Stability of linear control systems

Module overview. Stability is the base requirement for the design of a control system. This module gives a first insight into the design of closed-loop systems and the problem of their stability. It is shown by definitions what stability means. Then the most important stability criteria using the characteristic polynomial (Hurwitz, Routh) are introduced. Most emphasis is put on the Nyquist criterion, which can be used with Nyquist and Bode diagrams to design stable closed-loop systems with given stability margins. To simplify the stability test in the frequency domain with Nyquist or Bode diagrams several rules are given. All terms, techniques and rules are illustrated by examples.
Module objectives. When you have completed this module you should be able to:
  1. Understand the stability of linear dynamical systems.
  2. Understand the algebraic stability criteria for linear systems.
  3. Know how to test the stability of linear systems described by transfer functions.
  4. Know how to test the stability of linear systems described by frequency-response characteristics.
  5. Know how to test the stability of a closed loop from open-loop data.

Stable and unstable systems

The stability of control systems is an important property. Considering any bounded input signal

of a system, and if theoutput signal

of the system to such a signal is also bounded, then the system is called bounded-input-bounded-output stable. If the output signal does not show this property, the system is unstable. For illustration see Figure 5.1.


Figure 5.1: (a) Stable and (b) unstable system response

to a bounded input signal

Definition of stability and stability conditions

Because of its feedback structure a control system can become unstable, e.g. oscillations with increasing amplitudes in the signals can occur. In section 5.1 a signal-based definition of stability is established, which relies on the boundedness of the input-output signals. In this section we focus on a definition of stability for linear systems that is independent of the input-output signals. First the following definition is introduced:

A linear time-invariant system according to Eq. (3.3) is called ( asymptotically) stable, if its weighting function decays to zero, i.e. if





is valid. If the modulus of the weighting function increases with increasing to infinity, the system is called unstable.


A special case is a system where the modulus of the weighting function does not exceed a finite value as

or for which it approaches a finite value. Such systems are called critically stable. Examples are undamped

S and I elements, see sections 4.4.2 and 4.4.7.

This definition shows that stability is a system property for linear systems. If Eq. (5.1) is valid, then there exists no initial condition and no bounded input signal which drives the output to infinity. This definition can be directly applied to the stability analysis of linear systems by determining the value of the weighting function for

. If this value exists, and if it is zero, the system is stable. However, in most cases the weighting function is not given in an explicit analytic form and therefore it is costly to determine the final value. The transfer function of a system is often known and as it is theLaplace transform of the weighting function , there is an equivalent stability condition for according to Eq. (5.1). The analysis of this condition - see section A.5 - shows that for the stability analysis it is sufficient to check the poles of the transfer function of the system, that is the roots of its characteristic equation






Now the following necessary and sufficient stability conditions can be formulated:



Asymptotic stability

A linear system is only asymptotically stable, if for the roots of its characteristic equation


    for all    


is valid, or in other words, if all poles of its transfer function lie in the left-half plane.

A linear system is only unstable, if at least one pole of its transfer function lies in the right-half plane, or, if at least one multiple pole (multiplicity ) is on the imaginary axis of the plane.


Critical stability

A linear system is critically stable, if at least one singlepole exists on the imaginary axis, nopole of the transfer function lies in the right-half plane, and in addition no multiple poles lie on the imaginary axis.

It has been shown above that the stability of linear systems can be assessed by the distribution of the roots of the characteristic equation in the plane (Figure 5.2). For control problems there is often no need know these root with high precision. For a stability analysis it is interesting to know whether all roots of the characteristic equation lie in the left-half plane or not. Therefore simple criteria are available for easily checking stability, called stability criteria. These are partly in algebraic, partly in graphical form.


Figure 5.2: Stability of a linear system discussed by the distribution of the roots of the characteristic equation in the plane

Algebraic stability criteria

The algebraic stability criteria are based on the characteristic equation, Eq. (5.2), of the system to be analysed. They contain algebraic conditions as inequalities between coefficients , which are only valid if all roots of the polynomial lie in the left-half plane.

The Hurwitz criterion

A polynomial





with k complex conjugate pairs of roots and real roots can always be represented as





If all roots of the polynomial of are in the left-half plane then for all constants and in Eq. (5.4) are positive. From this follows that all coefficients of the polynomial , which are products and sums of positive numbers, are also positive. This result is formulated in the so-called Stodola criterion:

For the polynomial to have all roots with negative real parts it is necessary that





These conditions are also sufficient for and as can be easily verified by calculating the roots. However, for this is no longer the case.


Example 5.3.1   The polynomial with positive coefficients




fulfills the Stodola criterion, but not all the roots ,

have negative real parts.


A polynomial for which all roots

have negative real parts is called Hurwitzian. Therefore, according to the stability conditions introduced in section 5.2 a linear system is only asymptotically stable, if its characteristic polynomial is Hurwitzian. The Hurwitz criterion for the coefficients of a Hurwitz polynomial is as follows:

A polynomial is Hurwitzian, if and only if for all determinants










are positive.

The following schema of the coefficients can be used to build the Hurwitz determinants:





The Hurwitz determinants are characterised by the diagonal coefficients ,


) and by the increasing indices from left to right. Coefficients with indices larger than are set to zero. For applying this criterion all determinants until have to be calculated. Calculation of the last determinant is trivial.

While for a 2nd-order system the conditions of the determinants are automatically fulfilled as soon as the coefficients

are positive, for a 3rd-order system one obtains the Hurwitz conditions





It goes without saying that the determinant conditions will be only applied if the easily checkable conditions of Eq. (5.5) are fulfilled. The Hurwitz criterion is not only practical for the stability analysis of a system with given coefficients , but also of a system with free parameters. This is the task when the range of parameters must be determined for which the system is asymptotically stable. Therefore the following example is given.



Example 5.3.2   Figure 5.3 shows a control loop, for which the range of must be determined such that the closed loop is asymptotically stable.


Figure 5.3: Stability analysis of a simple control loop

The time constants and of both lag elements are known and positive. With the transfer function of the open loop




one obtains for the closed-loop transfer function 




and by substituting




The characteristic equation of the closed loop is




According to the Stodola and Hurwitz criteria the following conditions must be met for asymptotic stability:


, ,


must be positive. From this the lower limit follows.





must be valid.

With the coefficients given above it follows that




and for the upper limit of




The closed loop is asymptotically stable for the range




Routh criterion

For given coefficients of the characteristic equation the method of Routh, which is an alternative to the method of Hurwitz, can be applied, see section A.6. Here the coefficients

will be arranged in the first two rows of the Routh schema, which contains rows:




The coefficients

in the third row are the results from cross multiplication the first two rows according to




Building the cross products one starts with the elements of the first row. The calculation of these values will be continued until all remaining elements become zero. The calculation of the values are performed accordingly from the two rows above as follows:




From these new rows further rows will be built in the same way, where for the last two rows finally








follows. Now the Routh criterion is:

A polynomial is Hurwitzian, if and only if the following three conditions are valid:

all coefficients

are positive,

all coefficients

in the first column of the Routh schema are positive.



Example 5.3.3   




The Routh schema is:



As in the first row of the Routh schema a coefficient is negative the system is unstable.


For proving instability it is sufficient to build the Routh schema only until negative or zero value occurs in the first column. In the example given above the schema could have been stopped at the 5th row.

Another interesting property of the Routh scheme says, that the number of roots with positive real parts is equal to the number of changes of sign of the values in the first column.

Dear Guest,
Spend a minute to Register in a few simple steps, for complete access to the Social Learning Platform with Community Learning Features and Learning Resources.
If you are part of the Learning Community already, Login now!
Your rating: None Average: 3 (2 votes)

Posted by

Wed, 05/20/2009 - 10:39