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CONCEPT OF STABILITY

Stability is generally used notion which can be understood in many different ways.

Stability in mathematics solves the stability of solutions of differential equations and of trajectories.

Stability of dynamic systems in a sense of usual definition

a) If all the states of the unforced system converge to the equilibrium in origin than the system is assumed to be stable.

b) If the output will be bounded for every bounded input than the system is BIBO stable.

Try to think about strongness of two stability definitions above. BIBO (Bounded Input-Bounded Output) stability definition is often used for the stability analysis of linear time-invariant systems.

The notion of stability of nonlinear systems is much harder to define. We define different types of stability, from weak ones to strong ones.

Steability of linear continuous dynamic systems

A transfer function is a rational function in complex variable p = σ + γω

It is often convenient to factor polynomials in numerator and denominator and to write transfer function in terms of those factors

System zeros are defined as roots of the equation N(p) = 0.

System poles are defined as roots of the equation D(p) = 0.

Using partial fraction expansion of the transfer function above (assuming there are no repeated roots and no complex conjugate roots)

Each element of PFE can be converted to time domain using inverse Laplace transformation (or using dictionary of Laplace transform). The resulting impulse response is

It is apparent that the impulse response converges to zero for negative poles of the system.

• Continuous LTI system is said to be stable if and only if all poles are located in the open left half plane of the complex plane.

• The necessary and sufficient condition for the stability of the second-order polynomial is that all its coefficients are having an equal sign.

• For higher order polynomials it is only necessary condition but not sufficient.

Other methods for determining stability of continuous LTI systems

• For determining the stability, we can use existing algebraic criteria

• Routh-Schur stability criterion

• Hurwitz stability criterion

• or graphical criteria like Michailov-Leonhard criteria

Hurwitz stability criterion

Given a real characteristic polynomial

The n × n square matrix

Is called Hurwitz matrix of polynomial A(p) . The leading principal minors of H(p) can be computed.

LTI dynamical system of n-th order is stable if and only if subdeterminants (leading principal minors) Di where i = 1, 2, . . . n – 1 are nonzero and are having the same sign. Note that it is necessary to check whether all coefficients of characteristic polynomial are having the same sign.

Stability of linear discrete time systems

• the poles must be located inside of the unit circle

• there exist algebraic criteria which help to determine stability of discrete-time systems but it is easier to map content of unit circle to the left half plane and to use the same criteria as they were defined for continuous time systems - bilinear transformation