Lecture notes in Control system Engineering pdf

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ADVANCE CONTROL SYSTEM ENGINEERINGADVANCED CONTOL SYSTEMS      ADVANCED CONTROL SYSTEMS (PEEC5414)  Module‐I : (15 Hours) Discrete ‐ Time Control Systems :  Introduction:  Discrete  Time  Control  Systems  and  Continuous  Time  Control  Systems,  Sampling  Process.  Digital Control Systems: Sample and Hold, Analog to digital conversion, Digital to analog conversion.  The  Z‐transform:  Discrete‐Time  Signals,  The  Z‐transform,  Z‐transform  of  Elementary  functions,  Important properties and Theorms of the Z‐transform. The inverse Ztransform, Z‐Transform method  for solving Difference Equations.   Z‐Plane Analysis of Discrete Time Control Systems: Impulse sampling & Data Hold, Reconstruction  of Original signals from sampled signals: Sampling theorm, folding, aliasing. Pulse Transfer function:  Starred Laplace Transform of the signal involving Both ordinary and starred Laplace Transforms;  General procedures for obtaining pulse Transfer functions, Pulse Transfer function of open loop and  closed loop systems.  Mapping between the s‐plane and the z‐plane, Stability analysis of closed loop systems in the z‐ plane: Stability analysis by use of the Bilinear Transformation and Routh stability critgion, Jury  stability.                  Test. Book No. 1: 1.1; 1.2; 1.4; 2.1; 2.2; 2.3; 2.4; 2.5; 2.6; 3.2; 3.4; 3.5; 4.2; 4.3.  Module ‐II : (15 Hours) State Variable Analysis & Design:  Introduction: Concepts of State, State Variables and State Model (of continuous time systems):  State  Model  of  Linear  Systems,  State  Model  for  Single‐Input‐Single‐Output  Linear  Systems,  Linearization of the State Equation. State Models for Linear Continuous – Time Systems: State‐ Space Representation Using Physical Variables, State – space Representation Using Phase Variables,  Phase variable formulations for transfer function with poles and zeros, State – space Representation  using  Canonical  Variables,  Derivation  of  Transfer  Function  for  State  Model.  Diagonalization:  Eigenvalues and Eigenvectors, Generalized Eigenvectors.  Solution  of  State  Equations:  Properties  of  the  State  Transition  Matrix,  Computation  of  State  Transition Matrix, Computation by Techniques Based on the Cayley‐Hamilton Theorem, Sylvester’s  Expansion  theorm.  Concepts  of  Controllability  and  Observability:  Controllability,  Observability,  Effect of Pole‐zero Cancellation in Transfer Function. Pole Placement by State Feedback, Observer  Systems. State Variables and Linear Discrete – Time Systems: State Models from Linear Difference  Equations/z‐transfer Functions, Solution of State Equations (Discrete Case), An Efficient Method of  Discretization and Solution, Linear Transformation of State Vector (Discrete‐Time Case), Derivation  of z‐Transfer Function from Discrete‐Time State Model.       Book No. 2: 12.1 to 12.9.  Module ‐III : (12 Hours) Nonlinear Systems :  Introduction : Behaviour of Non linear Systems, Investigation of nonlinear systems.   Common Physical Non Linearities: Saturation, Friction, Backlash, Relay, Multivariable Nonlinearity.  The Phase Plane Method: Basic Concepts, Singular Points: Nodal Point, Saddle Point, Focus Point,  Centre  or  Vortex  Point,  Stability  of  Non  Linear  Systems:  Limit  Cycles,  Construction  of  Phase  Page 2   ADVANCED CONTOL SYSTEMS    Trajectories:  Construction  by  Analytical  Method,  Construction  by  Graphical  Methods.  The  Describing Function Method: Basic Concepts: Derivation of Describing Functions: Dead‐zone and  Saturation, Relay with Dead‐zone and Hysteresis, Backlash. Stability Analysis by Describing Function  Method: Relay with Dead Zone, Relay with Hysteresis, Stability Analysis by Gain‐phase Plots. Jump  Resonance. Liapunov’s Stability Analysis: Introduction, Liapunov’s Stability Critrion: Basic Stability  Theorems,  Liapunov  Functions,  Instability.  Direct  Method  of  Liapunov  &  the  Linear  System:  Methods of constructing Liapunov functions for Non linear Systems.  Book No. 2: 13.1 to 13.4; 15.1 to 15.10.    Text :   1. Discrete‐Time Control System, by K.Ogata, 2nd edition (2009), PHI.   2. Control Systems Engineering, by I.J. Nagrath and M.Gopal., 5th Edition (2007 / 2009), New Age  International (P) Ltd. Publishers.  Reference :  1. Design of Feedback Control Systems by Stefani, Shahian, Savant, Hostetter, Fourth Edition (2009),  Oxford University Press.  2. Modern Control Systems by K.Ogata, 5th Edition (2010), PHI.  3. Modern Control Systems by Richard C. Dorf. And Robert, H.Bishop, 11th Edition (2008), Pearson  Education Inc. Publication.  4.  Control  Systems  (Principles  &  Design)  by  M.Gopal,  3rd  Edition  (2008),  Tata  Mc.Graw  Hill  Publishing Company Ltd.  5. Control Systems Engineering by Norman S.Nise, 4th Edition (2008), Wiley India (P) Ltd.                                  Page 3   ADVANCED CONTOL SYSTEMS    Module 1: Discrete-Time Control systems Lecture Note 1 (Introduction) Continuous time Control System: In continuous time control systems, all the system variables are continuous signals. Whether the system is linear or nonlinear, all variables are continuously present and therefore known (available) at all times. A typical continuous time control system is shown in Figure below. (Closed loop continuous-time control system)  Discrete time Control System:  Discrete time control systems are control systems in which one or more variables can change only at discrete instants of time. These instants, which may be denoted by kT(k=0,1,2,…) specify the times at which some physical measurement is performed or the times at which the memory of a digital computer is read out.  Page 4   ADVANCED CONTOL SYSTEMS    (Block diagram of a discrete-time control system)  Continuous time control systems whose signals are continuous in time are described by differential equation, whereas discrete control systems that involve sampled data signals or digital signals and possibly continuous time signals as well are described by difference equation. Sampling Process:  Sampling is a process by which a continuous time signal is converted into a sequence of numbers at discrete time intervals. A sampling process is used whenever a control system involves a digital computer. Also a sampling process occurs whenever measurements necessary for control are obtained in an intermittent fashion. The sampling process is usually followed by a quantization process in which the sampled analog amplitude is replaced by a digital amplitude (represented by a binary number).Then the digital signal is processed by the computer and the output is sampled and fed to a hold circuit. The output of the hold circuit is a continuous time signal and is fed to the actuator for the control of the plant.      (Continuous-time analog signal)        (Continuous-time quantized signal)        Page 5   ADVANCED CONTOL SYSTEMS    (Sampled-data signal)    Lecture Note 2 (Digital Control Systems) Sample and Hold Circuits: A sampler in a digital system converts an analog signal into a train of amplitude-modulated pulses. The hold circuit holds the value of the sampled pulse signal over a specified period of time. The sample and hold is necessary in the A/D converter to produce a number that accurately represents the input signal at the sampling instant. The sample and hold circuit is an analog circuit in which an input voltage is acquired and then stored on a high quality capacitor as shown in figure below. Op-Amp 1 is an input buffer amplifier with a high input impedance and Op-Amp 2 is the output amplifier that buffers the voltage on the hold capacitor. The two modes of operation for a sample and hold circuit are tracking mode & hold mode. When the switch is closed the operating mode is the tracking mode in which the charge on the capacitor in the circuit tracks the input voltage. When the switch is open the operating Page 6   ADVANCED CONTOL SYSTEMS    mode is the hold mode in which the capacitor voltage holds constant for a specified time period.  (sample and hold circuit)  Digital to Analog Conversion: Digital-to-analog conversion is simple and effectively instantaneous. Properly weighted voltages are summed together to yield the analog output. For example, in Figure below, three weighted voltages are summed. The three-bit binary code is represented by the switches. Thus, if the binary number is 110 , the center and bottom switches are on, and the analog 2 output is 6 volts. In actual use, the switches are electronic and are set by the input binary code.   (Digital to Analog converter) Page 7   ADVANCED CONTOL SYSTEMS    Analog to Digital Conversion: The process by which a sampled analog signal is quantized and converted to a binary number is called analog to digital conversion. The A/D converter performs the operation of sample- and –hold, quantizing and encoding. The simplest type of A/D converter is the counter type. The basic principle on which it works is the clock pulses are applied to the digital counter in such a way that the output voltage of the D/A converter (that is part of the feedback loop in the A/D converter) is stepped up one least significant bit at a time, and the output voltage is compared with the analog input voltage once for each pulse. When the output voltage has reached the magnitude of the input voltage, the clock pulses are stopped. The counter output voltage is then the digital output. (Counter type analog to digital converter)     Page 8   ADVANCED CONTOL SYSTEMS    Lecture Note 3(The Z-Transform) Z-transform: Z-transform is a mathematical tool commonly used for the analysis and synthesis of discrete time control systems. The role of Z-transform in discrete time systems is similar to that of the Laplace transform in continuous systems. In considering Z-transform of a time function x(t), we consider only the sampled values of x(t), i.e.,x(0), x(T), x(2T)....... where T is the sampling period. For a sequence of numbers x(k) The above transforms are referred to as one sided z-transform. In one sided z-ransform, we assume that x(t) = 0 for t 0 or x(k) = 0 for k 0. In two sided z-transform, we assume that −1 t 1 or k =,±1,±2,±3, ........ The one sided z-transform has a convenient closed form solution in its region of convergence -1 for most engineering applications. Whenever X(z), an infinite series in z , converges outside the circle z = R, where R is the radius of absolute convergence it is not needed each time to specify the values of z over which X(z) is convergent. I.e. for z R convergent and for z R divergent. Page 9   ADVANCED CONTOL SYSTEMS    Z-Transforms of some elementary functions: (i) Unit step function is defined as: Unit step sequence is defined as   Assuming that the function is continuous from right, the Z-transform is: The above series converges if z 1. (ii) Unit ramp function is defined as: Again The Z-transform is:   The above series converges if z 1. (iii) Exponential function is defined as:   Again The Z-transform is:       Page 10   ADVANCED CONTOL SYSTEMS    Lecture Note 4(Theorems & properties of Z- Transform) Table for z- and s-transform: Important properties & theorems of z-transform: 1. Multiplication by a constant: , where 2. Linearity: If , then k 3. Multiplication by a : 4. Real shifting: and Page 11   ADVANCED CONTOL SYSTEMS    5. Complex shifting: 6. Initial value theorem:   7. Final value theorem:                         Page 12   ADVANCED CONTOL SYSTEMS    Lecture Note 5(Inverse Z-Transform) Inverse Z-transforms: When F(z), the Z-transform of f(kT) or f(k), is given, the operation that determines the corresponding x(kT) or x(k) is called inverse Z-transform. It should be noted that only the time sequence at the sampling instants is obtained from the inverse Z-transform. So the inverse Z-transform of F(z) yield a unique f(k) but does not yield a unique f(t).   The inverse Z- transform can be obtained by using (i) Power series method: Since F(z) is mostly expressed in the ratio of polynomial form We can immediately recognize the sequence of f(k) if F(z) can be written in form of series -1 with increasing powers of z .This is easily done by dividing the numerator by the denominator. I,e Here the values of f(k) for k=0,1,2,… can be determined by inspection. (ii) Partial fraction expansion method: To expand F(z) into partial fractions, we first factor the denominator polynomial of F(z) and find the poles F(z): We then expand F(z)/z into partial fractions so that each term is easily recognizable in a table of Z-transforms. I.e.   Page 13   ADVANCED CONTOL SYSTEMS    Where (iii) Inversion integral method: The formula for inversion integral method is given by Where C is a circle with its center at the origin of the z plane such that all poles of are inside it. Z-transform method for solving Difference Equation:  One of the most important applications of Z-transform is in the solution of linear difference equations. Let us consider that a discrete time system is described by the following difference equation.    Taking the Z-transform of both sides of the given difference equation, we obtain   Substituting the initial data and after simplifying we get Page 14   ADVANCED CONTOL SYSTEMS      Taking the inverse Z-transform we get       Lecture Note 6(Z-plane Analysis, sampling & hold)  Impulse Sampling: In case of an impulse sampling, the output of the sampler is considered to be a train of impulses that begin with t=0, with the sampling period equal to T and the strength of each impulse equal to sampled value of the continuous-time signal at the corresponding sampling instant as shown in figure below. Page 15   ADVANCED CONTOL SYSTEMS    (Impulse sampler) The sampler output is equal to the product of the continuous-time input x(t) and the train of unit impulses δ (t). T The train of unit impulses δ (t) can be defined as T Now the sampler output can be expressed as     In the impulse sampler the switch may be thought of closing instantaneously every sampling period T and generating impulses x(kT)δ(t-kT).The impulse sampler is a fictitious sampler and it does not exist in the real world. Data Hold: Data-hold is a process of generating a continuous-time signal h(t) from a discrete-time sequence x(kT).A hold circuit converts the sampled signal into a continuous-time signal, which approximately reproduces the signal applied to the sampler. The signal h(t) during the interval kT≤ t ≤ (k+1)T can be expressed as Since, So, Page 16   ADVANCED CONTOL SYSTEMS    If the data-hold circuit is an nth-order polynomial extrapolator, it is called an nth order hold. Zero-Order Hold: It is the simplest data-hold obtained by putting n=0 in eq(1), gives .It indicates that the circuit holds the value of h(kT) for kT≤ t (k + 1)T until the next sample h((k + 1)T) arrives. (sampler and zero order hold) The accuracy of zero order hold (ZOH) depends on the sampling frequency. When T→0, the output of ZOH approaches the continuous time signal. Zero order hold is again a linear device which satisfies the principle of superposition. (Impulse response of ZOH) Page 17   ADVANCED CONTOL SYSTEMS    The impulse response of a ZOH, as shown in the above figure can be written as               Lecture Note 7(Reconstruction of signal) Data Reconstruction:  To reconstruct the original signal from a sampled signal, there is a certain minimum frequency that the sampling operation must satisfy. Such a minimum frequency is specified by the sampling theorem.  Sampling Theorem: Page 18   ADVANCED CONTOL SYSTEMS    If the sampling frequency ω = 2 /T ( where T is the sampling period) and ω is the highest s 1 frequency component present in the continuous- time signal x(t),then it is theoretically possible that the signal x(t) can be constructed completely from the sampled signal x (t) if ω 2 ω . s 1 Considering the frequency spectrum of the signal x(t) as shown in the figure below   Then the frequency spectra of the sampled signal can be expressed as Page 19   ADVANCED CONTOL SYSTEMS      The above equation gives the frequency spectrum of the sampled signal x (t) as shown in figure below in which we can see that frequency spectrum of the impulse sampled signal is reproduced an infinite number of times and is attenuated by the factor 1/T.   Page 20   

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