HRRE materiály ke zkoušce

This transformation maps the inside of the unit circle in the z-plane into the left half of the w -plane. Although w -plane seems to be similar to p -plane, quantitatively it is not the same.

STABILITY OF CLOSED LOOP SYSTEMS

The feedback connected to the system can lead to stability but also to the instability of closed loop. The methods which can be used to determine the stability of closed loop systems

• compute transfer function of closed loop system and using known stability criteria

• using Nyquist stability criterion - stability is determined from the frequency response of the open loop transfer function F0(p)

  • Nyquist stability criterion - can be used regardless of the stability of the controlled system, must be solved solely in the complex plane.

  • Simplified Nyquist stability criterion - only for stable controlled systems, can be solved either in complex plane or in magnituds and phase frequency response.

• solving using Matlab

All closed loop transfer functions have got denominator equal

1 + F0(p)

Assume F0(p) = FR(p)FS(p) and FR(p) = q(p)/ p(p) and FS(p) = b(p) /a(p) .

The denominator of closed loop transfer functions is polynomial Δ(p) = a(p)p(p) + b(p)q(p)

which is called characteristic polynomial. By setting this polynomial equal to zero Δ(p) = 0 we obtain characteristic equation. For the stability of closed loop system it is necessary that all roots of characteristic equation lie in open left half plane.

NYQUIST STABILITY CRITERION

Points on imaginary axes p = jω are mapped to frequency response of F0(jω) and the encirclement through infinity is mapped to origin of F. Closed curve ΓF is therefore frequency response F0(jω) for ω ∈ (−∞, ∞).

To determine the stability of a system we:

• Start with a system whose characteristic equation is given by 1 + F0(s) = 0.

• Make a mapping from the p domain to the F domain where the path Γp of p encloses the entire right half plane.

• From the mapping we find the number N, which is the number of encirclements of the −1 + j0 point in F. (Note: This is equivalent to the number of encirclements of the origin in 1 + F0(p))

• We can factor F0(p) to determine the number of poles P that are in the right half plane.

• Since we know N and P, we can determine Z = N + P, the number of zeros of 1 + F0(p) in the right half plane (which is the same as the number of poles of Fw(p)).

• If Z > 0, the system is unstable.

If the open loop is asymptotically stable, then the closed loop is only asymptotically stable, if the frequency response locus of the open loop does neither revolve around or pass through the critical point (−1, j0).

The open loop has only poles in the left-half plane with the exception of a single or double pole at origin. In this case the closed loop is only stable, if the critical point (−1, j0) is on the left hand-side of the locus F0(ω) in the direction of increasing values of ω ∈ (0, ∞).

Example of conditionally stable system

TYPES OF CONTROLLERS

One degree of freedom controller