Prof. Wahied Gharieb Ali Abdelaal CSE 502: Control Systems (1) Topic# 3 Representation and Sensitivity Analysis Faculty of Engineering Computer and Systems Engineering Department Master and Diploma Students

2 Outline Transfer Functions Pulse Transfer Function Block Diagram Sensitivity Analysis

3 Transfer Functions Definitions G(s) U(s)Y(s) The transfer function G(s) of an LTI system is defined as the Laplace transform of the impulse response with all initial conditions set to zero. U(s)=L[  (t)]=1  G(s)=Y(s). Therefore, the transfer function G(s) can also be considered as the Laplace transform of the output when the input is the unit impulse function  (t). Because this fact, the transfer function is also called impulse response. Consider the transfer function G(s) is related to the Laplace transform of the input and the output through the following relation: If u(t) =  (t) = impulse Function

4 The transfer function G(s) can also be derived from the system differential equation as follows: Assuming zero initial conditions and taking the Laplace transforms of both sides, we get: Remember that : L[f’(t)]=SF(s)-f(0+) L[f n (t)]=s n F(s) – s n-1 f(0) – s n-2 f’(0) - …. – f n-1 (0) Transfer Functions

5 Properties 1. Transfer functions are defined only for a linear Time-Invariant (LTI) Systems. They are not defined for non-linear systems. 2. The transfer function between the input and the output is the Laplace transform of the impulse response. Also, it is the ratio of the Laplace transform of the output to the Laplace transform of the input. 3. All initial conditions of the system are zero. 4. The transfer function is dependent only on the system parameters Transfer Functions

6 Properties 5. The transfer function depends only the complex variable s and no other variables 6. If the order of the transfer function’s numerator is equal to that of the denominator, the transfer function is called proper. 7. If the order of the numerator is less than that of the denominator, the transfer function is called strictly proper transfer function. 8. If the order of the numerator is greater than that of the denominator, the transfer function is called improper. Transfer Functions

7 Example The characteristic equation of a linear system is defined as the equation obtained by setting the denominator polynomial of the transfer function to zero: Consider the following differential equation: By taking the Laplace transform (assuming zero initial conditions) Transfer Functions

8 Proper transfer function: Improper transfer function: Strictly proper transfer function: This transfer function is strictly proper and its characteristic equation is : The zeros are the roots of the numerator The poles are the roots of the denominator Examples

9 Pulse Transfer Function

10 Pulse Transfer Function

11 Block Diagram The block diagram is obtained after obtaining the differential equation & Transfer function of all components of a control system. The arrow head pointing towards the block indicates the i/p & pointing away from the block indicates the o/p. Suppose G(S) is the Transfer function then G(S) = C(S) / R(S) After obtaining the block diagram for each & every component, all blocks are combined to get a complete representation. It is then reduced to a simple form with the help of block diagram algebra.

12 Basic elements of a block diagram Blocks Transfer functions of elements inside the blocks Summing points Take off points Arrow A control system may consist of a number of components. A block diagram of a system is a pictorial representation of the functions performed by each component and of the flow of signals. The elements of a block diagram are block, branch point and summing point. Block Diagram

13 Block: In a block diagram all system variables are linked to each other through functional blocks. The blocks are used to identify many types of mathematical operations like integration, amplification, …etc. Summing point: Addition/subtraction is represented by a circle, called a summing point. As shown below, a summing point may have one or several inputs. Each input has its own appropriate plus or minus sign. A summing point has only one output and is equal to the algebraic sum of the inputs. Block Diagram

14 Advantages of Block Diagram Representation : It is always easy to construct the block diagram even for a complicated system Function of individual element can be visualized Individual & Overall performance can be studied Over all transfer function can be calculated easily Limitations of a Block Diagram Representation No information can be obtained about the physical construction Source of energy is not shown Block Diagram

15 Block diagram reduction technique: Because of the simplicity and versatility, the block diagrams are often used by control engineers to describe all types of systems. A block diagram can be used simply to represent the composition and interconnection of a system. Also, it can be used, together with transfer functions, to represent the cause-and-effect relationships throughout the system. Transfer Function is defined as the relationship between an input signal and an output signal to a device. Procedure to solve Block Diagram Reductions: Step 1: Reduce the blocks connected in series Step 2: Reduce the blocks connected in parallel Step 3: Reduce the minor feedback loops Step 4: Move the take off points and summing point for simplification Step 5: Repeat steps 1 to 4 till simple form is obtained Step 6: Obtain the Transfer Function for the overall system Block Diagram

16 Block diagram rules (1) Blocks in Cascade [Series]: When two blocks are connected in series,their resultant transfer function is the product of two individual transfer functions. (2) Combining blocks in Parallel: When two blocks are connected parallel as shown below,the resultant transfer function is equal to the algebraic sum (or difference) of the two transfer functions. This is shown in the diagram below. Block Diagram

17 (3) Eliminating a feed back loop: The following diagram shows how to eliminate the feed back loop in the resultant control system. (4) Moving a take-off point beyond a block: The effect of moving the takeoff point beyond a block is shown below. Block Diagram

18 (5) Moving a Take-off point ahead of a block: The effect of moving the takeoff point ahead of a block is shown below. Block Diagram

19 Block Diagram

20 Examples B(s)= H(s).Y(s) U(s)=R(s)-B(s) Y(s)=G(s)U(s)=G(s)R(S)-G(s)R(s)  Y(s)[1+G(s)H(s)]=G(s)R(s) + - y(t) Y(s) e(t)=r(t)-y(t) r(t) R(s) + - y(t) Y(s) e(t)=r(t).y(t) r(t) R(s) + - B(s) R(s) G(s) H(s) U(s) Y(s) Block Diagram

21 Block Diagram

22 Block Diagram

23 Block Diagram

24 Block Diagram

25 Block Diagram

26 Block Diagram

27 Block Diagram

28 Sensitivity Analysis

29 A change in the transfer function G(s) therefore causes a proportional change in the transform of the output Yo(s).This requires that the performance specifications of G(s) be such that any variation still results in the degree of accuracy within the prescribed limits. Sensitivity Analysis

30 The effect of changes of G(s) upon the transform of the output of the closed-loop control is reduced by the factor 1/[1+ G(s)] compared to the open-loop control. This is an important reason why feedback systems are used. Sensitivity Analysis

31 The closed-loop variation is reduced by the factor 1/[1+G(s)H(s)], H(s) may be introduced to provide an improvement in system performance to reduce the effect of parameter variations within G(s). Sensitivity Analysis

32 The approximation applies for those cases where [G(s)H(s)]>>1. It is seen that a variation in the feedback function has approximately a direct effect upon the output, the same as for the open-loop case. Thus, the components of H(s) must be selected as precision fixed elements in order to maintain the desired degree of accuracy and stability in the transform Y(s). Sensitivity Analysis

33 Sensitivity Analysis

34 Sensitivity Analysis