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ROOT LOCUS The root locus is one of the most powerful techniques used to analyse the stability of a closed- loop system. This technique is also used to design controllers with required time response characteristics. The root locus is a plot of the locus of the roots of the characteristic equation as the gain of […]
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ROUTH–HURWITZ CRITERION The stability of a sampled data system can be analysed by transforming the system characteristic equation into the s-plane and then applying the well-known Routh–Hurwitz criterion. A bilinear transformation is usually used to transform the left-hand s-plane into the interior of the unit circle in the z-plane. For this transformation, z is replaced […]
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PULSE TRANSFER FUNCTION AND MANIPULATION OF BLOCK DIAGRAMS The pulse transfer function is the ratio of the z-transform of the sampled output and the input at the sampling instants. Suppose we wish to sample a system with output response given by Equations (6.34) and (6.35) tell us that if at least one of the continuous […]
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JURY’S STABILITY TEST Jury’s stability test is similar to the Routh–Hurwitz stability criterion used for continuous- time systems. Although Jury’s test can be applied to characteristic equations of any order, its complexity increases for high-order systems. To describe Jury’s test, express the characteristic equation of a discrete-time system of order The necessary and sufficient conditions […]
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FACTORIZING THE CHARACTERISTIC EQUATION The stability of a system can be determined if the characteristic equation can be factorized. This method has the disadvantage that it is not usually easy to factorize the characteristic equation. Also, this type of test can only tell us whether or not a system is stable as it is. It […]
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DAMPING RATIO AND UNDAMPED NATURAL FREQUENCY USING FORMULAE In Section 7.4 above we saw how to find the damping ratio and the undamped natural frequency of a system using a graphical technique. Here, we will derive equations for calculating the damping ratio and the undamped natural frequency. The damping ratio and the natural frequency of […]
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This chapter is concerned with the various techniques available for the analysis of the stability of discrete-time systems. Suppose we have a closed-loop system transfer function where 1 + GH(z) = 0 is also known as the characteristic equation. The stability of the system depends on the location of the poles of the closed-loop transfer […]
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DAMPING RATIO AND UNDAMPED NATURAL FREQUENCY IN THE z-PLANE Damping Ratio As shown in Figure 7.9(a), lines of constant damping ratio in the s-plane are lines where ζ = cos α for a given damping ratio. The locus in the z-plane can then be obtained by the substitution z = esT . Remembering that we […]
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MAPPING THE s-PLANE INTO THE z-PLANE The pole locations of a closed-loop continuous-time system in the s-plane determine the behaviour and stability of the system, and we can shape the response of a system by positioning its poles in the s-plane. It is desirable to do the same for the sampled data systems. This section […]
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TIME DOMAIN SPECIFICATIONS The performance of a control system is usually measured in terms of its response to a step input. The step input is used because it is easy to generate and gives the system a nonzero steady-state condition, which can be measured. Most commonly used time domain performance measures refer to a second-order […]
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