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Control Engineering: Stability Analysis Techniques - Ruth-Hurwitz, Nyquist Criteria, Exams of Control Systems

International Islamic University (IIU)Control Systems

A control system is defined as a system of devices that manages, commands, directs, or regulates the behavior of other devices or systems to achieve a desired result. A control system achieves this through control loops, which are a process designed to maintain a process variable at a desired set point.

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2020/2021

Uploaded on 06/06/2021

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Department of Mechanical Engineering
University of Lahore Islamabad Campus
ME-03309 Control Engineering
Assignment#2. Spring-2021, 6st Semester
MUHAMMAD GHAFFAR KHAN 70066654
(PLO-3/CLO-4/Cog-3)
Q#1 Demonstrate the following with examples.
I. Ruth-Hurwitz stability criteria and its applications
II. Nyquist Stability Criterion
III. Nyquist Criterion for Systems with Minimum-Phase Transfer Functions
SOLUTION (PART 1):
Ruth-Hurwitz stability criteria and its applications
Routh Hurwitz criterion states that any system can be stable if and only if all the roots of
the first column have the same sign and if it does not has the same sign or there is a sign
change then the number of sign changes in the first column is equal to the number of
roots of the characteristic equation in the right half of the s-plane i.e. equals to the
number of roots with positive real parts.
Necessary but not sufficient conditions for Stability
We have to follow some conditions to make any system stable, or we can say that there
are some necessary conditions to make the system stable.
Consider a system with characteristic equation:
1. All the coefficients of the equation should have the same sign.
2. There should be no missing term.
If all the coefficients have the same sign and there are no missing terms, we have no
guarantee that the system will be stable. For this, we use Routh Hurwitz Criterion to
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Department of Mechanical Engineering

University of Lahore Islamabad Campus

ME-03309 Control Engineering

Assignment#2. Spring-2021, 6 st^ Semester

MUHAMMAD GHAFFAR KHAN 70066654

(PLO-3/CLO-4/Cog-3) Q#1 Demonstrate the following with examples. I. Ruth-Hurwitz stability criteria and its applications II. Nyquist Stability Criterion III. Nyquist Criterion for Systems with Minimum-Phase Transfer Functions

SOLUTION (PART 1):

Ruth-Hurwitz stability criteria and its applications

Routh Hurwitz criterion states that any system can be stable if and only if all the roots of the first column have the same sign and if it does not has the same sign or there is a sign change then the number of sign changes in the first column is equal to the number of roots of the characteristic equation in the right half of the s-plane i.e. equals to the number of roots with positive real parts. Necessary but not sufficient conditions for Stability We have to follow some conditions to make any system stable, or we can say that there are some necessary conditions to make the system stable. Consider a system with characteristic equation:

  1. All the coefficients of the equation should have the same sign.
  2. There should be no missing term. If all the coefficients have the same sign and there are no missing terms, we have no guarantee that the system will be stable. For this, we use Routh Hurwitz Criterion to

check the stability of the system. If the above-given conditions are not satisfied, then the system is said to be unstable. This criterion is given by A. Hurwitz and E.J. Routh. Advantages of Routh- Hurwitz Criterion

  1. We can find the stability of the system without solving the equation.
  2. We can easily determine the relative stability of the system.
  3. By this method, we can determine the range of K for stability.
  4. By this method, we can also determine the point of intersection for root locus with an imaginary axis. Limitations of Routh- Hurwitz Criterion
  5. This criterion is applicable only for a linear system.
  6. It does not provide the exact location of poles on the right and left half of the S plane.
  7. In case of the characteristic equation, it is valid only for real coefficients. The Routh- Hurwitz Criterion APPLICATION Consider the following characteristic Polynomial When the coefficients a0, a1, ......................an are all of the same sign, and none is zero. Step 1 : Arrange all the coefficients of the above equation in two rows: Step 2 : From these two rows we will form the third row:

s^4 + 2s^3 +6s^2 +4s+1 = 0 Solution Obtain the arrow of coefficients as follows Since all the coefficients in the first column are of the same sign, i.e., positive, the given equation has no roots with positive real parts; therefore, the system is said to be stable.

SOLUTION (PART 2):

Nyquist Stability Criterion

The Nyquist criterion represents a method of determining the location of the characteristic equation roots with respect to the left half and the right half of the s-plane. Unlike the root-locus method, the Nyquist criterion does not give the exact location of the characteristic equation roots. Let us consider that the closed-loop transfer function of a SISO system is where G(s)H(s) can assume the following form where the T’s are real or complex-conjugate coefficients, and Td is a real time delay Because the characteristic equation is obtained by setting the denominator polynomial of M(s) to zero, the roots of the characteristic equation are also the zeros of 1 + G(s)H(s). Or, the characteristic equation roots must satisfy In general, for a system with multiple number of loops, the denominator of M(s) can be written as Identification of Poles and Zeros

From the Nyquist plots, we can identify whether the control system is stable, marginally stable or unstable based on the values of these parameters. Gain cross over frequency and phase cross over frequency Gain margin and phase margin Phase Cross over Frequency The frequency at which the Nyquist plot intersects the negative real axis (phase angle is

  1. is known as the phase cross over frequency. It is denoted by ωpc Gain Cross over Frequency The frequency at which the Nyquist plot is having the magnitude of one is known as the gain cross over frequency. It is denoted by ωgc The stability of the control system based on the relation between phase cross over frequency and gain cross over frequency is listed below. If the phase cross over frequency ωpc is greater than the gain cross over frequency ωgc , then the control system is stable. If the phase cross over frequency ωpc is equal to the gain cross over frequency ωgc , then the control system is marginally stable. If phase cross over frequency ωpc is less than gain cross over frequency ωgc , then the control system is unstable. Gain Margin The gain margin GM is equal to the reciprocal of the magnitude of the Nyquist plot at the phase cross over frequency. GM=1Mpc Where, Mpc is the magnitude in normal scale at the phase cross over frequency. Phase Margin The phase margin PM is equal to the sum of 1800 and the phase angle at the gain cross over frequency. PM=1800+ϕgc Where, ϕgc is the phase angle at the gain cross over frequency.gc Where, ϕgc Where, ϕgc is the phase angle at the gain cross over frequency.gc is the phase angle at the gain cross over frequency.

The stability of the control system based on the relation between the gain margin and the phase margin is listed below. If the gain margin GM is greater than one and the phase margin PM is positive, then the control system is stable. If the gain margin GM is equal to one and the phase margin PM is zero degrees, then the control system is marginally stable. If the gain margin GM is less than one and / or the phase margin PM is negative, then the control system is unstable.

SOLUTION (PART 3):

Nyquist Criterion for Systems with Minimum-Phase Transfer Functions the Nyquist criterion to systems with L(s) that are minimum-phase transfer functions. The properties of the minimum-phase transfer functions are described in App. G and are summarized as follows:

  1. A minimum-phase transfer function does not have poles or zeros in the right-half s- plane or on the jω-axis, excluding the origin.
  2. For a minimum-phase transfer function L(s) with m zeros and n poles, excluding the poles at s = 0, when s = j ω and as ω varies from ∞ to 0, the total phase variation of L( jω) is (n − m)π/2 rad.
  3. The value of a minimum-phase transfer function cannot become zero or infinity at any finite nonzero frequency.
  4. A nonminimum-phase transfer function will always have a more positive phase shift as ω varies from ∞ to 0. Or, equally true, it will always have a more negative phase shift as ω varies from 0 to ∞. Because a majority of the loop transfer functions encountered in the real world satisfy condition 1 and are of the minimum-phase type, it would be prudent to investigate the application of the Nyquist criterion to this class of systems. As if turns out, this is quite simple. Because a minimum-phase L ( s ) does not have any poles or zeros in the right-half s -plane or on the jωω -axis (except at s = 0) P = 0, and the poles of Δ( s ) = 1+ L ( s ) also have the same properties. Thus, the Nyquist criterion for a system with L(s) being a minimum- phase transfer function is simplified to N= Application of the Nyquist Criterion to Minimum-Phase Transfer Functions