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Lecture One M.S.Salim PhD. Mechatronics Eng. Al-Nahrain University, IRAQ
Control Systems
Introduction
What Control Systems Are?
In modern usage the word system has many meanings. So let us begin by defining what we mean when we use this word in these lectures, first abstractly then slightly more specifically in relation to scientific literature.
Definition 1: A system is an arrangement, set, or collection of things connected or related in such a manner as to form an entirety or whole. Definition 2: A system is an arrangement of physical components connected or related in such a manner as to form and/or act as an entire unit. The word control is usually taken to mean regulate, direct, or command. combining the above definitions, we have:
Definition 3: A control system is an arrangement of physical components connected or related in such a manner as to command, direct, or
regulate itself or another system.
Examples Of Control Systems
Control systems abound in our environment. But before exemplifying this, we define two terms: input and output, which help in identifying, delineating, or defining a control system. The input is the stimulus, excitation or command applied to a control system, typically from an external energy source, usually in order to produce a specified response from the control system. The output is the actual response obtained from a control system. It may or may not be equal to the specified response implied by the input.
Inputs and outputs can have many different forms. Inputs, for example, may be physical variables, or more abstract quantities such as reference, set point, or desired values for the output of the control system. The purpose of the control system usually identifies or defines the output and input. If the output and input are given, it is possible to identify, delineate, or define the nature of the system components.
EX. The control system consisting of a person driving an automobile has
components which are clearly both manufactured and biological. The driver wants to keep the automobile in the appropriate lane of the roadway. He or she accomplishes this by constantly watching the direction of the automobile with respect to the direction of the road. In this case, the direction or heading of the road,
represented by the painted guide line or lines on either side of the lane may be considered as the input. The heading of the automobile is the output of the system. The driver controls this output by constantly measuring it with his or her eyes and brain, and correcting it with his or her hands on the steering wheel. The major components of this control system are the driver’s hands, eyes and brain, and the vehicle.
EX. part of the human temperature control system is the perspiration system
(نظام التعرق). When the temperature of the air exterior to the skin becomes too high the sweat glands secrete heavily, inducing cooling of the skin by evaporation. Secretions are reduced when the desired cooling effect is achieved, or when the air temperature falls sufficiently. The input to this system may be “normal” or comfortable skin temperature, a “setpoint,” or the air temperature, a physical variable. The output is the actual skin temperature.
Open-Loop And Closed-Loop Control Systems
Control systems are classified into two general categories: open-loop and closed-loop systems. The distinction is determined by the control action, that quantity responsible for activating the system to produce the output. The term control action is classical in the control systems literature, but the word action in this expression does not always directly imply change, motion, or activity. For example, the control action in a system designed to have an object hit a target is usually the distance between the object and the target.
Distance, as such, is not an action, but action (motion) is implied here, because the goal of such a control system is to reduce this distance to zero.
Definition
An open-loop control system is one in which the control action is independent of the output. A closed-loop control system is one in which the control action is somehow dependent on the output.
Two outstanding features of open-loop control systems are:
- Their ability to perform accurately is determined by their calibration. To calibrate means to establish or re establish the input-output relation to obtain desired system accuracy.
- They are not usually troubled with problems of instability, a concept to be subsequently discussed in detail. Closed-loop control systems are more commonly called feedback control systems, and are considered in more detail beginning in the next section.
EX. Most automatic toasters are open-loop systems because they are controlled by a timer. The time required to make ‘‘good toast” must be estimated by the user, who is not part of the system. Control over the quality of toast (the output) is removed once the time, which is both the input and the control action, has been set. The time is typically set by means of a calibrated dial or switch.
EX. The continuous, sinusoidally varying voltage o ( t ) or alternating current i(t ) available from an ordinary household electrical receptacle is a continuous-time (analog) signal, because it is defined at each and every instant of time t electrical power is available from that outlet.
EX. If a lamp is connected to the receptacle in above Ex. and it is switched on and then immediately off every minute, the light from the lamp is a discrete-time signal, on only for an instant every minute.
EX. The mean temperature T in a room at precisely 8 A.M. (08 hours) each day is a discrete-time signal. This signal may be denoted in several ways, depending on the application; for example T(8) for the temperature at 8 o’clock-rather than another time; T(l), T(2),... for the temperature at 8 o’clock on day 1, day 2, etc., or, equivalently, using a subscript notation, T,, c, etc. Note that these discrete-time signals are sampled values of a continuous-time signal, the mean temperature of the room at all times, denoted T( t).
The Control Systems Engineering Problem
Control systems engineering consists of analysis and design of control systems configurations. Analysis is the investigation of the properties of an existing system. The design problem is the choice and arrangement of system components to perform a specific task. Two methods exist for design:
- Design by analysis.
- Design by synthesis.
Design by analysis is accomplished by modifying the characteristics of an existing or standard system configuration, and Design by synthesis by defining the form of the system directly from its specifications. Choice and arrangement of system components to perform a specific task.
Control System Models Or Representations
To solve a control systems problem, we must put the specifications or description of the system configuration and its components into a form amenable to analysis or design. Three basic representations (models) of components and systems are used extensively in the study
- Mathematical models, in the form of differential equations, difference equations, and/or other mathematical relations, for example, Laplace- and z-transforms
- Block diagrams
- Signal flow graphs
Examples:
Q.1/ Identify the input and output for an automatic washing machine.
Ans./ Many washing machines operate in the following manner. After the clothes have been put into the machine, the soap or detergent, bleach, and water are entered in the proper amounts. The wash and spin cycle-time is then set on a timer and the washer is energized. When the cycle is completed, the machine shuts itself off. If the proper amounts of detergent, bleach, and water, and the appropriate temperature of the water are predetermined or specified by the machine manufacturer, or automatically entered by the machine itself, then the input is the time (in minutes) for the wash and spin cycle. The timer is usually set by a human operator. The output of a washing machine is more difficult to identify. Let us define clean as the absence of foreign substances from the items to be washed. Then we can identdy the output as the percentage of cleanliness. At the start of a cycle the output is less than 100%, and at the end of a cycle the output is ideally equal to 100% (clean clothes are not always obtained). For most coin-operated machines the cycle-time is preset, and the machine begins operating when the coin is entered. In this case, the percentage of cleanliness can be controlled by adjusting the amounts of detergent, bleach, water, and the temperature of the water. We may consider all of these quantities as inputs.
Other combinations of inputs and outputs are also possible.
Q2./ Identify a possible input and a possible output for a rotational generator of electricity.
Ans./ The input may be the rotational speed of the prime mover (e.g., a steam turbine), in revolutions per minute. Assuming the generator has no load attached to its output terminals, the output may be the induced voltage at the output terminals. Alternatively, the input can be expressed as angular momentum of the prime mover shaft, and the output in units of electrical power (watts) with a load attached to the generator.
Q.3/ how are the following open-loop systems calibrated: (a) automatic washing machine, (b) Automatic toaster, (c) voltmeter?
Ans./ Automatic washing machines are calibrated by estimating any combination of the following input quantities: (1) amount of detergent (المنظفات), (2) amount of bleach (التبيض) or other additives, (3) amount of water, (4) temperature of the water, ( 5 ) cycle-time. On some washing machines one or more of these inputs is (are) predetermined. The remaining quantities must be estimated by the user and depend upon factors such as degree of hardness of the water, type of detergent, and type or strength of the bleach or other additives. Once this calibration has been determined for a specific type of wash (e.g., all white clothes, very dirty clothes), it does not normally have to be re determined during the lifetime of the machine. If the machine breaks down and replacement parts are installed, recalibration may be necessary.
Ans./
A simple system that accomplishes this task is shown in Fig. Below:
At dusk, the photocell, which functions as a light-sensitive switch, closes the lamp circuit, thereby lighting the room. The lamp stays lighted until daylight, at which time the photocell detects the bright outdoor light and opens the lamp circuit.
Lecture Two M.S.Salim PhD. Mechatronics Eng.
Control Systems Terminology
Block Diagram
A block diagram is a short hand, pictorial representation of the cause-and-effect relationship between the input and output of a physical system. It provides a convenient and useful method for characterizing the ‘functional relationships among the various components of a control system. System components are alternatively called elements of the system. The simplest form of the block diagram is the single block, with one input and one output, as shown in Fig.
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. The output is the algebraic sum of the inputs. Any number of inputs may enter a summing point.
Block Diagrams Of Continuous (Analog) Feedback Control Systems
The blocks representing the various components of a control system are connected in a fashion which characterizes their functional relationships within the system. The basic configuration of a simple closed-loop (feedback) control system with a single input and a
The reference input r is an external signal applied to the feedback control system, usually at the first summing point, in order to command a specified action of the plant. It usually represents ideal (or desired) plant output behaviour.
The primary feedback signal b is a function of the controlled output c, algebraically summed with the reference input r to obtain the actuating (error) signal e , that is, r ± b = e. Note: An open-loop system has no primary feedback signal.
The actuating (or error) signal is the reference input signal r plus or minus the primary feedback signal b. The control action is generated by the actuating (error) signal in a feedback control system. Note: In an open-loop system, which has no feedback, the actuating signal is equal to r.
Negative feedback means the summing point is a subtracted, that is, e = r - b.
Positive feedback means the summing point is an adder, that is, e = r + b.
The term Controller in a feedback control system is often associated with the elements of the forward path, between the actuating (error) signal e and the control variable U. But it also sometimes includes the summing point, the feedback elements, or both, and some authors use the term controller and compensator synonymously. The context should eliminate ambiguity. The following five definitions are examples of control laws, or control algorithms.
- An on-off controller (two-position, binary controller) has only two possible values at its output U, depending on the input e to the controller.
EX.: A binary controller may have an output U = + 1 when the error signal is positive, that is, e > 0, and U = -1 when ≤ 0.
- A proportional (P) controller has an output U proportional to its input e , that is, u = Kpe , where Kp , is a proportionality constant.
- A derivative (D) controller has an output proportional to the derivative of its input e , that is, u = KD de/dt , where KD is a proportionality constant.
- An integral (I) controller has an output U proportional to the integral of its input e, that is, u = KI ∫ e ( t ) dt , where KI , is a proportionality constant.
- PD, PI, DI, and PID controllers are combinations of proportional ( P ), derivative ( D ), and integral ( I ) controllers.
EX.: The output U of a PD controller has the form: uPD = Kpe + KD (de/dt)
The output of a PID controller has the form:
uPID = Kpe+ KD(de/dt) + Kt ∫e(t)dt
Servomechanisms The specialized feedback control system called a servomechanism deserves special attention, due to its prevalence in industrial applications and control systems literature. A servomechanism is a power-amplifying feedback control system in which the controlled variable c is mechanical position, or a time derivative of position such as velocity or acceleration.
EX.:
An automobile power-steering apparatus is a servomechanism. The command input is the angular position of the steering wheel. A small rotational torque applied to the steering wheel is amplified hydraulically, resulting in a force adequate to modify the output, the angular position of the front wheels. The block diagram of such a system may be represented by Fig. 2-17. Negative feedback is necessary in order to return the control valve to the neutral position, reducing the torque from the hydraulic amplifier to zero when the desired wheel position has been achieved.
Regulators A regulator or regulating system is a feedback control system in which the reference input or command is constant for long periods of time, often for the entire time interval during which the system is operational. Such an input is often called a set point. A regulator differs from a servomechanism in that the primary function of a regulator is usually to maintain a constant controlled output, while that of a servomechanism is most often to cause the output of the system to follow a varying input.
Solved Questions
- Draw block diagrams for the following equation:
Sol. The + and - operations indicate the need for a summing point. The differentiation operation can be provide by combining two first derivative operations into one second derivative
- Draw a block diagram for the photocell light switch system described in Problem 5 / lecture 1. The light intensity in the room must be maintained at a level greater than or equal to a prespecified level. One way of describing this system is with two inputs, one input chosen as minimum reference room-light intensity r 1 , and the second as room sunlight intensity r 2. The output c is actual room-light intensity. The room is the plant. The manipulated variable (control signal) is the amount of light supplied to the room from both the lamp and the sun. The photocell and the lamp are the control elements because they control room-light intensity. Assume the minimum reference room-light intensity r 1 is equal to the intensity of room-light supplied by the lighted lamp alone. A block diagram for this system is given in Fig. blow:
Lecture Three M.S.Salim PhD. Mechatronics Eng.
Differential Equations, Difference Equations, and Linear Systems
Introduction A property common to all basic laws of physics is that certain fundamental quantities can be defined by numerical values. The physical laws define relationships between these fundamental quantities and are usually represented by equations. For example the scalar version of Newton's second law states that, if a force of magnitude f is applied to a mass of M units, the acceleration a of the mass is related to f by the equation f = Ma. Or Ohm's law states that, if a voltage of magnitude U is applied across a resistor of R units, the current i through the resistor is related to v by the equation v= Ri. Many nonphysical laws can also be represented by equations. Also, the compound interest law states that, if an amount P(o) is deposited for n equal periods of time at an interest rate I for each time period, the amount will grow to a value of P(n) - P(O)(1 + I ) n^.
Differential Equations And Difference Equations Two classes of equations with broad application in the description of systems are differential equations and difference equations. A differential equation is any algebraic or transcendental equality which involves either differentials or derivatives. Differential equations are useful for relating rates of change of variables and other parameters.
EX.: Newton's second law (Example 3.1) can be written alternatively as a relationship between force f, mass M, and the rate of change of the velocity U of the mass with respect to time t , that is, f= M(dv/dt).
EX.: Ohm's law can be written alternatively as a relationship between voltage U, resistance R, and the time rate of passage of charge through the resistor, that is, U = R(dq/dt).
A difference equation is an algebraic or transcendental equality which involves more than one value of the dependent variable(s) corresponding to more than one value of at least one of the independent variable(s). The dependent variables do not involve either differentials or derivatives. Difference equations are useful for relating the evolution of variables (or parameters) from one discrete instant of time (or other independent variable) to another.
A time-variable equation is an equation in which one or more terms depend explicitly on the independent variable time.
A time-invariant equation is an equation in which none of the terms depends explicitly on the independent variable time.
EX.: The difference equation ky( k+2)+ y ( k ) = v(k), where v and y are dependent variables, is time-variable because the term ky(k+2) depends explicitly on the coefficient k, which represents the time tk ,.
EX.: Any differential equation of the form:
where the coefficients a,, ao, a1, ...., an, b1,...bm are constants, is time-invariant. The equation depends implicitly on t, via the dependent variables U and y and their derivatives.
Linear and Nonlinear Differential And Difference Equations
A linear term is one which is first degree in the dependent variables and their derivatives. A linear equation is an equation consisting of a sum of linear terms. All others are nonlinear equations. If any term of a differential equation contains higher powers, products, or transcendental functions of the dependent variables, it is nonlinear. Such terms include (dy/dt)3, u(dy/dt) , and sinu, respectively. For example, (5/cos t)( d2 y / dt 2 ) is a term of first degree in the dependent variable y.
EX. : The ordinary differential equations (dy/dt)2 + y = 0 and d2y/dt2 + cosy= 0 are nonlinear because ( dy/dt)2 is second degree in the first equation, and cos y in the second equation is not first degree, which is true of all transcendental functions.
EX.: The difference equation y ( k + 2) + u(k + l ) y ( k + 1) + y ( k ) = u ( k ) , in which u and y are dependent variables, is a nonlinear difference equation because u(k+1)y(k+1) is second degree in u and y. This type of nonlinear equation is sometimes called bilinear in u and y.
Ex.: Determine a differential equation relating the voltage v(t) and the current i(t ) for 0 ≤ t for the electrical network given in Figure below. Assume the capacitor is un charged at t = 0, the current i is zero at t = 0, and the switch S closes at t = 0.
By Kirchhoffs voltage law, the applied voltage v ( t ) is equal to the sum of the voltage drops vR,vL, and vC across the resistor R , the inductor L, and the capacitor C, respectively. Thus
To eliminate the integral, both sides of the equation are differentiated with respect to time, resulting in the desired differential equation:
where an, an-1 ,... , a 0. are the coefficients of the characteristic equation and
The table is continued horizontally and vertically until only zeros are obtained. Any row can be multiplied by a positive constant before the next row is computed without disturbing the properties of the table. The Routh Criterion : All the roots of the characteristic equation have negative real parts if and only if the elements of the first column of the Routh table have the same sign. Otherwise, the number of roots with positive real parts is equal to the number of changes of sign.
EX.: S^3 + 6s^2 +12s +8 = 0
Since there are no changes of sign in the first column of the table, all the roots of the equation have negative real parts.
Often it is desirable to determine a range of values of a particular system parameter for which the system is stable. This can be accomplished by writing the inequalities that ensure that there is no change of sign in the first column of the Routh table for the system. These inequalities then specify the range of allowable values of the parameter.
sn^ an an- 2 an- 4 ....... sn-^1 an- 1 an- 3 an- 5 ......
. b1 b2 b . c1 c2 c ........ . ..... ..... ...... ....
S^3 1 12
S^2 6 8
S^1 64/6 0
S^0
EX.:
s
**3
**2
For no sign changes in the first column, it is necessary that the conditions 8 - K> 0, 1 + K> 0 be satisfied. Thus the characteristic equation has roots with negative real parts if - 1 < K < 8, the simultaneous solution of these two inequalities.
Hurwitz Stability Criterion
The Hurwitz criterion is another method for determining whether all the roots of the characteristic equation of a continuous system have negative real parts. This criterion is applied using determinants formed from the coefficients of the characteristic equation. It is assumed that the first coefficient, a,, is positive. The determinants Δi , i = 1,2,. ... n -1, are formed as the principal minor determinants of the determinant
The determinants are thus formed as follows:
S^3 1 3
S^2 3 1/k 0 S^1 (8-^ k)/3^0 S^0 1+k
