Introduction to Control Systems

For a simple introduction to Control Systems refer to the page below

https://www.facstaff.bucknell.edu/mastascu/eControlHTML/Intro/Intro1.html

Evaluation of Control Systems

Analysis of control system provides crucial insights to control practitioners on why and how feedback control works. Although the use of PID precedes the birth of classical control theory of the 1950s by at least two decades, it is the latter that established the control engineering discipline. The core of classical control theory are the frequency-response-based analysis techniques, namely, Bode and Nyquist plots, stability margins, and so forth.
In particular, by examining the loop gain frequency response of the system in Fig. 19.1.9, that is, L( jw) = Gc( jw)Gp( jw), and the sensitivity function 1/[1 + L(jw)], one can determine the following:

1. How fast the control system responds to the command or disturbance input (i.e., the bandwidth).
   
2. Whether the closed-loop system is stable (Nyquist Stability Theorem); If it is stable, how much dynamic variation it takes to make the system unstable (in terms of the gain and phase change in the plant). It leads to the definition of gain and phase margins. More broadly, it defines how robust the control system is.
   
3. How sensitive the performance (or closed-loop transfer function) is to the changes in the parameters of the plant transfer function (described by the sensitivity function).
   
4. ThefrequencyrangeandtheamountofattenuationfortheinputandoutputdisturbancesshowninFig.19.1.10 (again described by the sensitivity function).
   
   
   
   
   Digital Implementation
   Once the controller is designed and simulated successfully, the next step is to digitize it so that it can be pro- grammed into the processor in the digital control hardware. To do this:
   

   1. Determine the sampling period Ts and the number of bits used in analog-to-digital converter (ADC) and digital-to-analog converter (DAC).
      
   2. Convert the continuous time transfer function Gc(s) to its corresponding form in discrete time transfer func- tion Gcd(z) using, for example, the Tustin’s method, s = (1/T)(z − 1)/(z + 1).
      
   3. From Gcd(z), derive the difference equation, u(k) = g(u(k − 1), u(k − 2), . . . y(k), y(k – 1), . . .), where g is a linear algebraic function.
      After the conversion, the sampled data system, with the plant running in continuous time and the controller
      

   in discrete time, should be verified in simulation first before the actual implementation. The quantization error and sensor noise should also be included to make it realistic.
   The minimum sampling frequency required for a given control system design has not been established ana- lytically. The rule of thumb given in control textbooks is that fs = 1/Ts should be chosen approximately 30 to 60 times the bandwidth of the closed-loop system. Lower-sampling frequency is possible after careful tuning but the aliasing, or signal distortion, will occur when the data to be sampled have significant energy above theNyquist frequency. For this reason, an antialiasing filter is often placed in front of the ADC to filter out the high-frequency contents in the signal.
   

   Typical ADC and DAC chips have 8, 12, and 16 bits of resolution. It is the length of the binary number used to approximate an analog one. The selection of the resolution depends on the noise level in the sensor signal and the accuracy specification. For example, the sensor noise level, say 0.1 percent, must be below the accuracy spec- ification, say 0.5 percent. Allowing one bit for the sign, an 8-bit ADC with a resolution of 1/27, or 0.8 percent, is not good enough; similarly, a 16-bit ADC with a resolution. 0.003 percent is unnecessary because several bits are “lost” in the sensor noise. Therefore, a 12-bit ADC, which has a resolution of 0.04 percent, is appropriate for this case. This is an example of “error budget,” as it is known among designers, where components are selected economically so that the sources of inaccuracies are distributed evenly.
   Converting Gc(s) to Gcd(z) is a matter of numerical integration. There have been many methods suggested, some are too simple and inaccurate (such as the Euler’s forward and backward methods), others are too com- plex. The Tustin’s method suggested above, also known as trapezoidal method or bilinear transformation, is a good compromise. Once the discrete transfer function Gcd(z) is obtained, finding the corresponding difference equation that can be easily programmed in C is straightforward.
   
   Once the controller is designed and simulated successfully, the next step is to digitize it so that it can be pro- grammed into the processor in the digital control hardware. To do this:
   

   1. Determine the sampling period Ts and the number of bits used in analog-to-digital converter (ADC) and digital-to-analog converter (DAC).
   2. Convert the continuous time transfer function Gc(s) to its corresponding form in discrete time transfer func- tion Gcd(z) using, for example, the Tustin’s method, s = (1/T)(z − 1)/(z + 1).
   3. From Gcd(z), derive the difference equation, u(k) = g(u(k − 1), u(k − 2), . . . y(k), y(k – 1), . . .), where g is a linear algebraic function.
      
      After the conversion, the sampled data system, with the plant running in continuous time and the controller

   in discrete time, should be verified in simulation first before the actual implementation. The quantization error and sensor noise should also be included to make it realistic.
   The minimum sampling frequency required for a given control system design has not been established ana- lytically. The rule of thumb given in control textbooks is that fs = 1/Ts should be chosen approximately 30 to 60 times the bandwidth of the closed-loop system. Lower-sampling frequency is possible after careful tuning but the aliasing, or signal distortion, will occur when the data to be sampled have significant energy above theNyquist frequency. For this reason, an antialiasing filter is often placed in front of the ADC to filter out the high-frequency contents in the signal.
   

   Typical ADC and DAC chips have 8, 12, and 16 bits of resolution. It is the length of the binary number used to approximate an analog one. The selection of the resolution depends on the noise level in the sensor signal and the accuracy specification. For example, the sensor noise level, say 0.1 percent, must be below the accuracy spec- ification, say 0.5 percent. Allowing one bit for the sign, an 8-bit ADC with a resolution of 1/27, or 0.8 percent, is not good enough; similarly, a 16-bit ADC with a resolution. 0.003 percent is unnecessary because several bits are “lost” in the sensor noise. Therefore, a 12-bit ADC, which has a resolution of 0.04 percent, is appropriate for this case. This is an example of “error budget,” as it is known among designers, where components are selected economically so that the sources of inaccuracies are distributed evenly.
   Converting Gc(s) to Gcd(z) is a matter of numerical integration. There have been many methods suggested, some are too simple and inaccurate (such as the Euler’s forward and backward methods), others are too com- plex. The Tustin’s method suggested above, also known as trapezoidal method or bilinear transformation, is a good compromise. Once the discrete transfer function Gcd(z) is obtained, finding the corresponding difference equation that can be easily programmed in C is straightforward.
   
   Finally, the presence of the sensor noise usually requires that an antialiasing filter be used in front of the ADC to avoid distortion of the signal in ADC. The phase lag from such a filter must not occur at the crossover frequency (bandwidth) or it will reduce the stability margin or even destabilize the system. This puts yet another
   constraint on the controller design.
   
   
   

   ALTERNATIVE DESIGN METHODS 
   
   
   

   Nonlinear PID
   Using nonlinear PID (NPID) is an alternative to PID for better performance. It maintains the simplicity and intu- ition of PID, but empowers it with nonlinear gains. The need for the integral control is reduced, by making the proportional gain larger, when the error is small.
   
   
   

   Controllability and Observability. Controllability and observability are useful system properties and are defined as follows. Consider an nth order system described by
   x = Ax + Bu, z = Mx
   where A is an n × n matrix. The system is controllable if it is possible to transfer the state to any other state in finite time. This property is important as it measures, for example, the ability of a satellite system to reorient itself to face another part of the earth’s surface using the available thrusters; or to shift the temperature in an industrial oven to a specified temperature. Two equivalent tests for controllability are:
   The system (or the pair (A, B)) is controllable if and only if the controllability matrix C = [B, AB,..., An−1B] has full (row) rank n. Equivalently if and only if [siI − A, B] has full (row) rank n for all eigenvalues si of A.
   The system is observable if by observing the output and the input over a finite period of time it is possible to deduce the value of the state vector of the system. If, for example, a circuit is observable it may be pos- sible to determine all the voltages across the capacitors and all currents through the inductances by observ- ing the input and output voltages.
   
   
   

   Eigenvalue Assignment Design. Consider the equations: x ̇ = Ax + Bu, y = Cx + Du, and u = p + kx. When the system is controllable, K can be selected to assign the closed-loop eigenvalues to any desired locations (real or complex conjugate) and thus significantly modify the behavior of the open-loop system. Many algo- rithms exist to determine such K. In the case of a single input, there is a convenient formula called Ackermann’s formula
   K = −[0,..., 0, 1] C−1 ad(A)
   where C = [B, . . . , An−1B] is the n × n controllability matrix and the roots of ad(s) are the desired closed-loop  eigenvalues.
   
   Refer link below
   https://www3.nd.edu/~pantsakl/Publications/348A-EEHandbook05.pdf