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UNIT IV: MATHEMATICAL MODELS OF PHYSICAL SYSTEMS. Definition & classification of system - terminology & structure of feedback control theory Differential equation of physical systems - hydraulic and pneumatic systems Steady state errors - error constants
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UNIT IV: MATHEMATICAL MODELS OF PHYSICAL SYSTEMS • Definition & classification of system - terminology & structure of feedback control theory • Differential equation of physical systems - hydraulic and pneumatic systems • Steady state errors - error constants • Block diagram algebra - Signal flow graphs • Time response of first and second order system • Stability by Routh-Harwitz criterion -Simple problems.
OBJECTIVE • To give an introduction to the analysis of linear control systems. This will permit an engineer to exploit time domain and frequency domain tools to design and study linear control systems.
WHAT IS A CONTROL SYSTEM ? A control system is an interconnection of components forming a system configuration that will provide a desired system response. • A control system consists of subsystems and processes (or plants) assembled for the purpose of obtaining a desired output with desired performance, given a specified input. • Any quantity of interest in a machine or mechanism or other equipment is maintained or altered in accordance with a desired manner.
Terminologies • Controlled Variable The controlled variableis the quantity or condition that is measured and controlled. Normally, the controlled variable is the output of the system. • Manipulated Variable The manipulated variable is the quantity or condition that is varied by the controller so as to affect the value of the controlled variable. • Control Controlmeans measuring the value of the controlled variable of the system and applying the control signal to the system to correct or limit deviation of the measured value from a desired value.
Plant/process: A plant may be a piece of equipment, perhaps just a set of machine parts functioning together, the purpose of which is to perform a particular operation. Any physical object to be controlled (such as a mechanical device, a heating furnace, a chemical reactor or a spacecraft) can be called as a plant. Any operation to be controlled is called as a process. Examples are chemical, economic, and biological processes. • System A system is a combination of components that act together and perform a certain objective. A system need not be physical. Examples : physical, biological, economic ETC.,
TYPES OF CONTROL SYSTEMS: • Open loop and closed loop control. • CLASSIFICATION OF SYSTEMS: • Linear and Nonlinear Systems • Time varying and time in-variant systems • Continuous and Discrete systems • SISO and MIMO systems
Open loop control system • Those systems in which the output has no effect on the control action are called open-loop control systems. • In an open loop control system the output is neither measured nor feedback for comparison with the input. • Example : washing machine • Soaking, washing, and rinsing in the washer operate on a time basis. The machine does not measure the output signal, that is, the cleanliness of the clothes.
Open loop control of the speed of a rotating disk and the block diagram model
Closed loop control of the speed of a rotating disk and the block diagram model(dorf and Bishop)
Open loop control and Closed loop control of blood glucose(dorf and Bishop)
Advantages of open-loop control systems: 1. Simple construction and ease of maintenance. 2. Less expensive than a corresponding closed-loop system. 3. There is no stability problem. 4. Convenient when output is hard to measure or measuring the output precisely is economically not feasible. Disadvantages of open-loop control systems: 1. Disturbances and changes in calibration cause errors, and the output may be different from what is desired. 2. To maintain the required quality in the output, recalibration is necessary from time to time.
Advantages of closed-loop control systems: • Accurate even in the presence of non-linearity • Sensitivity to disturbances is made small. • Less affected by noise Disadvantages of closed-loop control systems: 1.More complex and costly. 2.Feedback in the closed loop may lead to oscillatory response 3.Feedback reduces overall gain of the system 4.Stability is major problem and proper care should be taken during design of a closed loop control system.
Linear and Non-linear control Systems • According to the method of analysis and design, classified as linear and non-linear control systems. • Linear System: When the magnitudes of signals in a control system are limited to ranges in which system components exhibit linear characteristics (i.e., the principle of superposition applies), then the system is linear. • The spring–mass system (damped oscillator) ,the operational amplifier in the presence of small (non-saturating) input signals, electrical circuits, models of small deviations from equilibria in solid and fluid mechanics, Signal-processing systems, including digital filters of the sort used in CD and MP3 players. • If the magnitude of signals are extended beyond the range the system becomes non linear.
Time Variant and Time-Invariant systems • Time Invariant System: When the parameters of a control system are stationary with respect to time during the operation of the system, the system is called a time-invariant system. • Examples: • The winding resistance of an electric motor will vary when the motor is first being excited and its temperature is rising. • A guided-missile control system in which the mass of the missile decreases as the fuel on board is being consumed during flight.
Continuous and Discrete control systems • A continuous-data system is one in which the signals at various parts of the system are all functions of the continuous time variable t. • Discrete-data control systems is one in which the signals at one or more points of the system are in the form of either a pulse train or a digital code.
Transfer Function • The transfer function of a linear, time-invariant, differential equation system is defined as the ratio of the Laplace transform of the output(response) to the Laplace transform of the input(driving function) under the assumption that all initial conditions are zero. • The transfer function of this system is the ratio of the Laplace transformed output to the Laplace transformed input when all initial conditions are zero. • If the highest power of s in the denominator of the transfer function is equal to n, the system is called nth order system.
Open Loop Transfer function and Feed-Forward transfer function • The ratio of the feedback signal B(s) to the actuating error signal E(s) is called the open-loop transfer function. • The ratio of the output C(s) to the actuating error signal E(s) is called the feed-forward transfer function, so that • If the feedback transfer function H(s) is unity, then the open-loop transfer function and the feed-forward transfer function are the same.
Close Loop Transfer Function The transfer function relating C(s) to R(s) is called the closed-loop transfer function.
Test inputs for steady-state error analysis anddesign vary with target type
Steady-state errors due to different test waveform and system type combinations
Non-Unity Feedback Systems • Example: Calculate the error constants and determine ess for a unit step, ramp and parabolic functions response of the following system.
Non-Unity Feedback Systems • Example: Calculate the error constants and determine ess for a unit step, ramp and parabolic functions response of the following system. • For step input,
Non-Unity Feedback Systems • Example: Calculate the error constants and determine ess for a unit step, ramp and parabolic functions response of the following system. • For ramp input,
Non-Unity Feedback Systems • Example: Calculate the error constants and determine ess for a unit step, ramp and parabolic functions response of the following system. • For parabolic input,
