Continuous And Discrete Time PID Control Of DC Motor Angular Velocity

PI and PID controllers

                                                         

Figure 1. 1: The electric circuit of the armature and the free-body diagram of the rotor

In this assignment, it deals with Continuous and discrete time PID control of DC motor angular velocity. The major control system considered in this assignment is the PID controller, which is simulated using MATLAB software. The system is then discretized and compared with the continuous time control.

Section 1: PI and PID controllers.

1) The strengths and limitations of PID control.

PID (Proportional+ Integral+ Derivative) controller provide a range of amendments because it contains three (3) key controls which includes P-control, I-control and D-control which may be altered. PID Controller control and handles system characteristics like settling time, percentage overshoot, stability, steady-state error, rise time, etc. Even if there are three control elements in the controller, it still has some disadvantage, because the implementation complexity increases in the system (Abu-Khalaf, et al., 2009). Though, each control element has different functions, the elements are exclusively dependent to each other; since single element can be varied by changing another element. Consequently, PID design is complex as compared to the designing P- controller, PD- controller or PI- controller (Anon., 2016). In this part, the strength and disadvantages of PID controller in terms of implementation of the controller, stabilization requirements, performances, robustness, energy consumptions and steady state error

a).Implementation of the controller.

During implementation of PID controller, one strength on implementing the PID controller is that it is easier to construct and design. The PID controllers can be an analogue circuit or a logic gate circuit or MCU or inductors and resistor circuit. Conversely, PID controller needs acceptable and a better sampling time for implementing which requires to be very accurate

b).Stabilization requirements.

The PID controllers to be stable it needs several factor such as Kp, Ki, and Kd. When the needs are all met the PID controller is more stable. The following table 1.1 illustrate how these element affect the PID percentage overshoot, rise time and steady state error.

Elements	Effect on Rise time	Effect on overshoot	Effect on steady state error
Kp	Reduces	increases	Reduces
Ki	Reduces	increases	Eliminates
Kd	No/small chage	Reduces	No effect

                                                                                         Table 1. 1Factors affecting PID controllers

In obtaining a very accurate PID controlled system, these requirements indicated in table 1.1 above must be met to be able to withstand external disturbances like noise, vibrations, etc. Failure to meet the requirement, the system becomes unstable.

Strengths and Limitations of PID Control

c).Performances

The performances of PID control systems is evaluated by its ability to overcome the disturbances effects referred to as the disturbance rejection of the control systems.

A small value for derivative value is required since it might result into unstable system due to the high sensitivity to disturbance such as noise and vibration. High value of derivative will result to oscillation of the system, thus unstable system.

The rise time or responses time of PID controller is required to be less than 2 percent of the system output and having a stable state. Moreover, speed of the peak time is required to be considerably faster in reaching the peak values for the given system.

d).Robustness

The robustness of a system can be attained when the stability and the performance of PID Controllers are not affected by a smaller differences in plant or the operating condition. The advantage of the PID controller is that, the robustness is achieved for system with less robust.

e).Energy consumptions.

The PID controller are designed to consume less power, for a system which is unstable it may consume a lot of power. But when PID contlol are introduce the system gains its stability, thus less energy is dissipated resulting to less power consumption

f).steady state errors

Steady state errors can be well-defined as the difference values between the exact output produced by the system and the desired output of the same system. PID controllers are used to minimize the steady state error in the control systems over time and the error rate (Novotecknik, 2009). For a PID, it reduces the error rate and sse of the system, When sse is zero, that means the desired output of the system is met. The integral components (Ki) sums the error term over time. The integral components increase continuously if there exist a small error. The phenomenon in which the integral component continue to increase is referred to as integral windup, which occurs when the integral actions reach to saturations and does not reduce the error to zero. The importance for the integrator are the ant windup operations for saturating the actuator (Owen, 2012).

2: Examples of PI-controlled system and PID controlled system.

PI Controller:

PI controllers are mostly used in eliminating the steady state error which may result from P-controllers. An example is the cruise control systems control which is used to control the speed/velocity of the vehicle, similarly by regulating the throttle positions. In fact cruise controls actuate the throttle valves by a cable connection to actuators in place of pressing the car pedals. Figure 1.2 below shows the pi cruise control (Deka & Haloi, June, 2014).

Implementation of the Controller

                                                                

The aim of cruise control systems are maintaining constant car speed in spite of having external disturbances such as change of road grade or wind. The control is accomplished by computing the car speed, therafter the speed is compared with  the reference/desired speed and automatically, the throttle is adjusted in accordance with control law (Deka & Haloi, June, 2014).

PID Controller:

PID controllers have the best control dynamic counting zero steady state errors, faster responses (shorter rise-time), higher stability and no oscillations as compared to other controllers such as PI-controller. An example used in PID control in many industries is a DC servo motor. The elementary components of normal servo motion systems are illustrated in figure 1.2 below. Yy use of standard Laplace notations. In the figure 1.2, servo drive close a current loops and are made simply as linear transfer functions  Obviously, the servo drive contains a peak current limit, thus the linear models are not completely accurate; nevertheless, it provides a sensible representations for the analysis. Basically, servo drives receives voltage commands that represent the preferred motor currents. The shaft torque, of the motor,  is directly associated to motor current,  by torque constant,  Equation (1.1) below shows the mentioned above relationships.

                                                                             

The servo-motors are made as torque constant, a viscous damping term, , and lump inertia. The lump inertia terms contains the servo-motor and inertia of the load. There is an assumption that the loads are firmly coupled in such a way that the torsional rigidity passes the natural mechanical resonance points further than the servo controller bandwidths. In this case, it become easier to model the total systems inertia as the sum of load inertia and the motor for a frequency that can be controlled. Slightly added complicated models are required if coupler dynamic is integrated.

The real motor positions,  typically estimated by by a resolve or an encoder couples directly to the motor shaft. Once more, the assumption made above also assumes that the feedback devices are firmly mounted in a way that mechanical resonant frequency can be ignored without any effect. Disturbances from external shaft torque,  is added to the generated torque by the current of the motor to give the torque available for accelerating the total inertia, J (Ziegler & Nichols, 2000).

Stabilization Requirements

There exist three (3) gains for adjusting in PID controllers, which acts on the position errors given in equation 1.2 below. The superscript * denotes a commanded values (Ziegler & Nichols, 2000):

The outputs of PID controllers are torque signals. The mathematical expressions in time domain is illustrated in equation 1.3 below (Ziegler & Nichols, 2000):

3).Modern alternatives to PID control

The two modern alternative to PID controls for slow process and system with uncertain parameters are Ziegler-Nicholas method and good gain control method. These two are lab methods used in tuning PID controller (WILLIAMSON, 2015).

The Ziegler–Nichols methods are exploratory methods whereby PID controllers are tuned through setting the D (derivative) and I (integral) gains to zero (WILLIAMSON, 2015).  The “P” (proportional) gain, K p is raised till it attains the final gain. This is the point at which the outputs of the control loops has consistent and stable oscillations. The maximum gain attained and the oscillation period are used to set the derivative, D, promotional, P and integral, I gains which depends on controller type used. This method can be used for simulations and it is also probably the most common to use in real life.

                                                     

The Good Gain method is used to give better stability to the control loop better stability than that of Ziegler-Nichols’ methods (OGATA, 2013). The Good Gain method, as simple as it is, can be used both on real processes (without any knowledge about the processes to be controlled), and in simulated systems. This method gives better stability and does not need the control loop to get into oscillations when tuning (OGATA, 2013). These are two benefits of this method as compared with the Ziegler-Nichols’ methods.

                                                                         

4.Reason for discretizing system in modern control.

The system which is digitized it has several advantages over time continuous system. The flexibility and capability of decision making in the control program is the chief benefits of digital control systems (Anon., n.d.)among others:

1. High accuracy, since digitized system is represented by 0s and 1s which results to a very small errors where noise and power supply drift are present
2. Low implementation error
3. High speed
4. Low cost

The modern control applications are using process control where the power of digital processing techniques are used to perform the desired control tasks (KUO & HASELMAN, 2014).  Although the majority of systems that need to be controlled are often analog nature, the modern digital control applications are using A/D- and D/A-conversions as the principal operations to achieve appropriate control of processes. This has brought the increasing need for discretization in modern control applications where sampling is carried out by point measurements (OGATA, 2013).

Performances

5.motor angular velocity control relevance to complex systems

Motor drives requires a rotor position sensors to correctly perform phase commutations and current controls. A constant supplies of position data is essential; therefore position sensors having high resolution, such as a resolver or a shaft encoders, is characteristically used (Gamazo-Real, et al., July, 2010). For complex systems, therefore, low-cost Hall-effects sensor are typically used. Moreover, accelerometers or electromagnetic variables reluctance sensors has widely applied in measuring motor position and speed (Deka & Haloi, June, 2014). The angular motion sensors based on magnetic fields sensing principle stand out due to several inherent advantages and sensing benefits (Deka & Haloi, June, 2014).

Conclusions:

The assignment consisted three (3) parts. In the first part, I was able to understand the major benefits and drawback of PID controllers. Some applications of PI and PID controller in day-todays life is discussed. Furthermore in this part, the advantages of using digital systems over continuous time systems. It is found that, discrete system are more proffered than continuous time system since discrete system has low error rate, low cost, high speed, etc. Also, we found how motor angular velocity control can be relevant to complex systems in day today life.

In part two (2), the control of angular velocity of a DC motor shaft by varying an input voltage u is illustrated. Simulation of continuous time PID control system of a DC motor is done using MATLAB software. The system stability, robustness, response time and other characteristics of system are simulated. A tuning PID controller is then simulated using different metric to determine which metrics is better and preferred to control the DC motor. It is found that, the Ziegler-Nichols method produces a faster response of the system having acceptable overshoot while Chien-Hrones-Reswick method of tuning the PID has a smaller overshoot having suitable system transient responses.

In the last part of the system, the DC motor is discretized and simulated in MATLAB to compare with the continuous time system. Different metrics of discrete system is used to determine the system response, robustness, stability and easiness of implementation and the error rate of the system. It is found that, using discrete system, the DC motor is more stable, more robust, less response time and less error rate as compared to continuous time system. Although, in both continuous and discrete system, choosing the PID controller should be chosen carefully to provide a stable and robust system

Robustness

Many industrial sectors today, direct current motors (also called DC motors) are used in different ways from automobiles to robotics small and medium-sized driving applications regularly features DC motors for the wide range of functionalities. A DC motor can be defined as an electric motor which runs on direct current. Common actuators in control systems are Direct Current motors. The DC motors provide direct rotary motion and, coupled with cables and drums or wheels, providing translational motion. The electric circuit of the armature and the free-body diagram of the rotor are shown in figure 1.1 below (Melkin, 2017):

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