Mathematical Modeling

CONTROL SYSTEMS MATHEMATICAL MODELING

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Introduction • Architects and structural engineers carry out extensive stress analysis on proposed designs and create architectural models. • Automobile body designers work with clay models. • Hydraulic engineers and shipbuilders carry out extensive modeling of a proposed design, followed by physical model testing. NTTF

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Introduction • A control system is conceived to ensure that some dynamic variable maintains a desire state with respect to time. • Before this control system constructed or assembled, it has to be designed and analyzed for operation in the field! • After a control system is in operation, it continues to require tuning and operational analysis from time to time. NTTF

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Introduction • It is thus necessary to develop a technique for the purpose of analysis, design, and tuning of control system & Transfer functions are used for analyzing the operation and performance of closed-loop.-control systems. • The transfer function of a control system depends on the characteristics of its components (or subsystems) and also on the way these components are connected together. NTTF

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Introduction • Behavior of some of these components such as electrical resistors and mechanical springs, can be described by linear algebraic expressions, whereas the majority of the components, such as inductors and capacitors, require integral or derivative terms to properly model their behavior. • A differential equation is required to model a simple series electrical circuit containing a resistor, a capacitor, and an inductor. • From this differential equation, a transfer function can be developed, which completely describes its dynamic behavior NTTF

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The mathematical model • The mathematical model for an element or a system is an equation or set of equations that define the relationship between the input and output (variables). • Operational behavior of an element depends on its characteristics. For a system, its behavior is dependent not only on its components, but also on how these components are linked together. NTTF

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Example • Ohm's law describes the relationship between current and voltage for a resistor. In other words, it can be stated that Ohm's law is a mathematical model for a resistor. – voltage = resistance x current – V = RI

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Nonlinear Behavior • Even though a number of physical components exhibit linear relationships between input and output, when viewed over a wide operating range these components exhibit nonlinear relationships. • Consider the case of a mechanical spring. The relationship between applied spring force and resultant deflection (extension or compression) is expressed as a linear equation. – force = spring stiffness x deflection – F = K x ΔL NTTF

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Nonlinear Behavior • But when the spring is subjected to a large amount of compression force, the spring seems to reach the limit of compression, with coils coming close to each other. • As this happens, the linear relationship between the applied force and resultant deflection no longer exists. • If the force is further increased, the coils are squeezed together with no further increase in deflection. NTTF

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Spring deflection • It can be said that the spring has reached the saturation limit. Graphically, this can be shown as in the Figure.

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Nonlinear Behavior • In our study of (classical) control systems, we assume that all elements, subsystems, and even complete systems can be described by linear algebraic and/or differential equations. • ‘This will be true’ only if every element is operated over a relatively narrow range of its entire span. • Thus, the preceding spring, even though it exhibits nonlinear behavior at the extreme deflection, can be considered to be linear over the middle (smaller) range. NTTF

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Operational amplifiers • Operational amplifiers also exhibit a similar behavior. • Output voltage of an amplifier (say a noninverting amplifier) is directly proportional to the applied input voltage. • This is true for all small-signal applications. • On the other hand, if the input-signal amplitude is sufficiently large or if the amplifier gain is set too high, the output voltage can reach the saturation limit (dictated by power supply), and a linear relationship no longer exists. NTTF

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Nonlinear Behavior • It is assumed throughout the rest of our course that every element can be described by a linear model. • This may be an inherent characteristic of the element, or it may be substantiated by an assumption that operation is over a small and linear range of the element.

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The Transfer Function • The transfer function of an element or a system is an s-domain expression describing the relationship that exists between input and output variables, assuming that all initial conditions are zero.

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Example • A mathematical model of a resistor element describing the relationship between current flow through it and the voltage applied across it is fairly simple and is a numerical constant (R). • Now consider a series RC circuit, where it is desired to express the relation between capacitor voltage Vc (output) and voltage applied to circuit e (input). NTTF

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Example • This relation is expressed through a differential equation,

– where both Vc and e are functions of time.

• It is not possible to isolate the capacitor voltage Vc from the differential equation, and thus it is not feasible to come up with a mathematical model directly linking the output and input variables using time-variable terms vc(t) and e (t). NTTF

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Example • But this relationship can be easily expressed using the s-domain terms Vc(s)and E(s).

• In other words, output (capacitor) voltage in a series RC circuit is related to input voltage through the model 1/ (RCs + 1); also known as the transfer function of the circuit. NTTF

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Example

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Transfer function • In a circuit there are several variables that may be of interest. • Before proceeding further, one and only one output variable has to be decided up on. • A mathematical expression (usually a differential equation) can then be set up. • Laplace transforms are applied to this differential equation, and a transfer function linking the desired output variable and the input variable can then be obtained. NTTF

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Transfer function • The derivation of a transfer function for a system depends on the desired output variable. • It is possible to obtain multiple transfer functions for a system relating different input and output variables. • Fortunately, in many control systems, only one output variable is of concern. • Hence a single transfer function is needed. NTTF

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Electrical Networks • A passive electrical system can contain a number of resistors, capacitors, and inductors. • If possible, all similar components can be lumped together as a single equivalent component for the purpose of analysis. • It is important that circuit operation remain unaffected by the lumped-parameter approach. • Otherwise the simplified model is incorrect and the components cannot be lumped together. NTTF

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Resistor • Current i (amperes) and voltage v (volts) through a resistor R (ohms) are governed by Ohm's law. • v = Ri • The corresponding transfer functions are as shown

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• Resistor TF current versus voltage

• Resistor TF voltage versus current.

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Capacitor • Current i (amperes) and voltage v (volts) through a capacitor C (farads) are governed by the following:

• Because integral and derivative terms are involved, Laplace transformation is applied.

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Capacitor transfer functions

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Inductor • Current i (amperes) and voltage v (volts) through an inductor L (henrys) are governed by the following

• Applying Laplace transformation to eliminate integral and derivative terms gives

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Inductor transfer functions

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Series RC Circuit

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• Series RC Circuit – Figure shows a series RC circuit. – Initially, there is no current flow in the Circuit and the switch is in the open position, as shown. – At time t=0, the switch is placed in the closed position and voltage e is applied. – Current starts to build up in the circuit and starts charging the capacitor. – Eventually, the capacitor is fully charged and full voltage appears across it. – An equation can be developed that fully describes the capacitor voltage (or current in the circuit) from start to final steady-state value.

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• The voltage in the circuit at time t is given by – capacitor voltage + resistor voltage = applied voltage – vc + vr= e

• The transfer function depends upon the desired output variable. • Two separate transfer functions will be developed: one for capacitor voltage and the other for resistor current. • Capacitor Voltage: – Because voltage across resistor is not a desired quantity, it needs to be replaced with a term containing a reference to capacitor voltage. NTTF

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• Capacitor Voltage: – Expanding the resistor voltage in terms of current gives vc+ Ri= e – Capacitor current i is same as the current through the resistor.

– This differential equation fully describes the behavior of capacitor voltage upon application of an external voltage e. NTTF

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• Capacitor Voltage: Transfer Function: – By taking the Laplace transform of both sides of the equation and simplifying, a transfer function of the RC circuit can be obtained.

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• Capacitor Voltage: Transfer Function: – Using the transfer function, a black box approach can be developed, where output (capacitor) voltage can be determined for any applied input voltage

• Resistor Current – The transfer function related to current in the circuit can be developed similarly. – Again starting with an equation for voltage across the circuit, vc +vr= e NTTF

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• Resistor Current – Here the variable of interest is the current through the resistor. – Neither the resistor voltage across R nor the capacitor voltage, vc, is the desired variable; both need to be replaced with terms containing circuit current. – Capacitor voltage is related to capacitor current through

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• Resistor Current – Substituting it in the equation,

– This differential equation fully describes the current flow in the circuit from time t = 0 to steady-state condition upon application of an external voltage e. – Taking the Laplace transform of both sides of the equation and simplifying to develop the transfer function gives NTTF

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• Resistor Current: Transfer Function

– This transfer function allows the output current to be determined at any instant as a result of an applied (input) voltage

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Series RL Circuit

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• Series RL Circuit – Figure shows a series RL circuit. – Initially, there is no current flow in the circuit and the switch is in open position as shown. – At time t = 0, the switch is placed in the closed position and voltage e is applied. – Current starts to increase from an initial condition of 0 A.

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– It eventually reaches a maximum value and remains in this steady state condition. – An equation relating inductor voltage (or current in the circuit), from start to final steady-state value, can be developed – Equating the voltage drops in the circuit gives • inductor voltage + resistor voltage = applied voltage • vL + vR = e

• Inductor Voltage – The term vL will be retained, whereas the vR term will be changed to relate to inductor voltage. NTTF

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• Inductor Voltage cont.. – Expanding the vR term, vL+ Ri = e – Since the resistor current is same as the inductor current and inductor current is related to inductor voltage through – substituting the value of current i gives

– This integral equation describes the behavior of inductor voltage upon application of an external voltage e. NTTF

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• Inductor Voltage: Transfer Function – By taking the Laplace transform of both sides of the equation and simplifying

Ls V L( s ) = X E (s) Ls + R NTTF

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• Inductor Voltage: Transfer Function Ls = TF Ls + R

– This transfer function relates inductor voltage to the applied input voltage

Applied voltage E(s)

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Ls Ls + R

Inductor voltage VL(s)

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