Stability
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1 Introduction to Stability

There are a number of different types of stability.

1. BIBO Stable
2. Marginally Stable
3. Conditionally Stable
4. Uniformly Stable
5. Asymptoticly Stable
6. Unstable

2 Poles and Stability

When the poles of the closed-loop transfer function of a given system are located in the right-half of the S-plane (RHP), the system is unstable. When the poles of the system are located in the left-half plane (LHP), the system is stable. A number of tests deal with this particular facet of stability: The Routh-Hurwitz Criteria, the Root Locus, and the Nyquist Stability Criteria all test whether there are poles of the transfer function in the RHP.

If the system is a multivariable, or a MIMO system, then the system is stable if and only if every pole of every transfer function in the transfer function matrix has a negative real part. For these systems, it is possible to use the Routh-Hurwitz, Root Locus, and Nyquist methods described later, but these methods must be performed once for each individual transfer function in the transfer function matrix.

3 Poles and Eigenvalues

Note:
Every pole of G(s) is an eigenvalue of the system matrix A. However, not every eigenvalue of A is a pole of G(s).

The poles of the transfer function, and the eigenvalues of the system matrix A are related. The eigenvalues of the system matrix A are the poles of the transfer function of the system. In this way, if the eigenvalues of a system in the state-space domain, we can use the Routh-Hurwitz, and Root Locus methods as if the system were represented by a transfer function instead.

On a related note, eigenvalues and all methods and mathematical techniques that use eigenvalues to determine system stability only work with time-invariant systems. In systems which are time-variant, the methods using eigenvalues to determine system stability fail.

4 State-Space and Stability

The state-space system is stable if the eigenvalues of the system matrix A have negative real parts.

5 Marginal Stablity

When the poles of the system in the complex s-Domain exist on the complex frequency axis (the vertical axis), or when the eigenvalues of the system matrix are imaginary (no real part), the system exhibits oscillatory characteristics, and is said to be marginally stable. A marginally stable system may become unstable under certain circumstances, and may be perfectly stable under other circumstances.

6 Open Loop Stability Criteria[1]

Another way to determine closed loop stability is to evaluate the frequency response of the open loop transfer function KG(jω)and then perform a test on that response.

Suppose that we have a system where in stability results if K >α where

K is the control gain and
α is the neutral stability point (i.e. the closed loop roots are on the imaginary axis)

At the point of neutral stability the following root locus conditions hold

 $LaTeX: \left | KG\left(j\omega \right) \right |=1$ (6.22a)

 $LaTeX: \angle G\left(j\omega \right)=180^\circ$ (6.22b)

A Bode plot of a system that is neutrally stable will satisfy the conditions of Eqn. (6.22a & b). Does increasing K (from the neutral stability point) increase or decrease the system's stability? At the frequency ω where the phase {{Eqn eqnValue=$LaTeX: \angle G\left(j\omega \right)=-180^\circ$, | number=|}} the magnitude

 $LaTeX: \left | KG\left(j\omega \right) \right | < 1.0$ for stable values of K

and

 $LaTeX: \left | KG\left(j\omega \right) \right | > 1.0$ for unstable values of K

The following stability criteria are only valide when the open loop gain crosses magnitude = 1 only once.

6.1 For systems where an increasing K leads to instability

Stability conditions are

 $LaTeX: \left | KG\left(j\omega \right) \right | < 1$ at $LaTeX: \angle G\left(j\omega \right)=-180^\circ$

6.2 For systems where an increasing K leads from instability to stability

Stability conditions are

 $LaTeX: \left | KG\left(j\omega \right) \right | > 1$ at $LaTeX: \angle G\left(j\omega \right)=-180^\circ$

7 References

• Franklin, G. F., Emami-Naeini, A., and Powell, J. D. 1993 Feedback Control of Dynamic Systems. 3rd. Addison-Wesley Longman Publishing Co., Inc. ISBN 0201527472

7.1 Notes

1. Franklin, pp. 359-361