2_math - Spring 2006 Process Dynamics, Operations, and...

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Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 2: Mathematics Review 2.0 context and direction Imagine a system that varies in time; we might plot its output vs. time. A plot might imply an equation, and the equation is usually an ODE (ordinary differential equation). Therefore, we will review the math of the first-order ODE while emphasizing how it can represent a dynamic system. We examine how the system is affected by its initial condition and by disturbances, where the disturbances may be non-smooth, multiple, or delayed. 2.1 first-order, linear, variable-coefficient ODE The dependent variable y(t) depends on its first derivative and forcing function x(t). When the independent variable t is t 0 , y is y 0 . 0 0 y ) t ( y ) t ( Kx ) t ( y dt dy ) t ( a = = + (2.1-1) In writing (2.1-1) we have arranged a coefficient of +1 for y. Therefore a(t) must have dimensions of independent variable t, and K has dimensions of y/x. We solve (2.1-1) by defining the integrating factor p(t) = ) ( exp ) ( t a dt t p (2.1-2) Notice that p(t) is dimensionless, as is the quotient under the integral. The solution + = t t 0 0 0 dt ) t ( a ) t ( x ) t ( p ) t ( p K ) t ( p ) t ( y ) t ( p ) t ( y (2.1-3) comprises contributions from the initial condition y(t 0 ) and the forcing function Kx(t). These are known as the homogeneous (as if the right-hand side were zero) and particular (depends on the right-hand side) solutions. In the language of dynamic systems, we can think of y(t) as the response of the system to input disturbances Kx(t) and y(t 0 ). 2.2 first-order ODE, special case for process control applications The independent variable t will represent time. For many process control applications, a(t) in (2.1-1) will be a positive constant; we call it the time constant τ . 0 0 y ) t ( y ) t ( Kx ) t ( y dt dy = = + τ (2.2-1) The integrating factor (2.1-2) is revised 2005 Jan 11 1
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Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 2: Mathematics Review τ = τ = t e dt exp ) t ( p (2.2-2) and the solution (2.1-3) becomes () dt ) t ( x e e K e y ) t ( y t t t t t t 0 0 0 τ τ τ τ + = (2.2-3) The initial condition affects the system response from the beginning, but its effect decays to zero according to the magnitude of the time constant - larger time constants represent slower decay. If not further disturbed by some x(t), the first order system reaches equilibrium at zero. However, most practical systems are disturbed. K is a property of the system, called the gain . By its magnitude and sign, the gain influences how strongly y responds to x. The form of the response depends on the nature of the disturbance. Example : suppose x is a unit step function at time t 1 . Before we proceed formally, let us think intuitively. From (2.2-3) we expect the response y to decay toward zero from IC y 0 . At time t 1 , the system will respond to being hit with a step disturbance. After a long time, there will be no memory of the initial condition, and the system will respond only to the disturbance input. Because this is constant after the step, we guess that the response will also become constant.
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2_math - Spring 2006 Process Dynamics, Operations, and...

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