Signals and Systems 1 Lecture 8 Dr. Ali. A. Jalali September 6, 2002

Signals and Systems 1 Lecture # 8 Differential Equation Model EE 327 fall 2002

Differential Equation Model 1. Many physical systems are described by linear differential equations. 2. Reducing differential equations to algebraic equations 3. Homogeneous solution, exponential solution and natural frequencies. 4. Particular solution, system function and poles- zeros 5. Total solution, initial condition and steady- state 6. Conclusion

The Nth-order Differential Equation Model 1. One important way of modeling LTI systems is by means of linear constant-coefficient equations. 2. Example for DE model: Analog Filter 2nd-order ED

The Nth-order Differential Equation Model Instrument servo 1. Example for DE model: 3 rd -order ED

The Nth-order Differential Equation Model 1. Example for DE model: Electric Network 2nd-order ED

The Nth-order Differential Equation Model 1. Example for DE model: Electric Network 2nd-order ED

The Nth-order Differential Equation Model X(t) system input, Y(t) system output and our practical restriction order 1. The general linear constant-coefficient Nth-Order DE for SISO systems are: OR:

Initial Condition Solution of Differential Equation This is Characteristic Equation. 1. The character equation of DE of system is found by substituting a trial solution into the homogeneous DE: The result is: Since the cannot be zero it can be factored out. The remaining term must satisfy the algebraic equation.

Initial Condition Solution of Differential Equation This is Characteristic Equation Can be written as: Or as factored form: Characteristic roots are: Where may be real or complex (conjugate pairs).

Initial Condition Solution of Differential Equation The solution of the homogeneous DE of: For a given set of initial conditions: Is called the initial condition (IC) solution and for simple (non repeating) roots is of the form: or:

Initial Condition Solution of Differential Equation In IC solution: are coefficients that must be determined in order to satisfy the given set of initial conditions and are the characteristic roots. Multiple roots: If the CE contains multiple roots indicated by factor terms of the form will appear in IC solution.

System Stability in term of IC response A casual system is stable if the IC response decays to zero as: This happens if and only if all the ‘s are negative real, or if there are any complex roots, their real parts must be negative. Real Img

Natural Frequency The roots of the characteristic polynomial are called natural frequencies. These are frequencies which there is an output in the absence of an input. Also the roots of the characteristic polynomial are called eigenvalues.

Example: The flexible shaft subjected to an applied torque with the shaft angle described by the DE: 1- Find CE? 2- Catachrestic roots? 3- Is this system stable? 4- Give the algebraic form of the initial condition response? 5- Find the constants in the IC solution if and

Example Solution 1- Find CE? The homogeneous differential equation is: Yielding the CE as: 2- CR? Using the quadratic formula, CR are: 3- Is this system stable? Both roots are negative real so the system is stable. 4- Algebraic form of IC response? 5- Constant IC solution? and Solving gives: and Thus

MATLAB Comment Function roots in MATLAB: If the CE is 5 th order as: >>p=[ ]; >>r=roots(p) The results are: >> p=[ ];roots(p) ans = i i i i >> p=[ ];roots(p) ans = i i i i

The Unit Impulse Response Model 1. The unit impale function is defined implicitly by its sifting property: where f(t) is assumed to be continuous at Approximation to impulse function: EE 327 fall 2002 Signals and Systems 1

The Unit Impulse Response Model 1. Using the sifting property lead the following product function. Approximation of the sifting property The value of integral is: EE 327 fall 2002 Signals and Systems 1

Unit Impulse Response The response of an LTI system to an input of unit impulse function is called the unit impulse response. Important: When determining the unit impulse response h(t) of an LTI system, it is necessary to make all initial conditions zero. ( output due to input not energy stored in system ) EE 327 fall 2002 Signals and Systems 1 LTI x(t)=  (t)y(t)= h(t)

Convolution If the unit impulse response h(t) of a linear continuous system is known, the system output y(t) can be found for any input x(t). EE 327 fall 2002 Signals and Systems 1 Approximation by pulse Sifting property of pulse

Convolution Integral EE 327 fall 2002 Signals and Systems 1 1. The convolution integral is one of the most important results used in the study of the response of linear systems. 2. If we know the unit impulse response h(t) for a linear system, by using the convolution integral we can compute the system output for any known input x(t). 3. In the following integration integral h(t) is the system’s unit impulse response.

Convolution Evaluation EE 327 fall 2002 Signals and Systems 1 1. The convolution integral can be evaluated in three distinct ways. a) Analytical method, b) Graphical method, c) Numerical convolution We will discuss about these and about convolution properties in class. (see class notes)