definition of stability

8/7/2019 Definition of Stability

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Definition of StabilityStabilityis an important concept in linear systems we all want to fly in airplanes withstable control systems! Although many of us have an intuitive feel for the idea ofstability, we need a working definition that will allow us to classify systems as eitherstable or unstable.Working with a partner, draft a definition of stability.The definition should be specific enough that you can test whether a system with impulseresponse g(t) (or equivalently, transfer function G(s)) is stable or not.Note: There are dozens, and maybe hundreds, of definitions of stability. There is nowrong answer!When you are finished, press 1 on your PRS remote.Definition of StabilitySolutionGu(t) y(t)

The LTI system G is Bounded Input / Bounded Output(BIBO) stable if every bounded input u(t) produces a bounded output y(t).Basically, this definition says that every nice input produces a nice output.BIBO Stability IConsider the systems F, G, and H, with impulse responses given byf(t) = _(t)e2t g(t) = _(t) h(t) = _(t)etWhich of the systems are BIBO stable?1. F, G, and H2. F and G3. F only4. G and H

5. H only6. none of the aboveBIBO Stability IConsider the systems F, G, and H, with impulse responses given by f(t) = _(t)e2tg(t) = _(t)h(t) = _(t)etWhich of the systems are BIBO stable?1. F, G, and H2. F and G3. ~ F only4. G and H5. H only

6. none of the aboveThe correct answer is:BIBO Stability IIConsider the system G with impulse response given by g(t) =11 + t_(t)Is the system G BIBO stable?1. Yes2. No3. Dont knowBIBO Stability IIConsider the system G with impulse response given by g(t) =11 + t_(t)


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Is the system G BIBO stable?1. Yes2. ~ No

3. Dont knowThe correct answer is:BIBO Stability IIIConsider the system G with impulse response given by g(t) =1pt_(t)Is the system G BIBO stable?1. Yes2. No3. Dont knowBIBO Stability IIIConsider the system G with impulse response given by g(t) =1pt_(t)Is the system G BIBO stable?

1. Yes2. ~ No3. Dont knowThe correct answer is:

Stability of Linear SystemsWe consider systems described by the state equations for continuous time systems, and

(1)(2)

In the Circuits and Systems course, these equations were solved in the frequency domain by using

Laplace transforms for continuous time systems and z-transforms for discrete time systems. Thisapproach is limited to linear systems. Since we are eventually interested in nonlinear systems, we will

perform the analysis in the time domain.

The free solution (u = 0) can be represented as(3)

where(4)

for continuous time systems and

(5) for discrete time systems. Note that in general, x is a vector and A, B, C and D are matrices.

Definition 1:

a) System is asymptotically stable if for all x0 we have

(6)


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b) System is (simply) stable if for all x0 there exists C such that for all t

(7)

c) System is weakly unstable if it is not stable and if for all x0 there exist C and n such that for all t

(8)d) System is strongly unstable if it is neither stable nor weakly unstable.

Example 1: Discharging a capacitor

Fig.1

State equation:(9)

Solutions:

(10) System is asymptotically stable if R > 0 System is stable if R = (open circuit)

System is strongly unstable if R


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Note that the system is never weakly unstable.

Example 2: Free frictionless motion

Fig.2State equations:

(11)

Solutions:(12)

System is weakly unstable, with n = 0.

Example 3: Population growth model

State equation:

(13)

Solution:

(14) Asymptotically stable if 0 < a < 1 Stable for a = 1

Strongly unstable for a > 1

Note that the system is never weakly unstable.


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Example 4: Counter with adjustable increment

The system uses the first state variable to model the increment. State equations:

(15)Solutions:

(16) System is weakly unstable

Conditions for stability, continuous time systems

Form of the free solutions:

(17)

Try to diagonalize the matrix A by a coordinate change:

(18)This is most of the time possible, but not always. Suppose temporarily that it is possible, i.e. that A is

diagonalizable. Then i are the eigenvalues of A and the i-th column of S the eigenvector for i.We can represent the matrix exponential as

(19)

and thus


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(20)Hence, the solutions become

(21)

and in transformed coordinates

(22)This leads directly to the following proposition.

Proposition 1:

If A is diagonalizable The system is

asymptotically

stable if all

eigenvalues isatisfy Rei < 0

The system is stable if all eigenvalues i satisfy Rei 0.

The system is strongly unstable if Rei > 0 for at least one

eigenvalue i.

Note that if the system has a diagonalizable matrix A it cannot beweakly unstable. If we take example 2, the matrix A is

(23)This matrix cannot be diagonalized. Otherwise, it would have two

linearly independent eigenvectors with eigenvalue 0 (since it is atriangular matrix, all eigenvalues are diagonal elements).But if x is an

eigenvector with eigenvalue 0, we must have(24)


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and thus all eigenvectors lie in the 1-dimensional subspace x2 = 0, i.e. there are no two linearlyindependent eigenvectors.

Jordan normal form:

Any matrix A can be reduced to the Jordan normal form:

(25)

where

(26)and the Jordan blocks Jihave the form

(27)Now

(28)and

(29)

After some calculations, oneobtains


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(30)(31)

(32)

The solution x(t) is a linear combination of the terms in this matrix.From this the following theorem follows:

Theorem 1: The system is asymptotically stable if all eigenvalues i satisfy Rei < 0 The system is stable, if all eigenvalues i satisfy

- either Rei < 0- or Rei = 0 and the corresponding Jordan block is of dimension 1

The system is weakly unstable if all eigenvalues i satisfy Rei 0 and if there is at least oneeigenvalue i with Rei = 0 and Jordan block with dimension higher than 1.

The system is strongly unstable if there is at least one eigenvalue i with Rei > 0.

Conditions for stability, discrete time systems

Everything is similar as in the case of continuous time systems. Form of the free solutions:


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(33)Jordan normal form

(34)

(35)

(36)(37)

The solution x(t) is a linear combination of the terms in this matrix. From this the following theoremfollows:

Theorem 2:

The system is asymptotically stable if all eigenvalues i satisfy |i| < 1 The system is stable, if all eigenvalues i satisfy

- either |i| < 1

- or |i| = 1 and the corresponding Jordan block is of dimension 1 The system is weakly unstable if all eigenvalues i satisfy |i| 0 and if there is at least one

eigenvalue i with |i| = 1 and Jordan block with dimension higher than 1. The system is strongly unstable if there is at least one eigenvalue i with |i| > 1

Remarks:

1. If the initial conditions x0 coincide with an eigenvector of A, the trajectory x(t) lies on the

straight line through x0 and the origin.2. A system that has no eigenvalue with Rei = 0 (continuous time) resp. |i| = 1 (discrete time) is

called hyperbolic. Hyperbolic systems are robust against parameter changes, and non-hyperbolic systems are fragile.


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3. There are two kinds of multiplicities of an eigenvalue i:

- The algebraic multiplicity is the number of times it appears on the diagonal of J. This is

equal to the sum of the dimensions of all Jordan blocks with i.

- The geometric multiplicity is the number of linearly independent eigenvectors witheigenvalue i. This is equal to the number of Jordan blocks with eigenvalue i.

Connection with frequency domain analysis

Continuous time systems

Example of dimension 1:

(38)

Laplace transform

(39)

The constant a is at the same time eigenvalue of the (1x1) state matrix and pole of the Laplacetransformed free solution, i.e. natural frequency. We now show that this is the case in general.Dimension n:

(40)Laplace transform

(41)This is a system of n linear equations for the components ofX(p).

Reminder: Cramers rule for the solution of linear equations


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(42)Applied to our case:

(43)

Both the numerator and the denominator are polynomials in p. The zeros of the denominator are the

natural frequencies of the system. At the same time they are the eigenvalues of the matrix A.

Discrete time systems:

Everything is similar to the case of continuous time systems:

(44)Again, the numerator and the denominator are polynomials in z. The zeros of the denominator are the

natural frequencies and the eigenvalues of A


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Summary:

Poles of the Laplace/z-transform of the free solution = Natural frequencies

= Eigenvalues of the state matrix A

definition of stability

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