Illustrations In this chapter we describe a general process for designing a control system. A control system consisting of interconnected components is designed to achieve a desired purpose. To understand the purpose of a control system, it is useful to examine examples of control systems through the course of history. These early systems incorporated many of the same ideas of feedback that are in use today. Modern control engineering practice includes the use of control design strategies for improving manufacturing processes, the efficiency of energy use, advanced automobile control, including rapid transit, among others. We also discuss the notion of a design gap. The gap exists between the complex physical system under investigation and the model used in the control system synthesis. The iterative nature of design allows us to handle the design gap effectively while accomplishing necessary tradeoffs in complexity, performance, and cost in order to meet the design specifications. Chapter 1: Introduction to Control Systems

Illustrations Introduction System – An interconnection of elements and devices for a desired purpose. Control System – An interconnection of components forming a system configuration that will provide a desired response. Process – The device, plant, or system under control. The input and output relationship represents the cause-and- effect relationship of the process.

Illustrations Introduction Multivariable Control System Open-Loop Control Systems utilize a controller or control actuator to obtain the desired response. Closed-Loop Control Systems utilizes feedback to compare the actual output to the desired output response.

Illustrations History Watt’s Flyball Governor (18 th century) Greece (BC) – Float regulator mechanism Holland (16 th Century)– Temperature regulator

Illustrations History

Illustrations History 18th Century James Watt’s centrifugal governor for the speed control of a steam engine. 1920s Minorsky worked on automatic controllers for steering ships. 1930s Nyquist developed a method for analyzing the stability of controlled systems 1940s Frequency response methods made it possible to design linear closed-loop control systems 1950s Root-locus method due to Evans was fully developed 1960s State space methods, optimal control, adaptive control and 1980s Learning controls are begun to investigated and developed. Present and on-going research fields. Recent application of modern control theory includes such non-engineering systems such as biological, biomedical, economic and socio-economic systems

Illustrations (a) Automobile steering control system. (b) The driver uses the difference between the actual and the desired direction of travel to generate a controlled adjustment of the steering wheel. (c) Typical direction- of-travel response. Examples of Modern Control Systems

Illustrations Examples of Modern Control Systems

Illustrations Examples of Modern Control Systems

Illustrations Examples of Modern Control Systems

Illustrations Examples of Modern Control Systems

Illustrations Examples of Modern Control Systems

Illustrations The Future of Control Systems

Illustrations The Future of Control Systems

Illustrations Control System Design

Illustrations

Design Example

Illustrations CVN(X) FUTURE AIRCRAFT CARRIER Design Example

Illustrations Design Example

Illustrations Design Example

Illustrations Design Example

Illustrations

Sequential Design Example

 Block diagram is a shorthand, graphical representation of a physical system, illustrating the functional relationships among its components. OR  A Block Diagram is a shorthand pictorial representation of the cause-and-effect relationship of a system.

 The simplest form of the block diagram is the single block, with one input and one output.  The interior of the rectangle representing the block usually contains a description of or the name of the element, or the symbol for the mathematical operation to be performed on the input to yield the output.  The arrows represent the direction of information or signal flow.

 The operations of addition and subtraction have a special representation.  The block becomes a small circle, called a summing point, with the appropriate plus or minus sign associated with the arrows entering the circle.  Any number of inputs may enter a summing point.  The output is the algebraic sum of the inputs.  Some books put a cross in the circle.

 System components are alternatively called elements of the system.  Block diagram has four components: ◦ Signals ◦ System/ block ◦ Summing junction ◦ Pick-off/ Take-off point

 In order to have the same signal or variable be an input to more than one block or summing point, a takeoff point is used.  Distributes the input signal, undiminished, to several output points.  This permits the signal to proceed unaltered along several different paths to several destinations.

 Consider the following equations in which x 1, x 2, x 3, are variables, and a 1, a 2 are general coefficients or mathematical operators.

 Block diagram is a shorthand, graphical representation of a physical system, illustrating the functional relationships among its components. OR  A Block Diagram is a shorthand pictorial representation of the cause-and-effect relationship of a system.

 The simplest form of the block diagram is the single block, with one input and one output.  The interior of the rectangle representing the block usually contains a description of or the name of the element, or the symbol for the mathematical operation to be performed on the input to yield the output.  The arrows represent the direction of information or signal flow.

 The operations of addition and subtraction have a special representation.  The block becomes a small circle, called a summing point, with the appropriate plus or minus sign associated with the arrows entering the circle.  Any number of inputs may enter a summing point.  The output is the algebraic sum of the inputs.  Some books put a cross in the circle.

 System components are alternatively called elements of the system.  Block diagram has four components: ◦ Signals ◦ System/ block ◦ Summing junction ◦ Pick-off/ Take-off point

 In order to have the same signal or variable be an input to more than one block or summing point, a takeoff point is used.  Distributes the input signal, undiminished, to several output points.  This permits the signal to proceed unaltered along several different paths to several destinations.

 Consider the following equations in which x 1, x 2, x 3, are variables, and a 1, a 2 are general coefficients or mathematical operators.

 Consider the following equations in which x 1, x 2,..., x n, are variables, and a 1, a 2,..., a n, are general coefficients or mathematical operators.

 Draw the Block Diagrams of the following equations.

 We will now examine some common topologies for interconnecting subsystems and derive the single transfer function representation for each of them.  These common topologies will form the basis for reducing more complicated systems to a single block.

Any finite number of blocks in series may be algebraically combined by multiplication of transfer functions. That is, n components or blocks with transfer functions G 1, G 2,..., G n, connected in cascade are equivalent to a single element G with a transfer function given by

 Multiplication of transfer functions is commutative; that is, GiGj = GjGi for any i or j.

Figure: a) Cascaded Subsystems. b) Equivalent Transfer Function. The equivalent transfer function is

 Parallel subsystems have a common input and an output formed by the algebraic sum of the outputs from all of the subsystems. Figure: Parallel Subsystems.

Figure: a) Parallel Subsystems. b) Equivalent Transfer Function. The equivalent transfer function is

 The third topology is the feedback form. Let us derive the transfer function that represents the system from its input to its output. The typical feedback system, shown in figure: Figure: Feedback (Closed Loop) Control System. The system is said to have negative feedback if the sign at the summing junction is negative and positive feedback if the sign is positive.

Figure: a)Feedback Control System. b)Simplified Model or Canonical Form. c) Equivalent Transfer Function. The equivalent or closed-loop transfer function is

The control ratio is the closed loop transfer function of the system. The denominator of closed loop transfer function determines the characteristic equation of the system. Which is usually determined as:

The system is said to have negative feedback if the sign at the summing junction is negative and positive feedback if the sign is positive.

1.Open loop transfer function 2.Feed Forward Transfer function 3.control ratio 4.feedback ratio 5.error ratio 6.closed loop transfer function 7.characteristic equation 8.closed loop poles and zeros if K=10.

Reduction techniques 1. Combining blocks in cascade 2. Combining blocks in parallel

Reduction techniques 3. Moving a summing point behind a block

5. Moving a pickoff point ahead of a block 3. Moving a summing point ahead of a block 4. Moving a pickoff point behind a block Reduction techniques

6. Eliminating a feedback loop 7. Swap with two neighboring summing points Reduction techniques

The letter P is used to represent any transfer function, and W, X, Y, Z denote any transformed signals.

However in this example step-4 does not apply. However in this example step-6 does not apply.

Example-8: For the system represented by the following block diagram determine: 1.Open loop transfer function 2.Feed Forward Transfer function 3.control ratio 4.feedback ratio 5.error ratio 6.closed loop transfer function 7.characteristic equation 8.closed loop poles and zeros if K=10.

◦ First we will reduce the given block diagram to canonical form

1.Open loop transfer function 2.Feed Forward Transfer function 3.control ratio 4.feedback ratio 5.error ratio 6.closed loop transfer function 7.characteristic equation 8.closed loop poles and zeros if K=10.

 Example-9: For the system represented by the following block diagram determine: 1.Open loop transfer function 2.Feed Forward Transfer function 3.control ratio 4.feedback ratio 5.error ratio 6.closed loop transfer function 7.characteristic equation 8.closed loop poles and zeros if K=100.

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Example 13: Find the transfer function of the following block diagrams.

Solution: 1. Eliminate loop I 2. Moving pickoff point A behind block Not a feedback loop

3. Eliminate loop II

When R1 = 0, When R2 = 0,

 Introduction to Signal Flow Graphs ◦ Definitions ◦ Terminologies  Mason’s Gain Formula ◦ Examples  Signal Flow Graph from Block Diagrams  Design Examples

 Alternative method to block diagram representation, developed by Samuel Jefferson Mason.  Advantage: the availability of a flow graph gain formula, also called Mason’s gain formula.  A signal-flow graph consists of a network in which nodes are connected by directed branches.  It depicts the flow of signals from one point of a system to another and gives the relationships among the signals.

Consider a simple equation below and draw its signal flow graph: The signal flow graph of the equation is shown below;

 Branches :- line joining two nodes is called branch. Branch Dummy Nodes:- A branch having one can be added at i/p as well as o/p. Dummy Nodes

Out put node Input node b x4x4 x3x3 x2x2 x1x1 x0x0 h f g e d c a  Input node:- It is node that has only outgoing branches.  Output node:- It is a node that has incoming branches.

 Any path from i/p node to o/p node. Forward path

 A closed path from a node to the same node is called loop.

Self loop:- A feedback loop that contains of only one node is called self loop. Self loop

The product of all the gains forming a loop Loop gain = A 32 A 23

Path & path gain Path:- A path is a traversal of connected branches in the direction of branch arrow. Path gain:- The product of all branch gains while going through the forward path it is called as path gain.

 it is a path to o/p node to i/p node.

 when the loops are having the common node that the loops are called touching loops.

 when the loops are not having any common node between them that are called as non- touching loops.

 it is a node that has incoming as well as outgoing branches. Chain node

SFG terms representation input node (source) Chain node forward path path loop branch node transmittance input node (source) Output node

 The block diagram reduction technique requires successive application of fundamental relationships in order to arrive at the system transfer function.  On the other hand, Mason’s rule for reducing a signal-flow graph to a single transfer function requires the application of one formula.  The formula was derived by S. J. Mason when he related the signal-flow graph to the simultaneous equations that can be written from the graph.

 The transfer function, C(s)/R(s), of a system represented by a signal-flow graph is;  Where  n = number of forward paths.  Pi = the i th forward-path gain.  ∆ = Determinant of the system  ∆i = Determinant of the ith forward path

∆ is called the signal flow graph determinant or characteristic function. Since ∆=0 is the system characteristic equation. ∆ = 1- (sum of all individual loop gains) + (sum of the products of the gains of all possible two loops that do not touch each other) – (sum of the products of the gains of all possible three loops that do not touch each other) + … and so forth with sums of higher number of non-touching loop gains ∆i = value of Δ for the part of the block diagram that does not touch the i-th forward path (Δi = 1 if there are no non-touching loops to the i-th path.)

Example1: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph Therefore, There are three feedback loops Continue……

∆ = 1- (sum of all individual loop gains) There are no non-touching loops, therefore Continue……

∆ 2 = 1- (sum of all individual loop gains)+... Eliminate forward path-2 ∆ 2 = 1 ∆ 1 = 1- (sum of all individual loop gains)+... Eliminate forward path-1 ∆ 1 = 1 Continue……

 Example2 － － － C(s) R(s) G1G1 G2G2 H2H2 H1H1 G4G4 G3G3 H3H3 E(s) X1X1 X2X2 X3X3 R(s) －H2－H2 1 G4G4 G3G3 G2G2 G1G1 1 C(s) －H1－H1 －H3－H3 X1X1 X2X2 X3X3 E(s) Continue……

R(s) －H2－H2 1 G4G4 G3G3 G2G2 G1G1 1 C(s) －H1－H1 －H3－H3 X1X1 X2X2 X3X3 E(s)

Continue……