# What is Control theory: Definition and 70 Discussions

Control theory deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control stability; often with the aim to achieve a degree of optimality.
To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or set point (SP). The difference between actual and desired value of the process variable, called the error signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are controllability and observability. This is the basis for the advanced type of automation that revolutionized manufacturing, aircraft, communications and other industries. This is feedback control, which involves taking measurements using a sensor and making calculated adjustments to keep the measured variable within a set range by means of a "final control element", such as a control valve.Extensive use is usually made of a diagrammatic style known as the block diagram. In it the transfer function, also known as the system function or network function, is a mathematical model of the relation between the input and output based on the differential equations describing the system.
Control theory dates from the 19th century, when the theoretical basis for the operation of governors was first described by James Clerk Maxwell. Control theory was further advanced by Edward Routh in 1874, Charles Sturm and in 1895, Adolf Hurwitz, who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development of PID control theory by Nicolas Minorsky.
Although a major application of mathematical control theory is in control systems engineering, which deals with the design of process control systems for industry, other applications range far beyond this. As the general theory of feedback systems, control theory is useful wherever feedback occurs - thus control theory also has applications in life sciences, computer engineering, sociology and operation research.

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1. ### The use of Riccati equations in optimal control theory

I know that linear control theory, in the form ##\dot{x}=Ax+Bu##, ##\dot{u}=Cx+Du##, can be put in the form of a matrix Riccati equation. But is there really an advantage to doing so?
2. ### How do PI controllers achieve steady state?

Summary: A question about how PI controllers reach steady state Hello PhysicsForums, I need a little help understanding how a PI controller works when operating in steady state. Here is the equation in State Space form: Where In standard form, the equation is: Where The question I’m...
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### Courses How useful is control theory in computational neurosciences?

Hello, I have the opportunity of choosing an elective course on control theory before embarking for my master's program in computational neurosciences and I am weighing the pros and cons. I am looking for advice from PF members who specialized in this field. Since I have chosen enough courses...
4. ### Engineering Why Are Some System States Indistinguishable in Observability Analysis?

https://en.wikipedia.org/wiki/Observability I am studying observability and I try to get some intuition on the topic. Given the observable matrix, we can find the null space. However, the vectors of the null space are states but this differs from the definition of what a state vector of a...
5. ### Engineering Decomposition minimal phase & all pass

Hey everybody! I have put G1 = (1-s)/(2-10s) & G2 = (2-10s)/ (2 +10s) but than I read that all poles and zeroes should be inside the unit circle, and I don't know how to move the Zero S_01 = 1 to the unit circle
6. ### Systems Non-Proper Transfer Fns: Causality?

Prove that if a system's transfer function is not proper (order of numerator greater than order of denominator) then the system is not causal.
7. ### Operators for (Nonlinear) Dynamical Systems

Koopman Generators, Liouville Operators, and Transfer Functions! We talk about the role of operators in Dynamical Systems theory. This requires a discussion of Hilbert spaces, Densely Defined Operators, and Occupation Kernels.
8. ### Control Theory: Derivation of Controllable Canonical Form

Hi, I was recently being taught a control theory course and was going through a 'derivation' of the controllable canonical form. I have a question about a certain step in the process. Question: Why does the coefficient ## b_0 ## in front of the ## u(t) ## mean that the output ## y(t) = b_0 y_1...
9. ### Pole placement design, Control Theory

Summary:: Control Theory root equation pole Hi, I ran into a simple question but somehow I can't get it right. My work this far: ## G_0(s) = G(s) \cdot K \cdot \frac{1}{T_I s} = \frac{k}{\tau s +1} \cdot \frac{2\beta \tau -1}{k} \cdot \frac{2\beta^2 \tau}{Kks} = \frac{2\beta^2\tau}{s(\tau s...
10. ### Engineering What is the difference equation of the system?

Hey guys, what is the difference equation of the signl seen in the figure ? my suggestion is: y(k) = 2u(k) + 2u(k-1) + 2u(k-2) - y(k-6)
11. ### How to draw the transfer behavior of a fuzzy controller

Hey Guys, i try to solve the question mentioned above. But I'm hones...i have no idea to solv this. I need a kind advice. I tryed to find help in the internet but it's impossible. You can see the hole exercise in the figure.
12. ### Engineering How to draw a block diagram of a discrete element with its impulse response

Hello friends, I have a problem with a exercise sheet. Given is the impulse response of a discret element. The task is to draw the block diagram. But I think that the solution in the sheet is wrong. Because based on the difference equation (Exercise.pdf) there should be 3 delay elements. I have...
13. ### Control Theory - Nyquist Plot - Speed of Response from Low Frequency Asymptote

[Moderator: moved from a homework forum. This does not sound like homework.]Homework Statement:: Why is it the case that when the low-frequency response is to the right of the M = 1 line that the 'speed of response is slow'? Relevant Equations:: M-cirlces Hi, Hope you are doing well and...
14. ### Does the transfer function exist for an unforced system?

Homework Statement c is the coefficient of damping, k is the spring constant, and m is the mass. if the cart is displaced from rest by distance X_0 and released at t=0 with initial velocity v_0=1.Homework EquationsThe Attempt at a Solution Ok this might be a stupid question because I wasn't...
15. ### Control theory: process given, PID settings

Homework Statement Problem: We want a closed loop control loop for the given process (see bode plot), the control loop requirements are: Steady state error needs to be zero. The gain margin needs to be infinite. The phase margin needs to be 40 degrees. Questions: Which controller will be...
16. ### Why are we still teaching Routh-Hurwitz criterion in Control

I am a TA for Automatic control class, I can't find an answer for my students about why we are still studying Routh-Hurwitz criterion as almost anyone can use computer software to find the roots of the closed-loop system and determine stability instead of doing these tedious hand calculations
17. ### A Causality in differential equations

Hello, I am studying control theory. And I have encountered something I have never considered or thought about. Consider a system with y as the output differential equation and u as the input. any(n) + ... + a1y(1) + a0y = bmu(m) + ... + b1u(1) + b0u Here, the subscripts indicate...
18. ### Best way to learn control theory for mechanical engineers

Hello, I have begun to teach myself Control Theory. I am looking for a book that is focused for mechanical engineers. I do not mind examples in electrical engineering, but they bore me (no offense). Also, I find some books begin with Laplace Transforms. Yet I found this online lecture...
19. ### Disadvantages of unnecessary increase in PID gain

hi, I read somewhere that if we increase the PID gains unnecessarily, it will cause more energy consumption of the controller and it will lead to noise and distortion in the practical. May someone please give me more explanation on this?? that how it lead to noise or distortion in practical??
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### A Transformation of the neighborhood of a branch point

Hi all, I was trying the understand theory behind Fourier and Laplace Transform (especially in the context of control theory) by reading the book "Complex Variables and the Laplace Transform for Engineers" written by "Wilbur R. LePage". In section 6-10 of the book the author touches on the...
21. ### Probability theory and statistics for Robotics and ME

I study control theory and robotics. Recently I figured out that I have a much deeper understanding of probability and statistics compared to my colleagues. Is this 'talent' valuable in my field and if so, where? We used this theory to define white noise, but nothing more...as of now. Also I am...
22. ### I Exploring the Identity Matrix in Multivariable Control Theory

Hello everyone. Iam working on a course in multivariable control theory and I stumbled over the Identity Matrix. I understand what the identity matrix is, though the use of it is a mistery... I was reading about going from state space to transfer functions and I found this expressions...
23. ### Understanding Root Locus Method: Stability and Response of Control Systems

Good afternoon people; recently I decided to give a study break from control engineering; now I'm back. I'm studying again control theory and I see some good results. By the way I got stuck in a topic called root locus method. Now my issue is not about how to sketch the root loci graph (BTW I...
24. ### Control theory: block diagram, problem (detailed below)

I´m taking a course on control engineering and I have a test next Tuesday so I need to study the basics which are: Laplace transform, simplification of block diagrams, and analysis of transient and steady state responses. Right now I am dealing with the second one. I know the basic rules of...
25. ### Engineering How Control Theory can be useful to Mechanical Engineers?

I'm a student of Mechanical Engineering and I've been taking classes on Control Theory. Looks like a cool and very interesting field of study, but it seems to be more inclined towards Electrical Engineering (in fact, looks like most jobs on control go to electrical engineers). I wonder how a...
26. ### Learning control theory first job out of college

Hello everyone, so my dilemma is this: I graduated with a degree in physics which had a curriculum that did not cover control theory. I recently got a job as an EE and am now likely to be tasked with designing controllers. How can I most quickly and or thoroughly gain a good applicable...
27. ### Alternative Methods vs Iterative Design — Control Theory

QUESTION (2/2) 1) What are some alternatives to iterative design in control theory? 2) I have a certain plant transfer function PTF(s) that is higher order than two and non-unity numerator. I want certain characteristics such as a certain damping ratio (zeta). So I want approximate it as...
28. ### Analytical Expressions for Frequency Response

Hi, I am struggling to understand the maths in the text shown below. (1) How does: r(t) = Acos(wt)+Bsin(wt) become; sqrt((A^2)+(B^2))cos(wt - arctan(B/A))? (2) He states that we can represent the input as a phasor and in one of three ways but doesn't say which one. (3) He then focus on the...
29. ### State Space Control Theory Controllability/Observability

I'm studying for an upcoming final and this is a question from a practice exam we were given. It looks simple but for some reason I just can't wrap my brain around it. The question is attached and gives a state space representation of a system (A, B, C matrices) and asks what part of the system...
30. ### Unstable 4th order system (control theory)

Homework Statement Hello everyone! To simplify, I have a system whose transfer function looks like this: G = (s^2-1) / (s^4-s^2-1) It's unstable and I have to find a way to stabilize it using PID controller and/or lead/lag compensators. Homework EquationsThe Attempt at a Solution So far I...
31. ### Engineering Job Market for Control Engineers?

I've been giving a look at some of the electives in my college - I'm a student of MechE - and became interested in Control Systems, specially after I borrowed some books in the subject in the library and read a little bit more about it. I wonder how big is the field of application of this...
32. ### Deviation after completion of the doctorate

At the moment I'm in the final stages of my doctorate in mathematics. (My background is a BS in physics and an MS in mathematics.) My focus and interest have been in applied functional analysis in general and various kinds of abstract and concrete delay equations in particular. These are...
33. ### Help me understand the usefulness of control theory please

I am looking at control theory but I don't understand how this is used in real systems. Lets take an example of a cruise control system on a car which is also a control system. I can use an embedded computer connected to a hall effect sensor which measures the speed of the vehicle and also...
34. ### Response of a system (Control theory)

Hi everyone. I'm studying for the exam of control theory, and now I'm having an hard time with the response of a system, in particular when we have oscillations. Suppose you have a system, with a transfer function, say, G(S), in the form: G(S) = 1 -------------------...
35. ### How can I control two outputs with a single PID-controller?

Overview I'm trying to levitate a constrained permanent magnet with 2 electromagnets. I'm having trouble conceptualizing the control system for such an operation. Setup The permanent magnet is fixed onto a horizontal pendulum and is repelled by an electromagnet above and repelled by an...
36. ### Control Theory State-Space method with derivative input

Hi, I'm reading Ogata's Modern Control Engineering, and when he talks about the representation of a differential equation in state space he divides the method in two. The first one is when the input of the differential equation involves no derivative term, for example: x'(t)+x(t)=u(t) The...
37. ### Controling mirror movement with piezoelectric stage

I am moving a small glued mirror (~10 gram) attached to a piezoelectric stage: http://www.physikinstrumente.com/product-detail-page/s-330-300700.html The mirror moves back and forth for a few milliseconds using a PID control loop, the range is about 2mrad. I want to increase the angular...
38. ### How does control theory help engineers understand and predict system behavior?

Hello Everybody, Im a third year MechE student and I am focusing my studies on thermodynamics and control theory. I am getting really deep into the controls right now and after taking some introductory courses there are so many things that puzzle me even though I understand the actual theory...
39. ### How Does Unity Position Feedback Affect System Dynamics in Control Theory?

Homework Statement With unity position feedbck, i.e. make K2=0, plot root locus as a function of pitch gain (K1). By imposing 2nd order system approximation, estimate settling time, rise time, peak time of the closed-loop system with 20% overshoot. Pic of system...
40. ### Computed Torque Control vs. PID

I'm working on a pantograph device, which is not a linear plant, and implemented 2 different control schemes. First one is computed torque; For this control technique, I modeled nonlinear terms in the equation of motion and canceled them by injecting their model within the control input, which...
41. ### Finding a State-Space Representation for Helicopter Pitch Angle Control

I need help finding a state-space representation for the given system and locally linearizing it... Objective is to control the pitch angle theta, as an output, of the helicopter by regulating the rotor angle u, as an input. This is what I've done, but apparently it isn't correct, and I...
42. ### What causes droop in control theory?

In control theory, we have the proportional controller which is of the form- u(t) = ke where e is the error between the current measured value and the required value. The P controller is said to droop. To avoid this, an integral...

Hi everyone, I have this question for you: In Instrumentation, when connecting the input of one circuit to the output of another circuit,(for instance, connecting the output of a sensor to the input of an ampliﬁer circuit), a loading problem may arise, could you explain to this problem ? And...
44. ### Find State Transition Matrix (time variant)

Homework Statement find the state transition matrix of a time varying system where: dX/dt = A*X with A = [-1 , exp(-t - (t^2)/2) ; ; 0 , t] (Matlab format - sorry but its easier) Homework Equations How to go about solving such problems in a systematic way? The Attempt at a Solution...
45. ### Control Theory (EE): How to set up a transmittance given wn only?

Homework Statement How do we set up a second-order plant transmittance with the only information available are: One pole is at a position where the undamped natural frequency (ωn = 0 rad/sec), and the other pole is at a position where ωn = 2 rad/sec? The question asks to build that...
46. ### Control Theory prerequisites and learning

Hello, I have recently been doing a lot of research around designing quadcopters/hexcopters and so forth. I have come to the conclusion that it may be fun to learn bait of control theory on the side and it sounds useful for later on when I go to university. Do you know what the prerequisite are...
47. ### Digital PID Controller: Windows OS, Processor Config/Software?

I am trying to understand the process of a digital pid controller that operates with/in a Windows operating system. By process I mean the sequence of steps a computer software system takes to adjust the output of the system whether the process be cruise control, temperature, pressure etc. So I'm...
48. ### Should I study D. S. Processing or control theory in greater depth?

I'm studying CE and math. My computer engineering course is basically electronic/electrical engineering with extra software. I have a choice between the two subject combinations below. Additionally, 1. I'm studying two mechatronic control theory subject, and a mathematical control theory...
49. ### Feedback and Control Theory applied to Cosmology and Particle Physics

The radical Smolin thread left me wondering if other similar ideas could not be applied to other areas of Cosmology and even Particle Physics. Every complex system I can think of in our lives, and in nature, forms part of a complex multifaceted control system or process, with various feedback...
50. ### Can someone explain control structures in simpler terms in the paper on DES?

Could someone refer a book or a tutorial that explains the concept of "control structure" - In section 3 of the following paper http://link.springer.com/article/10.1007/BF01797154# in simpler terms? Thanks in advance Evrin