MECH431_topic2_response - K G(s Chapter 5 Transient and...

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K G(s) + - MECH 431 System response Slide 1 Chapter 5 Transient and Steady-State response analysis
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K G(s) + - MECH 431 System response Slide 2 Introduction In analyzing & designing control systems, we need a basis of comparison of performance. This basis is set up by specifying particular input signals and comparing the responses of the various systems to these inputs. By studying the response of a system to a known input, we can infer and predict its response to other systems ( e.g., impulse response transfer function for SISO).
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K G(s) + - MECH 431 System response Slide 3 Typical test signals Common test signals include: Step function, Ramp function, Impulse function, Sinusoidal function. Choosing which of these systems to use depends on the form of the actual signal that the system will be subjected to during normal operation.
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K G(s) + - MECH 431 System response Slide 4 Transient & steady state responses The time response of a system consists of: Transient response, Steady state.
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K G(s) + - MECH 431 System response Slide 5 Absolute Stability A control system is in equilibrium if in the absence of any input or disturbance, it stays the same. A linear-time-invariant system is : 1) Stable if the output eventually comes back to its equilibrium state after being subjected to an input. 2) Critically stable if the oscillations of the output continues forever. 3) Unstable if the output diverges without bound when it is subjected to an input. Unstable also means that the original equations do not apply anymore.
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K G(s) + - MECH 431 System response Slide 6 Steady state error The transient response of a control system often involves energy storage , which leads to a delayed transient response before it can reach the final state. The steady state error is the discrepancy between the output and desired input once it reaches steady state. The steady state error indicates the accuracy of the system.
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K G(s) + - MECH 431 System response Slide 7 First order systems A first order system is a system that contains one energy-storage element and one dissipating-energy element. An example would be a thermometer, where the mass of the mercury stores thermal energy, and the body of the thermometer dissipates heat via convection.
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K G(s) + - MECH 431 System response Slide 8 First-Order systems Consider a thermometer having a mass m , specific heat c , heat transmission area A , and convection heat transfer coefficient h . At steady state: Heat in –Heat out = Energy stored hA ( T e T i ) dt 0 = mcdT i External temperature Internal temperature Heat transmission area Convection heat transfer coefficient Specific heat mass
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K G(s) + - MECH 431 System response Slide 9 First-Order systems Taking the Laplace transform of this differential equation: Therefore the resistance to heat transfer, along with the mass and thermal capacity, determine the time constant (or delay) in the sensor’s temperature change.
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K G(s) + - MECH 431 System response Slide 10 Unit step response (1 st order) To find the unit step response, multiply the transfer function of the system by the Laplace transform of the unit step: + - R(s) E(s) C(s) R(s) C(s)
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K G(s) + - MECH 431 System response Slide 11 Unit step response (1 st order)
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