Differential and Difference LTI systems

Transcription

1 Signals and Systems Lecture: 6 Differential and Difference LTI systems Differential and difference linear time-invariant (LTI) systems constitute an extremely important class of systems in engineering. They are used in circuit analysis, filter design, controller design, process modeling, and in many other applications. We will review the classical solution approach for such systems. Fig. 1 shows that differential systems form a subset of the set of continuous-time LTI systems. A consequence of this set diagram is that any differential system has an impulse response. The same is true for difference systems. We will show techniques to compute their impulse response. 1. Response of Causal LTI systems described by differential equations Differential systems form the class of systems for which the input and output signals are related implicitly through a linear, constant coefficient ordinary differential equation. Example 1: Consider a first-order differential equation relating the input x(t) to the output y(t): 1000 dy y(t) = x(t) (1) dt This equation could represent the evolution of the velocity y(t) in m/s of a car of mass m = 1000 kg, subjected to an aerodynamic drag force proportional to its speed Dy(t), where D = 300 / and in which x(t) is the tractive force in newton s applied on the car (Fig. 2). According to Newton s law, the sum of forces accelerates the car so that we can write = Dy(t) + x(t), where the derivative of the velocity is the car s acceleration. Rearranging this equation, we obtain (1). Dr. Ayman Elshenawy Elsefy Page 1

2 Signals and Systems Lecture: 6 Given the input signal x(t), that is, the tractive force, we would normally have to solve the differential equation to obtain the output signal (the response) of the system, that is, the speed of the car. In general an N th -order linear constant coefficient differential equation has the form: d y(t) d x(t) a dt = a dt This can be expanded to: (2) a d y(t) dt + + a dy(t) d x(t) + a dt y(t) = b dt + + b dx(t) + b dt x(t) (3) The constant coefficients a and b are assumed to be real, and some of them may be equal to zero, although it is assumed that without loss of generality. The order of the differential equation is defined as the order of the highest derivative of the output present in the equation. To find a solution to a differential equation of this form, we need more information than the equation provides. We need N initial conditions (or auxiliary conditions) on the output variable y(t) and its derivatives to be able to calculate a solution. The complete solution y (t) (General Solution) to the differential equation (2) is given by the sum of two components: The Homogeneous solution y (t) or Natural response or Zero-state response of (a solution with the input signal set to zero and usually) The particular solution y (t) or Forced response (an output signal that satisfies the differential equation and usually has the same form as the input signal), also called the forced response of the system. General Solution y (t) = y (t) Homogenous Solution + y (t) Particular Solution Dr. Ayman Elshenawy Elsefy Page 2

3 Signals and Systems Lecture: 6 Example 2: Consider the LTI system described by the causal linear constant coefficient differential Equation dy y(t) = x(t) dt Calculate the output (car velocity) of this system, if the input tractive force signal x(t) = 5000 e u(t)n. This input signal could correspond to the driver stepping on the gas pedal from a standstill and then rapidly easing the throttle. As stated above, the solution is composed of a homogeneous response and a particular solution of the system: y(t) = y (t) + y (t) y (t) == (satisfies ) ==> y (t) = 0 y (t) == (satisfies ) ==> 1000 dy dt y (t) = x(t) Step 1: for particular solution assume output is the same input signal form y (t) = Ce 2000 Ce Ce = 5000 e then c = ==> y (t) = 2.941e Step 2: the homogenous solution y (t) assume y (t) = Ae in exponential form Then 1000 Ase + 300Ae = Ae (s + 0.3) = 0 s = 0.3 y (t) = Ae. y(t) = y (t) + y (t)= Ae e, A can be chosen as any value How to choose the value of A? Now, because we have not yet specified an initial condition on y(t), this response is not completely determined, as the value of A is still unknown. Strictly speaking, for causal LTI systems defined by linear constant-coefficient differential equations, the initial conditions must be y(0) = ( ) = ( ) = = ( ), what is called initial rest. That is, if at least one initial condition is nonzero, then strictly speaking, the system is nonlinear. In practice, we Dr. Ayman Elshenawy Elsefy Page 3

4 Signals and Systems Lecture: 6 often encounter nonzero initial conditions and still refer to the system as being linear. In, initial rest implies that y(0) = 0, so that y(t) = Ae e y(0) = A = 0 A = Thus if t > 0 the car velocity is given by y(t) = 2.941(e. e ), t > 0 If t < 0 The condition of initial rest and the causality of the system imply that y(t) = 0 since y(t) = 0, t < 0. Therefore, we can write the speed of the car as follows: y(t) = 2.941(e. e )u(t) This speed signal is plotted in Fig. 3, and we can see that the car is moving forward, as expected. The above remark on initial rest is true in general for causal LTI systems, as we now show. A linear system is causal if its output depends only on past and present values of x(t), but for a linear system y = Sx, the output to the zero input is zero,. Since we assumed that and that y(t) only depends on past or current values of the input. The condition of initial rest means that the output of the causal system is zero until the time when the input becomes nonzero. 2. An Impulse response of a differential LTI system The general from of causal LTI system of order N: d y(t) d x(t) a = b dt dt The impulse response of such a system can be calculated as the following: (4) Dr. Ayman Elshenawy Elsefy Page 4

5 Signals and Systems Lecture: 6 We have seen that the impulse response of an LTI system is the derivative of its step response S(t), that is h(t) = ( ). Thus, we can obtain the impulse response of an LTI differential system by first calculating its step response and then differentiating it. This method is useful when the right-hand side of the differential equation does not have derivatives of the input signal. Example 3: Calculate the impulse response for the following system d y(t) d y(t) dt y(t) = x(t) dt Let x(t) = u(t) unit step input, then the characteristic polynomial of the system is p(s) = s + 3 s + 2 = (s + 1)(s + 2), p(s) is zero when s = 1 or s = 2 Then the homogenous solution is y (t) = A e + A e Assume x(t) = 1 and y (t) = K then substitute ( ) + 3 ( ) Then [k] + 3 [k] + 2[k] = 1 Then 2K = 1 and k = + 2y (t) = 1 and y (t) = By adding the y (t) and y (t), we obtain the overall step response for t > 0: s(t) = y(t) = y (t) + y (t) = A e + A e And by finding the initial conditions for s(t), At t = 0 S(0) = 0 A + A + = 0 At t = 0 ( ) = 0 2A A = 0 s(t) at t=0 By solving the above two equations we find that: A = and A = 1 Then s(t) = e e + u(t), By differentiating the two sides, using chain rule then h(t) = d dt s(t) = d dt 1 2 e e + 1 u(t) 2 = (e + e ) u(t) + e e + u(t) = s(t) = (e + e ) u(t) + e e + δ(t) = (e + e ) u(t) Dr. Ayman Elshenawy Elsefy Page 5

6 Signals and Systems Lecture: 6 Because e e + δ(t) = 0 3. Causal LTI systems described by difference equations In a causal LTI difference system, the discrete-time input and output signals are related implicitly through a linear constant-coefficient difference equation. In general, an N -order linear constant coefficient difference equation has the form: a y[n k] = b x[n k] This can be expanded to a y[n N]+.. +a y[n 2] + a y[n 1] + a y[n] = b x[n] + b x[n 1] + + b x[n M] The constant coefficients a and b are assumed to be real, and although some of them may be equal to zero, it is assumed that without loss of generality. The order of the difference equation is defined as the longest time delay of the output present in the equation. To find a solution to the difference equation, we need more information than what the equation provides. We need N initial conditions (or auxiliary conditions) on the output variable (its N past values) to be able to compute a specific solution. a. Finding the response of the Causal LTI systems using general solution method Dr. Ayman Elshenawy Elsefy Page 6

7 Signals and Systems Lecture: 6 Example 4: using the general solution find y[n] that achieves the following difference equation using the general solution method y[n] = 5 3 ( 0.5) ( 0.8) Dr. Ayman Elshenawy Elsefy Page 7

8 Signals and Systems Lecture: 6 b. Finding the response of the Causal LTI systems using Recursive solution method Suppose that the system is initially at rest and that x[n] has nonzero values starting at n = 0. Then, the condition of initial rest means that y[ 1] = y[ 2] =... = y[ N] = 0, and one can start computing y[n] recursively. Example 5: using the recursive solution find y[n] that achieves the following difference equation. Note that because this system is LTI, it is completely determined by its impulse response. Thus, the response to the unit impulse that we obtain here numerically by recursion is actually the impulse response of the system. For a simpler first-order system, it is often easy to find the general term describing the impulse response h[n] for any time step n. For example, you can check that the causal LTI system described by the difference equation, Dr. Ayman Elshenawy Elsefy Page 8

9 Signals and Systems Lecture: 6 4. Characteristic Polynomial and stability for LTI Differential system The BIBO stability of differential and difference systems can be determined by analyzing their characteristic polynomials. A causal LTI differential system is BIBO stable if and only if the real part of all of the zeros of its characteristic polynomial is negative (we say that they lie in the left half of the complex plane). Example 7: Dr. Ayman Elshenawy Elsefy Page 9

10 Signals and Systems Lecture: 6 Example 8: Dr. Ayman Elshenawy Elsefy Page 10

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16 01/03/1439 Block Diagram Representation. Is an interconnection of elementary operations that act on the input signal. A more detailed representation of the system than the impulse response or differential (difference) equation description since it describes how the system s internal computations or operations are ordered. Block diagram representations consists of an interconnection of three elementary operations on signals; (1) Scalar Multiplication. (2) Addition. (3) Integration for continuous-time LTI system. 1 1

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18 01/03/1439 S1 followed by S2 S2 followed by S1 3

19 01/03/1439 S1 followed by S2 S2 followed by S1 S1 followed by S2 S2 followed by S1 4

20 01/03/1439 S1 followed by S2 S2 followed by S1 5

21 01/03/1439 Second Order Difference Equation. 6

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23 01/03/1439 Find Impulse Response: Find Impulse Response: 1

24 01/03/1439 Find Impulse Response: 2