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ENIN 811System Analysis and SynthesisSystem Analysis and Synthesis
Dr. Wei Peng
Lecture 6:
Stability
ENIN 811 2014 Winter 1 Dr. Wei Peng
1 Introduction1 Introduction
System are designed to perform some System are designed to perform some task or to process signals. If a system is not stable, the system may burn out, disintegrate or saturate when a signal disintegrate, or saturate when a signal , no matter how small, is applied. Therefore an unstable system is useless in practice and stability is a basic in practice and stability is a basic requirement for all system.
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1 Introduction1 Introduction
The response of linear systems can The response of linear systems can always be decomposed as the zero-state response and the zero-input response. We will introduce the BIBO (boundedWe will introduce the BIBO (bounded-input bounded-output) stability for the zero-state response, and marginal and asymptotic stability for the zero input asymptotic stability for the zero input response.
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1 Introduction1 Introduction
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2 Input-Output Stability of LTI Systems
Consider a SISO linear time-invariant (LTI) system described byy y
( ) ( ) ( ) ( ) ( ) == tt dgtudtguty (causal)( ) ( ) ( ) ( ) ( ) == dgtudtguty 00 (causal) where g(t) is the impulse response and the
t i li ti i i t l d system is linear time-invariant, causal, and relaxed at t=0.
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2 Input-Output Stability of LTI Systems
Definition: A system is said to be BIBO stable (bounded-input bounded-output stable) if every ( p p ) ybounded input excites a bounded output.
Bounded input ( )
2 Input-Output Stability of LTI Systems
Theorem 5.1 A SISO system described by (5.1) is BIBO stable if and only if g(t) is absolutely y g( ) yintegrable in [0,) , i.e. there exists a constant M such thatM such that
( )
2 Input-Output Stability of LTI Systems
Proof: (i) let u(t) be a input with utu )( Proof: (i) let u(t) be a input with for all
tt
mutu )(0t
( ) ( ) ( ) ( ) ( )= tt dgtudgtuty 00 ( ) ( )
2 Input-Output Stability of LTI Systems
(ii) BIBO stable ?(ii) BIBO stable ?We know Suppose( )
2 Input-Output Stability of LTI Systems
Theorem If a input-output system with impulse response, has:p p ,
(1) Step input (or a constant input), the output converges to a constant functionconverges to a constant function.
(2) Sinusoidal input, the output converges to a sinusoidal function.
(3) Any input not converging to 0 the output (3) Any input not converging to 0, the output will converge to the unstable input mode.
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2 Input-Output Stability of LTI Systems
Th 5 2 If t ith i l Theorem 5.2 If a system with impulse response g(t) is BIBO stable, then, as t :1. The output excited by u(t)=a, for t 0,
approaches g(0)*aapproaches g(0) a.2. The output excited by u(t) = sin0t, for t 0,
approaches
with is the Laplace transform of g(t), i.e.
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2 Input-Output Stability of LTI Systems
Proof: If u(t)=a for all t0 then (5 1) becomesProof: If u(t)=a for all t0, then (5.1) becomes
Which implies:
If u(t) = sin0t, then (5.1) becomes:( ) 0 , ( )
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2 Input-Output Stability of LTI Systems
Thus we have as t Thus we have, as t ,
Replacing s by jw in (5.2) yields
So we have
d and
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2 Input-Output Stability of LTI Systems
Substituting these into (5 3) givesSubstituting these into (5.3) gives
Theorem 5.3A SISO t ith ti l t f A SISO system with proper rational transfer function is BIBO stable if and only if every
l f h i l pole of has a negative real part or, equivalently, lies inside the left-half s-plane
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Pole and Reduce FormPole and Reduce Form A rational function is a ratio of polynomials A rational function is a ratio of polynomials
N(s)/D(s) If the numerator N(s) and the denominator D(s)
have no roots in common, then the rational function N(s)/D(s) is in Minimal FormExample (s 2)/(s2 4) is not in reduced form Example. (s-2)/(s2 - 4) is not in reduced form, because s = 2 is a root of both numerator and denominator. We can rewrite this in reduced form as
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Pole and Reduce FormPole and Reduce Form Poles: For a rational function in Minimal Form Poles: For a rational function in Minimal Form
the poles are the values of s where the d i t i l t denominator is equal to zero
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ExampleExample
Consider the positive feedback system shown Consider the positive feedback system shown in above figure, Its impulse response was computed as
where the gain a can be positive or negative. W hWe have
and
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SolutionSolutionThus we conclude that the positive feedback Thus we conclude that the positive feedback
system is BIBO stable if and only if the gain a has a magnitude less than 1 has a magnitude less than 1.
The transfer function of the system was d computed as
It is an irrational function of s and Theorem It is an irrational function of s and Theorem 5.3 is not applicable. In this case, it is simpler to use Theorem 5 1 to check its simpler to use Theorem 5.1 to check its stabilityENIN 811 2014 Winter Dr. Wei Peng18
BIBO Stability of MIMO SystemsBIBO Stability of MIMO Systems
Theorem 5 M1Theorem 5.M1 A multivariable system with impulse
response matrix is BIBO stable if and only if every is absolutely integrable in [0, ).
Theorem 5.M3Theorem 5.M3 A multivariable system with proper rational
transfer matrix is BIBO stable if and transfer matrix is BIBO stable if and only if has a negative real part.
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BIBO Stability of a State Space EquationBIBO Stability of a State Space Equation
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Poles of G(s) and Eigenvalues of APoles of G(s) and Eigenvalues of A
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Poles of G(s) and Eigenvalues of APoles of G(s) and Eigenvalues of A
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3 Internal Stability 3 Internal Stability
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3 Internal Stability 3 Internal Stability
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3 Internal Stability 3 Internal Stability
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Marginal stability(stability in the sense of Lyapunov)
(1) Definition: Every finite excites a bounded state response
0x( )txp(2) A d ffi i bl di i
( ) 0,00
Asymptotic stabilityAsymptotic stability
(1) Definition: marginal stable + ( )( ) 0lim =txmarginal stable + ( )
(2) A necessary stable condition: ( ) 0lim txt
( ) .0Re
Minimal Polynomial Minimal Polynomial
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Example Example
Consider
BecauseBecause
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Solution Solution
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4 Lyapunov Theory4 Lyapunov TheoryMotivations:Motivations: In the previous section, we determine stability of
LTI systems by analytically solving an ODE LTI systems by analytically solving an ODE (ordinary differential equation). Then we determine stability according to the obtained solution For stability according to the obtained solution. For general nonlinear systems, it is difficult, or impossible to get the desired analytic solution impossible, to get the desired analytic solution. Then how to determine stability?Lyapunov (direct) method direct means that we Lyapunov (direct) method --direct means that we doesnt need to solve the ode first. We just directly determine stability from the ODEdirectly determine stability from the ODE.
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Homogeneous systemHomogeneous system
Homogeneous system and equilibrium stateThe system with no control input is called The system with no control input is called
homogeneous system. Usually, the time varying homogeneous system can be varying homogeneous system can be described by
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4 Lyapunov Theory4 Lyapunov Theory
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Equilibrium StateEquilibrium State
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Stability in the Sense of LyapunovStability in the Sense of Lyapunov
Definition 5.3 (Stability in the sense of Lyapunov) Suppose that xe is a isolated equilibrium state. pp e qWe say that xe is stable in the sense of Lyapunovif for any given there exists a real number if, for any given , there exists a real number r such that the motion excited by initial state x which satisfiesby initial state x0 which satisfies
(5.15)meets with ( ); tttt
( )
(5 16)ENIN 811 2014 Winter Dr. Wei Peng36
( ) 000 ,,; ttxtxt e (5.16)
Stability in the Sense of Lyapunovy y p
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Asymptotical StabilityAsymptotical Stability Definition 5 4 (Asymptotical stability) A isolated Definition 5.4 (Asymptotical stability) A isolated
equilibrium state xe of dynamic system (5.13) is ll d t ti ll t bl if called asymptotically stable if
(i) xe is stable in the sense of Lyapunov;( ) e y p ;(ii) for given in definition 5.3 and any , there exists a real number such that there exists a real number , such that the motion excited by initial state x0which satisfies (5.15) also meets with
( ) )(; tTttxtxt + (5 17)ENIN 811 2014 Winter Dr. Wei Peng38
( ) ),,(,,; 0000 tTttxtxt e + (5.17)
Asymptotical StabilityAsymptotical Stability(5 17) is equivalent to(5.17) is equivalent to
The concept of Definition 5.4 can be explained by the Fig.5.2
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Global Asymptotical StabilityGlobal Asymptotical Stability
Definition 5.5 (Global asymptotical stability) if a motion excited by stability) if a motion excited by any un-zero state vector x0 in state
fspace is bounded and satisfies
then we say that original equilibrium state xe =0 is whole asymptotically stablestable.
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Global Asymptotical StabilityGlobal Asymptotical Stability
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Example -- Cannot Use Eigenvalues in LTV System
Eigenvalues can not be used to determine the Eigenvalues can not be used to determine the stability of LTV systems
Consider: xe
xtAxt
== 1)(2
&Consider: xxtAx ==
10)(
Then:
( ) ( ) ( ) ( ) ttt eeettt 5.001 ( ) ( ) ( ) ( )=== tettt 00,,121 ENIN 811 2014 Winter Dr. Wei Peng42
Example p-- Cannot Use Eigenvalues in LTV System
( )( ) ( ) ( ) ( )
===
t
ttt
eeee
ttt0
5.00,,121
B l t f ( t t) g ith t b d
e0Because element of (et e-t) grows without bound,
the equation is neither asymptotically stable nor marginally stable. This example shows that even though the eigenvalues can be defined for A(t) at g g ( )every t , the concept of eigenvalues is not useful in the time-varying casein the time varying case.
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Global Asymptotical StabilityFor Dynamic System
Theorem 5 5 (global asymptotical stability) For Theorem 5.5 (global asymptotical stability) For dynamic system (5.13), if there is a scalar f ti V( t) ith fi t d ti function V(x, t) with first-order continuous partial derivative and V(0,t)=0 satisfy:
(i) V(x,t) is positive definite and bounded, that is there exit two continuous non-decrease is, there exit two continuous non decrease scalar functions and with and such that for any t t and any and such that for any t t0 and any x0, following inequalities holding
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Global Asymptotical StabilityFor Dynamic System
(ii) the derivative of V(x,t) with respect to tis negative definite and bounded that is negative definite and bounded, that is, there is a continuous non-decrease
fscalar function with such that for any t t0 and x0, we havet at o a y t t0 a d 0, e a e
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Global Asymptotical StabilityFor Dynamic System
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Global Asymptotical Stability forTime-Invariant System
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Global Asymptotical Stability forTime-Invariant System
Theorem 5.6 For time-invariant system (5.18), if there exists a continuous scalar function V(x) with first-order derivative and V(0)=0,satisfies(i) V(x) is positive definite;(ii) is negative definite;(ii) is negative definite;(iii) as , we have then the original equilibrium state 0 is globally asymptotically stable.
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ExampleExample Consider time-invariant system Consider time-invariant system
Obviously the originalObviously the original
is its solo equilibrium state Consider a scalar is its solo equilibrium state. Consider a scalar function of
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Solution Solution we know that (i) V(x) is positive definite;we know that (i) V(x) is positive definite;(ii) By the computation of the derivative, we have
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Solution Solution
(iii) as ,
we have
by Theorem 5 6 we know that the original by Theorem 5.6, we know that the original equilibrium state is globally asymptotically t blstable.
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Global Asymptotical Stability forTime-Invariant System
Theorem 5.7 For time-invariant system (5.18), if there exists a continuous scalar function V(x) with first-order derivative and V(0)=0, satisfiessatisfies(i) V(x) is positive definite;(ii) is negative semi-definite;(iii) the set contains no
trajectories other than trivial trajectory(iv) as( )
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Global Asymptotical Stability forTime-Invariant System
then the original equilibrium state 0 is globallyasymptotically stableasymptotically stable. An equivalent expression of (iii)
The solution of with respective to x(t) is not any trajectory (solution) of (5.18) (t) s ot a y t ajecto y (so ut o ) o (5 8)excited by any initial state except to 0.
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Semi DefinitionSemi-Definition
A continuously differentiable function f(x) is A continuously differentiable function f(x) is negative semidefinite ( or positive semidefinite ) on a neighborhood of the origin, if f(0) = 0, and f(x) 0 ( 0) for every non-zero x.( ) ( ) y
if f(0) = 0, and f(x) < 0 ( > 0) for every non-zero x. f(x) is negative definite (positive definite).zero x. f(x) is negative definite (positive definite).
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Example Example
Consider time-invariant nonlinear system
Obviously, 0 is the solo equilibrium state.Select we haveSelect , we have(i) V(x) is positive definite;(ii)
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Solution Solution
(ii)(ii)
is negative semi-definite (why?). So, at t l l d th t 0 ipresent, we can only conclude that 0 is
stable. Is it still asymptotically stable? We need to verify condition (iii) in Theorem 5.7.
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Solution Solution
(iii) Suppose that we have(iii) Suppose that , we have(a) x1 is arbitrary but x2 = 0(b) x1 is arbitrary but x2 = -1
For situation (a), suppose that isa solution of original system excited by someinitial state this impliesinitial state, this implies
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SolutionSolutionFor situation (b) we suppose that is a For situation (b), we suppose that is a
solution of original system excited by some initial state and we have state and we have
which is a contradictive result.Now we can conclude that 0 is asymptotically stable
since condition (iii) of Theorem 5.7 is satisfied.since condition (iii) of Theorem 5.7 is satisfied.(iv) Obviously, condition (iv) is also met with, so we say
that 0 is global asymptotically stablethat 0 is global asymptotically stable.ENIN 811 2014 Winter Dr. Wei Peng58
Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
This part we apply the Lyapunov Theorem to a This part we apply the Lyapunov Theorem to a special case: linear system of
Definition 5 6 Definition 5.6 We call square matrix A stable if all the eigenvalues of A have negative real parts
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
Theorem 5 8Theorem 5.8All eigenvalues of A have negative real parts if and only if for any given positive definite symmetric matrix N, the Lyapunov equationsymmetric matrix N, the Lyapunov equationATM+MA= -N (5.19)
has a unique symmetric solution M and M is positive definite.positive definite.
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
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Lyapunov Theorem for Linear System
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
For 3) Suppose there are two solutions MFor 3), Suppose there are two solutions M1and M2. Then we haveAT(M1-M2)+(M1-M2)A=0Multiplying from left side and from Multiplying from left side and from right side of above equation leads to
Integrating above equation from 0 to yields
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
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Complex Conjugate TransposeComplex Conjugate Transpose
where A =
E l Example:
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
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Lyapunov Theorem for Linear SystemLyapunov Theorem for Linear System
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