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Overview

Control systems are frequently represented by block diagram circuits, where each block represents a transfer function.  Since transfer functions are something that we have been examining with regards to our frequency domain discussion, it is not surprising to discover that the tools from previous lectures can be used with regards to the interpretation of these block diagrams.

Any system may be analysed for stability, although we normally think of this as something associated with control circuits.  This section will examine how the previously learned tools can be used to perform stability evaluations.

Topic 1 –  Transfer Functions

• A transfer function is another way that we can look at modelling dynamic systems
  • Transfer functions model the input-output relationships of components or systems
  • Differential equations for the system are linear, and time-invariant
  • Output is called the response function
  • Input is called the driving function or forcing function

Let us say that we have a dynamic system described by the differential equation:

$\text{ax” + bx’ + cx = f(t)}$ 

If we Laplace Transform this:

$\text{[a}{\text{s}}^2\text{+ bs + c]X(s) = F(s)}$ 

The transfer function would be

$\text{X(s)/F(s) = X(s)/ [a}{\text{s}}^2\text{+ bs + c]X(s) = 1/ [a}{\text{s}}^2\text{+ bs + c]}$ 

The above can fairly easily be inverse transformed back to the time domain with tools we have already studied.   For more complex relationships, MATLAB can help us evaluate this using the residue command.

Application 1

Topic 2 –  Control System Block Diagram Transfer Functions

Suppose that we have a control system with a feedback loop like the following, with input signal A(s), output signal C(s), feedback signal B(s), transfer function between input and output of G(s) and feedback loop transfer function of H(s):

Where  $\text{C(s) = G(s)D(s)}$

And  $\text{D(s) = A(s) – B(s) = A(s) – H(s)C(s)}$

Which can be combined to  $\text{C(s) = G(s) [A(s) – H(s)C(s)]}$

There are also some standard simplifications (as shown in the following table) that can be utilized to simplify block diagrams and aid us in arriving at a transfer function.

(Ogata)

Application 1

Application 2

Topic 2 – Stability Analysis

Unstable systems are never a positive thing for designers.  There are a number of different types of instabilities that engineering designers must be wary of:

• Mechanical
  • Nonlinear spring stiffness
  • Prop-whirl flutter
• Electrical
  • Control system instability
  • Microphone feedback

Instability occurs when a constant input results in an output that grows in an unbounded fashion.  A stable system will tend toward a bounded response, at the very least, and often a response that evolved into the input signal.  The figure below shows these characteristics.

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We will exam two methods for assessing instability:

• Routh Criteria
• Root Locus

First, let us recall what the zeros and roots of a transfer function would be.  A typical form for a complex transfer function would be something like this:

$\text{G(s)}=\frac{\text{K (s + 3) (s + 20)}}{{\text{s (s - 4)(s + 6)(s + 15)}}^2}$ 

The poles of this function are values of s at which G(s) equals infinity because the denominator of the function goes to zero. (i.e. s = 4, 0, -6, -15 and -15 in this example – NOTE -15 is referred to as a repeated pole).

The zeros of this function are values of s at which G(s) equals 0 because the numerator of the function goes to zero.  (i.e. -3 and -20 in this example)

Technically, the function should have as many zeros as poles (in which case for the example the zeros are -2, -10 and three repeated zeros at infinity).

We will examine stability in the complex x-y plane.  For a linear closed-loop system the stability can be determined from the location of closed-loop poles in the s plane.  If the poles are in the left-half of the s plane, the system will be stable.   If the poles are on the right-half of the s plane the system is unstable and the response either increases monotonically, or oscillates with increasing amplitude with time.  If no saturation occurs, the system response will grow unbounded and fail.  These responses are represented in the following figure.  Poles that lie on the vertical axis yield constant oscillation, and are therefore technically stable, but the presence of noise can increase the amplitude of oscillation determined by the noise power level.

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Topic 3 – Routh Criteria Stability Analysis

The Routh stability criteria provides information on absolute stability.  It determines the number of closed-loop poles in right-half of the s plane.  It involves finding the characteristic equation for a transfer function, but does not require factoring the polynomial. Let us assume that the polynomial characteristic equation has this general form:

${\text{a}}_0{\text{s}}^{\text{n}}+{\text{a}}_1{\text{s}}^{\text{n − 1}}+\text{.......}+{\text{a}}_{\text{n − 1}}\text{s}+{\text{a}}_{\text{n}}=0$ 

For any coefficients that are zero or negative in the presence of at least one positive coefficient, the polynomial has a root, or roots that are imaginary, or have positive real parts, in which case the system is unstable.

When all coefficients are positive we must use another assessment approach.  We write the characteristic polynomial in rows and columns in the following form.

Etc, following the same pattern.

The number of roots with positive real parts is equal to the number of changes in sign of the coefficients of the first column of this array.  In order for the system to be stable, all coefficients of the characteristic polynomial must be positive and all coefficients in the first column of the above array must also be positive.

Application 1

Topic 4 – Root Locus Stability Analysis

Deriving and solving the differential equations for a control system yields an accurate solution of the system's performance in time.  This approach is only feasible for simple systems.  Additionally, it is not easy to determine from this solution alone what system parameters should be changed to improve response.  We can use this approach when we want to predict system performance without actually solving differential equations.

The process is to plot the roots of the characteristic equation of the closed loop system.We recognize that the transient response of a closed-loop system is closely related to location of poles. It helps us to understand how the poles move in the s plane as the system gain is varied.  Roots of the characteristic equation are plotted for all values of a system parameter from zero to infinity.  Note that the parameter is usually the system gain.

Consider a general closed-loop system with gain K

$\text{A(s)/B(s) = KG(s)/[1 + KG(s)H(s)]}$ 

Stability is evaluated by examining the roots of the characteristic equation of the closed loop system

$\text{1 + KG(s)H(s) = 0}$

Rearrange the equation into the following form so the parameter of interest appears as the multiplicative factor (K in this example):

$1+\frac{\text{K(s+}{\text{z}}_1\text{)(s+}{\text{z}}_2\text{)….(s+}{\text{z}}_{\text{m}}\text{)}}{\text{(s+}{\text{p}}_1\text{)(s+}{\text{p}}_2\text{)….(s+}{\text{p}}_{\text{n}}\text{)}}=0$ 

The parameter of interest is the gain K, where k>0. Once the characteristic equation is expressed in the above form, the following general procedure for constructing the root loci.

• Locate the poles and zeros of G(s)H(s) on the s plane
  • Root-locus branches start from open-loop poles
  • Root-locus branches terminate at zeros (finite zeros or zeros at infinity)
  • Find the number of separate root loci
  • Points corresponding to K = 0 are open-loop poles
  • Root-locus plot will have as many branches as there are roots of the characteristic equation
• Determine the root loci on the real axis
  • Determined by open-loop poles and zeros lying on the axis
  • If the total number of real poles and real zeros to the right of a test point is odd, then that point lies on a root locus
  • Root locus and its compliment form alternate segments along the real axis
• Determine the asymptotes of root loci
  • When a test point s is located far from the origin, the angle of each complex quantity may be considered to be the same
  • Thus, one open-loop zero then cancels the effects of one open-loop pole and vice versa
  • Root loci for very large values of s must be asymptotic to straight lines whose angles (slopes) are given by
    • angles of asymptotes = ± 180^o (2k + 1)/(n-m) where k = 0,1,2 and n and m are the numbers of poles and zeros respectively. All asymptotes intersect on the real axis (due to symmetry)
  • Abscissa of the intersection of the asymptotes and the real axis is found from σ_a = -[(p₁+p₂+……+p_n) – (z₁+z₂+…..z_m)]/(n-m)  
• Find the breakaway and break-in points
  • Correspond to points in s plane where multiple roots of the characteristic equation occur
  • Breakaway and break-in points either lie on the real axis or occur in complex conjugate pairs
  • Root locus between two adjacent open-loop poles on the real axis. There exists at least one breakaway point between the two poles
  • Root locus between two adjacent zeros (one zero may be located at negative infinity) on the real axis. There always exists at least one break-in point between the two zeros.
  • Root locus between an open-loop pole and a zero (finite or infinite) on the real axis. There may exist at no breakaway or break-in points or there may exist both breakaway and break-in points
  • Suppose the characteristic equation is given by B(s) + KA(s) = 0
  • Breakaway and break-in points correspond to multiple roots of the characteristic equation
  • They can be determined from the roots of dK/ds = - [B’(s)A(s) – B(s)A’(s)]/A²(s) = 0
  • Breakaway and break-in points must be roots of this equation
  • Not all roots of this equation are breakaway or break-in points
  • Real roots on the root-locus portion of the real axis are actual breakaway/break-in points
  • Real roots not on the root-locus portion of the real axis are neither
  • Two roots that are a complex-conjugate pair must be checked against the corresponding K value, for a value of K which is positive indicates the point is an actual breakaway or break-in point
• Determine the angle of departure of the root locus from a complex pole, or the angle of arrival at a complex zero
  • For a reasonably accurate sketch of the root loci we need the directions of the root loci near the complex poles and zeros
  • To find the angle of departure from a complex pole subtract from the sum of all the angles of vectors from all other poles and zeros to the complex pole in question
  • Angle of departure from a complex pole = 180^o – ∑(angles of vectors to a complex pole in question from other poles) + ∑(angles of vectors to a complex pole in question from zeros)

Click the images above to enlarge

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• Find the points where the root loci may cross the imaginary axis
  • Let s = iω in the characteristic equation
  • Equate both the real and imaginary parts to zero, solving for ω and K
  • Calculated values of ω yield the frequencies at which the root loci cross the imaginary axis
  • K value corresponding to each crossing frequency gives the gain at the crossing point
• Sketch the root loci
  • Take a series of test points in the broad neighbourhood of the iω-axis and origin
  • Most important part of the root loci is not on the real axis nor the asymptotes
  • Most important region is around the iω-axis and origin, and the shape of the root loci in this region of the s plane must be obtained with sufficient accuracy
• Determine the closed-loop poles
  • A particular point on each root-locus branch is a closed-loop pole if the value of K at that point satisfies the magnitude condition
  • Magnitude condition enables one to determine value of the gain K at any specific point on the locus:
  • $\text{K}=\frac{\text{(product of lengths between point s to poles)}}{\text{(product of lengths between point s to poles)}}$
  • Value may be obtained graphically or analytically

Application 1

Application 2

Application 3

Summary

• Control systems are often described/represented by block diagrams using transfer functions.
• These block diagrams can be simplified mathematically, and MATLAB can assist with this.
• System stability can be examined using system transfer function relationships using various analyses.
• The Routh stability criteria and root locus stability analysis are two such.
• MATLAB can also assist with these stability studies.

References and Bibliography

Ogata, K. (2004).  System Dynamics, 4th edition. Upper Saddle River, USA: Pearson Prentice Hall.