 dB6 dB4 dB

Imag

inar

y Ax

is

-4

-3

-2

-4

-3

-2

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-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0-5

Real Axis-2 0 2 4 6 8 10

-5

Real Axis

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Gain and phase stability marginsGain and phase stability marginsThe margins give us information of how close the curves are g gto encircle the point (-1,0).

Im{C(s)W(s)} ωcg and ωcp are the gain and phase crossover frequencies

Unit circle

Re{C(s)W(s)}C( jω cg )W ( jω cg ) =1

arg C( jω cp )W ( jω cp ){ }=180º{ }

{ }jWjCPm cgcg

1

)()(argº180 ωω+=

[ ] [ ]dBjWjC

dBGmcpcp )()(

1log20ωω

=

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Pm – phase stability marginGm – gain stability margin

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Nonlinear systems versus linearNonlinear systems versus linearNonlinear system in state-space form

Linear system in state-space form

Nonlinear systems versus linear • Superposition principle is not valid for nonlinear systems• Superposition principle is not valid for nonlinear systems• The principle of frequency preservation does not apply to nonlinear

systemsLi d l b i ti f li t

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• Linear models can be seen as approximations of nonlinear systems linearization

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Stability for equilibria via linearizationStability for equilibria via linearizationA nonlinear system in state-space form

Can be described, by using linearization, in the vicinity of the equilibrium (x0,u0) by the linear system( 0, 0) y y

where z=x-x0 and v=u-u0.

If all eigenvalues of A have strictly negative real part, then (x0,u0) is an asymptotically stable equilibrium.If any of the eigenvalues of A has strictly positive real part then (x0 u0) isIf any of the eigenvalues of A has strictly positive real part then (x0,u0) is an unstable equilibrium If none of the eigenvalues of A has positive real part but there are eigenvalues on the imaginary axis then the equilibrium can be either

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eigenvalues on the imaginary axis then the equilibrium can be either stable or unstable.

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Stability of equilibria via linearizationStability of equilibria via linearizationTo establish stability properties of equilibria there is no need in studying the solutions of the nonlinear system in question. Stability properties are defined by the stability properties of the linearized system and linear theory is sufficient.y

If an equilibrium is asymptotically stable then it is surrounded by an attraction domain. All solutions that start within the attraction domain converge to the equilibrium. The size of an attraction domain is typically difficult to estimate.

x

Ω

x(0) x0

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SummarySummaryThere many notions for stability.y yStability is a system property in LTI systems and defined by the eigenvalues of the system matrix in state space form or the denominator of the transfer function, i.e. system poles.The Nyquist stability criterion provides useful information about stability of closed-loop LTI systems.Nonlinear systems are much more difficult to analyze than lilinear ones. Linearization of nonlinear systems can be used to investigate stability of equilibria In some cases though it is necessary tostability of equilibria. In some cases though, it is necessary to analyze the nonlinear system, anyway.

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