Introduction to the Controllability of a System - ECE-180D-WS-2024/Wiki-Knowledge-Base GitHub Wiki

Introduction

Automation, as a hallmark to the modern industries, promotes the evolution of designing and the pursuit of accuracy and efficiency. To achieve an automated process, all the industries should carry on in a precise, predictable and repeatable manner. The revolution of automation leads to the embryonic concept of a controller as it monitors a process and compares the output with the set value and this process repeats. And through a set of controllability theories which nowadays has been becoming the backbone of all sorts of technological inventions, control systems have become the core of many technologies. But before humans get divulged to the knowledge base of a control system, the opening question comes to, what is a system and how does it get controlled by humans? In the big picture, a control system is a network of electrical and electro-mechanical devices that’s used to regulate the behavior of a dynamic system with control loops. The following figure shows the simple block diagram of a control system.

Motivation of Controllability Theory

The controllability theory aims to model any physical system using a "dynamical" model. Here the word "dynamical" means changes over time or the evolution of states. Examples of physical systems include the mechanical system's motion (spring-damper cart), dynamics of current in a circuit, and dynamics of heat flows. However, controllability theory isn't limited to modeling physical systems but systems like opinion dynamics in a social network or the evolution of an epidemic in a population. Some might wonder why it is so important to model physical systems using a dynamic model; Well, the answer is quite obvious: without controllability theory, we couldn't do many things we do today. For example, if we are given a motor that spins and thus moves the wheel of the car. If we don't use controllability to design a controller to control the input to the motor, the behavior of the motor will be unpredictable. Say we give it some input voltage, the motor might spin very fast or very slow, but there is no way we would tell how it is going to operate without modeling it with controllability theory. Once we apply the controllability theory, we can design a controller that we use to control the input voltage to produce any desired speed of the motor. Hence, the motivation or the goal of controllability theory is to model any system using a "dynamical" model, design a controller, and produce desired outputs.

Open-Loop and Closed-Loop Control Systems

Control Systems can be classified as open loop control systems and closed loop control systems based on the feedback path.

In open loop control systems, output is not fed-back to the input, where the control action is independent of the desired output. An input is applied to a controller and it produces an actuating signal or controlling signal. This signal is given as an input to a plant or process which is to be controlled. Hence, the plant produces an output, which is controlled.

Likewise, In closed loop control systems, output is fed back to the input, where the control action is dependent on the desired output. The error detector produces an error signal, which is the difference between the input and the feedback signal. This feedback signal is obtained from the block (feedback elements) by considering the output of the overall system as an input to this block. Instead of the direct input, the error signal is applied as an input to a controller. So, the controller produces an actuating signal which controls the plant. In this combination, the output of the control system is adjusted automatically till we get the desired response.

In this tutorial, we will start from the beginning of controllability theories and concentrate on the key aspects of a control system which are stability, tracking, regulation, and controllability.

From Time domain to Frequency domain

The above introduction wipes off the dust of the structure of a control system. Generally, in a signal processing task, engineers usually study a control signal (Open-loop or Closed-Loop) from the two pivotal domains: time domain and frequency domain.

In signal processing, convolution is a mathematical way of combining two signals to form a third signal, which in this case would be the output signal of the system itself. For an time domain input signal that’s perceived by a linear time invariant (Usually abbreviated as LTI) system, it would first convolute with the impulse response of the system, and the output signal would be the convolutional product of those two signals.

The Laplace transform, inspired by the complexity of time-domain convolution, can transform a signal from time domain to frequency domain. From time domain to frequency domain, there is a duality property between those two domains as convolution in time domain becomes multiplication in frequency domain.

By applying Laplace’s Transform, we simplify the procedure of signal processing and also engineers are benefited from filtering out the random noise in the time domain and get a more intuitive sense by analyzing a signal in the frequency domain.

Moreover, the reason why we prefer working in the frequency domain instead of the time domain is that the calculations in the frequency domain are much simpler than the calculations in the time domain. For instance, when solving a 2-degree system of equations, we need to do lots of integrations and calculations in the time domain; However, we can take the same system of equations and apply the Laplace transform. We can then avoid integrations and only need to to some algebra calculations to get the same result, which is much faster than doing it in the time domain. Besides that, convolution calculations are often very complicated as they require an integration of two functions multiplied together; however, due to the duality between the domains, convolution in the time domain is equivalent to multiplication in the frequency domain. Furthermore, block diagrams are easier to draw in the frequency domain as shown in the diagram above.

Stability

Due to all the irreducible errors brought by the external factors (Noise; Temperature; etc…) , when our system interacts with our input controlling signal, there might be an issue of instability in the stage of signal processing. To stabilize our system, we would introduce the first controllability theory which is known as the Routh-Hurwitz stability criterion.

This matrix is constructed based on the coefficient of the transfer function. And Routh-Hurwitz stability criterion tells us that for the transfer function of a system (Product of System response and Control Gain), if the determinant of this matrix is negative, the output of this system is going to be bounded, meaning that the system will be able to stabilize its final output. Routh-Hurwitz stability criterion is significant because it determines whether the poles of our control system would be negative or not in corresponding to the convergence of our output when it is converted back to the time domain with inverse Laplace Transform. Technically, that’s called the LHP stability(the Left Half Plane of complex planes) in terms of control theories. (a < 0)

A negative pole will lead to the convergence of an exponential function. (a < 0)

A positive pole will lead to the divergence of an exponential function. (a > 0)

Tracking

The intactness of our input system where there shouldn’t be a loss or defection during the processing is crucial. By inputting a reference input to our system we will be able to examine the discrepancies between our input reference signal and the output of the system for which this error would need to be approximately close to 0.

The above equation has a name called Final Value Theorem. It tells us that when the time approaches infinity, the time response of our output should be equal to the frequency response of our output when frequency approaches 0. This equation is usually used to determine the final value in the time domain by applying just the zero frequency component to the frequency domain representation of a system. Based on this equation we will be able to retrieve an estimation of the accuracy of our system.

Regulation

For the unavoidable error caused by disturbance present in the system between our reference input and output, we would need to perform some regulation to compensate for this error. For example, we can adjust our input by adding in a regulation factor (such as a small signal which we can easily control or manage upon desire). From a technical perspective, regulation improves the robustness of a control system and also provides a framework for consistency in system operations as it helps define and maintain a standard set of procedures, protocols, and specifications, reducing variability and improving predictability in the system's behavior.

Controllability

Finally, to design an input (controller) such that it’s going to stabilize the system. We will first check if the system is still controllable by finding the determinant of the controllability matrix. The controlling parameter, K, is the knot that engineers can design in the input stage to ensure the controllability of the system.

The controllability matrix is constructed by the concatenation of eigenvectors which are the product of the knot B and the characteristics function A derived based on the state of the system. The determinant of the controllability matrix tells us that if it is not equal to 0, the system is going to be controllable.

Now we know how to determine whether a system is controllable or not, the next step is then to design a controller K such that it can take an arbitrary initial state of the state to any destination state we want. The way to design it is fairly simple: u shown below represents the state feedback, which is often the form of u = x1 + x2 + x3 + ... + xn where the x's are the states. Therefore, we could introduce a controller K into u to control the system as the following: u = x1k1 + x2k2 + x3k3 + ... + xnkn. After that, we can then find the characteristic polynomial by computing the determinant of M – λIn where M = A + BK. Once we have the characteristic polynomial, we can then find the constraints on K by using the Routh-Hurwitz stability criterion. Finally, any values of K that meet the constraints is a valid controller for the given system.

Application of the controllability theories to a system

Below is an application of the controllability theories to a closed-loop feedback sound-controlled system. In this project, I use a PID controller as my gain to my input signal, and based on the Routh-Hurwitz stability criterion I conclude on the legitimate combination of my PID factors. After that, I perform a series of analyses on the system to track and regulate its behavior to guarantee the output that I am expected to get. Below is the Bode plot analysis of the system. And later, I used a controllability matrix to regulate the knot of my input.

%Define variables
s = tf('s');
t = 0:0.001:1500;
sigma = 5;
tau1 = 0.1;
tau2 = 0.1;
Ro = 1;
Wo = 1;

%Controller
%Note1: kp < 400(1+kd)
%Note2: Ki < 5kp(1+kd) - 0.0125kp^2
%Note3: Ki > 0   
diffp = 6300;
diffi = 7874;
Kd = 15;
Kp = 400*(1+Kd) - diffp;
Ki = 5*Kp*(1+Kd) - 0.0125*Kp^2 - diffi;

%Plotting
A = s^2*(tau1*s+1)*(tau2*s+1);
B = (Kp*s+Kd*s^2+Ki)*tau1*tau2*sigma;
C = (tau1*s+1)*s;
sysE = Ro-(Ro*B+Wo*C)/(A+B);
err = step(sysE,t);
figure(1);
plot(t,err,'LineWidth',2)
grid on
xlabel('t [s]');
ylabel('e(t), closed-loop');