Control theory

Centrifugal governor in a Boulton & Watt engine of 1788

Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of the centrifugal governor, conducted by the physicist James Clerk Maxwell in 1868, entitled On Governors.[5] A centrifugal governor was already used to regulate the velocity of windmills.[6] Maxwell described and analyzed the phenomenon of self-oscillation, in which lags in the system may lead to overcompensation and unstable behavior. This generated a flurry of interest in the topic, during which Maxwell's classmate, Edward John Routh, abstracted Maxwell's results for the general class of linear systems.[7] Independently, Adolf Hurwitz analyzed system stability using differential equations in 1877, resulting in what is now known as the Routh–Hurwitz theorem.[8][9]

A notable application of dynamic control was in the area of manned flight. The Wright brothers made their first successful test flights on December 17, 1903 and were distinguished by their ability to control their flights for substantial periods (more so than the ability to produce lift from an airfoil, which was known). Continuous, reliable control of the airplane was necessary for flights lasting longer than a few seconds.

By World War II, control theory was becoming an important area of research. Irmgard Flügge-Lotz developed the theory of discontinuous automatic control systems, and applied the bang-bang principle to the development of automatic flight control equipment for aircraft.[10][11] Other areas of application for discontinuous controls included fire-control systems, guidance systems and electronics.

Sometimes, mechanical methods are used to improve the stability of systems. For example, ship stabilizers are fins mounted beneath the waterline and emerging laterally. In contemporary vessels, they may be gyroscopically controlled active fins, which have the capacity to change their angle of attack to counteract roll caused by wind or waves acting on the ship.

The Space Race also depended on accurate spacecraft control, and control theory has also seen an increasing use in fields such as economics and artificial intelligence. Here, one might say that the goal is to find an internal model that obeys the good regulator theorem. So, for example, in economics, the more accurately a (stock or commodities) trading model represents the actions of the market, the more easily it can control that market (and extract "useful work" (profits) from it). In AI, an example might be a chatbot modelling the discourse state of humans: the more accurately it can model the human state (e.g. on a telephone voice-support hotline), the better it can manipulate the human (e.g. into performing the corrective actions to resolve the problem that caused the phone call to the help-line). These last two examples take the narrow historical interpretation of control theory as a set of differential equations modeling and regulating kinetic motion, and broaden it into a vast generalization of a regulator interacting with a plant.

A block diagram of a negative feedback control system using a feedback loop to control the process variable by comparing it with a desired value, and applying the difference as an error signal to generate a control output to reduce or eliminate the error.

Example of a single industrial control loop; showing continuously modulated control of process flow.

Fundamentally, there are two types of control loops: open loop control and closed loop (feedback) control.

In open loop control, the control action from the controller is independent of the "process output" (or "controlled process variable" - PV). A good example of this is a central heating boiler controlled only by a timer, so that heat is applied for a constant time, regardless of the temperature of the building. The control action is the timed switching on/off of the boiler, the process variable is the building temperature, but neither is linked.

In closed loop control, the control action from the controller is dependent on feedback from the process in the form of the value of the process variable (PV). In the case of the boiler analogy, a closed loop would include a thermostat to compare the building temperature (PV) with the temperature set on the thermostat (the set point - SP). This generates a controller output to maintain the building at the desired temperature by switching the boiler on and off. A closed loop controller, therefore, has a feedback loop which ensures the controller exerts a control action to manipulate the process variable to be the same as the "Reference input" or "set point". For this reason, closed loop controllers are also called feedback controllers.[12]

The definition of a closed loop control system according to the British Standard Institution is "a control system possessing monitoring feedback, the deviation signal formed as a result of this feedback being used to control the action of a final control element in such a way as to tend to reduce the deviation to zero." [13]

Likewise; "A Feedback Control System is a system which tends to maintain a prescribed relationship of one system variable to another by comparing functions of these variables and using the difference as a means of control."[14]

To overcome the limitations of the open-loop controller, control theory introduces feedback. A closed-loop controller uses feedback to control states or outputs of a dynamical system. Its name comes from the information path in the system: process inputs (e.g., voltage applied to an electric motor) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with sensors and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.

Closed-loop controllers have the following advantages over open-loop controllers:

• disturbance rejection (such as hills in the cruise control example above)
• guaranteed performance even with model uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
• unstable processes can be stabilized
• reduced sensitivity to parameter variations
• improved reference tracking performance

In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed feedforward and serves to further improve reference tracking performance.

A common closed-loop controller architecture is the PID controller.

The output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.

This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).

A simple feedback control loop

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations:

${\displaystyle Y(s)=P(s)U(s)}$

${\displaystyle U(s)=C(s)E(s)}$

${\displaystyle E(s)=R(s)-F(s)Y(s).}$

Solving for Y(s) in terms of R(s) gives

${\displaystyle Y(s)=\left({\frac {P(s)C(s)}{1+P(s)C(s)F(s)}}\right)R(s)=H(s)R(s).}$

The expression ${\displaystyle H(s)={\frac {P(s)C(s)}{1+F(s)P(s)C(s)}}}$ is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If ${\displaystyle |P(s)C(s)|\gg 1}$, i.e., it has a large norm with each value of s, and if ${\displaystyle |F(s)|\approx 1}$, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input.

A block diagram of a PID controller in a feedback loop, r(t) is the desired process value or "set point", and y(t) is the measured process value.

A proportional–integral–derivative controller (PID controller) is a control loop feedback mechanism control technique widely used in control systems.

A PID controller continuously calculates an error value e(t) as the difference between a desired setpoint and a measured process variable and applies a correction based on proportional, integral, and derivative terms. PID is an initialism for Proportional-Integral-Derivative, referring to the three terms operating on the error signal to produce a control signal.

The theoretical understanding and application dates from the 1920s, and they are implemented in nearly all analogue control systems; originally in mechanical controllers, and then using discrete electronics and later in industrial process computers. The PID controller is probably the most-used feedback control design.

If u(t) is the control signal sent to the system, y(t) is the measured output and r(t) is the desired output, and e(t) = r(t) − y(t) is the tracking error, a PID controller has the general form

${\displaystyle u(t)=K_{P}e(t)+K_{I}\int e(\tau ){\text{d}}\tau +K_{D}{\frac {{\text{d}}e(t)}{{\text{d}}t}}.}$

The desired closed loop dynamics is obtained by adjusting the three parameters K_P, K_I and K_D, often iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in process control). The derivative term is used to provide damping or shaping of the response. PID controllers are the most well-established class of control systems: however, they cannot be used in several more complicated cases, especially if MIMO systems are considered.

Applying Laplace transformation results in the transformed PID controller equation

${\displaystyle u(s)=K_{P}e(s)+K_{I}{\frac {1}{s}}e(s)+K_{D}se(s)}$

${\displaystyle u(s)=\left(K_{P}+K_{I}{\frac {1}{s}}+K_{D}s\right)e(s)}$

with the PID controller transfer function

${\displaystyle C(s)=\left(K_{P}+K_{I}{\frac {1}{s}}+K_{D}s\right).}$

As an example of tuning a PID controller in the closed-loop system H(s), consider a 1st order plant given by

${\displaystyle P(s)={\frac {A}{1+sT_{P}}}}$

where A and T_P are some constants. The plant output is fed back through

${\displaystyle F(s)={\frac {1}{1+sT_{F}}}}$

where T_F is also a constant. Now if we set ${\displaystyle K_{P}=K\left(1+{\frac {T_{D}}{T_{I}}}\right)}$, K_D = KT_D, and ${\displaystyle K_{I}={\frac {K}{T_{I}}}}$, we can express the PID controller transfer function in series form as

${\displaystyle C(s)=K\left(1+{\frac {1}{sT_{I}}}\right)(1+sT_{D})}$

Plugging P(s), F(s), and C(s) into the closed-loop transfer function H(s), we find that by setting

${\displaystyle K={\frac {1}{A}},T_{I}=T_{F},T_{D}=T_{P}}$

H(s) = 1. With this tuning in this example, the system output follows the reference input exactly.

However, in practice, a pure differentiator is neither physically realizable nor desirable[15] due to amplification of noise and resonant modes in the system. Therefore, a phase-lead compensator type approach or a differentiator with low-pass roll-off are used instead.

The field of control theory can be divided into two branches:

• Linear control theory – This applies to systems made of devices which obey the superposition principle, which means roughly that the output is proportional to the input. They are governed by linear differential equations. A major subclass is systems which in addition have parameters which do not change with time, called linear time invariant (LTI) systems. These systems are amenable to powerful frequency domain mathematical techniques of great generality, such as the Laplace transform, Fourier transform, Z transform, Bode plot, root locus, and Nyquist stability criterion. These lead to a description of the system using terms like bandwidth, frequency response, eigenvalues, gain, resonant frequencies, zeros and poles, which give solutions for system response and design techniques for most systems of interest.
• Nonlinear control theory – This covers a wider class of systems that do not obey the superposition principle, and applies to more real-world systems because all real control systems are nonlinear. These systems are often governed by nonlinear differential equations. The few mathematical techniques which have been developed to handle them are more difficult and much less general, often applying only to narrow categories of systems. These include limit cycle theory, Poincaré maps, Lyapunov stability theorem, and describing functions. Nonlinear systems are often analyzed using numerical methods on computers, for example by simulating their operation using a simulation language. If only solutions near a stable point are of interest, nonlinear systems can often be linearized by approximating them by a linear system using perturbation theory, and linear techniques can be used.[16]

Mathematical techniques for analyzing and designing control systems fall into two different categories:

• Frequency domain – In this type the values of the state variables, the mathematical variables representing the system's input, output and feedback are represented as functions of frequency. The input signal and the system's transfer function are converted from time functions to functions of frequency by a transform such as the Fourier transform, Laplace transform, or Z transform. The advantage of this technique is that it results in a simplification of the mathematics; the differential equations that represent the system are replaced by algebraic equations in the frequency domain which is much simpler to solve. However, frequency domain techniques can only be used with linear systems, as mentioned above.
• Time-domain state space representation – In this type the values of the state variables are represented as functions of time. With this model, the system being analyzed is represented by one or more differential equations. Since frequency domain techniques are limited to linear systems, time domain is widely used to analyze real-world nonlinear systems. Although these are more difficult to solve, modern computer simulation techniques such as simulation languages have made their analysis routine.

In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs, and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a point within that space.[17][18]

Control systems can be divided into different categories depending on the number of inputs and outputs.

• Single-input single-output (SISO) – This is the simplest and most common type, in which one output is controlled by one control signal. Examples are the cruise control example above, or an audio system, in which the control input is the input audio signal and the output is the sound waves from the speaker.
• Multiple-input multiple-output (MIMO) – These are found in more complicated systems. For example, modern large telescopes such as the Keck and MMT have mirrors composed of many separate segments each controlled by an actuator. The shape of the entire mirror is constantly adjusted by a MIMO active optics control system using input from multiple sensors at the focal plane, to compensate for changes in the mirror shape due to thermal expansion, contraction, stresses as it is rotated and distortion of the wavefront due to turbulence in the atmosphere. Complicated systems such as nuclear reactors and human cells are simulated by a computer as large MIMO control systems.

Every control system must guarantee first the stability of the closed-loop behavior. For linear systems, this can be obtained by directly placing the poles. nonlinear control systems use specific theories (normally based on Aleksandr Lyapunov's Theory) to ensure stability without regard to the inner dynamics of the system. The possibility to fulfill different specifications varies from the model considered and the control strategy chosen.

List of the main control techniques

• Adaptive control uses on-line identification of the process parameters, or modification of controller gains, thereby obtaining strong robustness properties. Adaptive controls were applied for the first time in the aerospace industry in the 1950s, and have found particular success in that field.
• A hierarchical control system is a type of control system in which a set of devices and governing software is arranged in a hierarchical tree. When the links in the tree are implemented by a computer network, then that hierarchical control system is also a form of networked control system.
• Intelligent control uses various AI computing approaches like artificial neural networks, Bayesian probability, fuzzy logic,[20] machine learning, evolutionary computation and genetic algorithms or a combination of these methods, such as neuro-fuzzy algorithms, to control a dynamic system.
• Optimal control is a particular control technique in which the control signal optimizes a certain "cost index": for example, in the case of a satellite, the jet thrusts needed to bring it to desired trajectory that consume the least amount of fuel. Two optimal control design methods have been widely used in industrial applications, as it has been shown they can guarantee closed-loop stability. These are Model Predictive Control (MPC) and linear-quadratic-Gaussian control (LQG). The first can more explicitly take into account constraints on the signals in the system, which is an important feature in many industrial processes. However, the "optimal control" structure in MPC is only a means to achieve such a result, as it does not optimize a true performance index of the closed-loop control system. Together with PID controllers, MPC systems are the most widely used control technique in process control.
• Robust control deals explicitly with uncertainty in its approach to controller design. Controllers designed using robust control methods tend to be able to cope with small differences between the true system and the nominal model used for design.[21] The early methods of Bode and others were fairly robust; the state-space methods invented in the 1960s and 1970s were sometimes found to lack robustness. Examples of modern robust control techniques include H-infinity loop-shaping developed by Duncan McFarlane and Keith Glover, Sliding mode control (SMC) developed by Vadim Utkin, and safe protocols designed for control of large heterogeneous populations of electric loads in Smart Power Grid applications.[22] Robust methods aim to achieve robust performance and/or stability in the presence of small modeling errors.
• Stochastic control deals with control design with uncertainty in the model. In typical stochastic control problems, it is assumed that there exist random noise and disturbances in the model and the controller, and the control design must take into account these random deviations.
• Energy-shaping control view the plant and the controller as energy-transformation devices. The control strategy is formulated in terms of interconnection (in a power-preserving manner) in order to achieve a desired behavior.
• Self-organized criticality control may be defined as attempts to interfere in the processes by which the self-organized system dissipates energy.

Many active and historical figures made significant contribution to control theory including

• Pierre-Simon Laplace invented the Z-transform in his work on probability theory, now used to solve discrete-time control theory problems. The Z-transform is a discrete-time equivalent of the Laplace transform which is named after him.
• Irmgard Flugge-Lotz developed the theory of discontinuous automatic control and applied it to automatic aircraft control systems.
• Alexander Lyapunov in the 1890s marks the beginning of stability theory.
• Harold S. Black invented the concept of negative feedback amplifiers in 1927. He managed to develop stable negative feedback amplifiers in the 1930s.
• Harry Nyquist developed the Nyquist stability criterion for feedback systems in the 1930s.
• Richard Bellman developed dynamic programming since the 1940s.[23]
• Andrey Kolmogorov co-developed the Wiener–Kolmogorov filter in 1941.
• Norbert Wiener co-developed the Wiener–Kolmogorov filter and coined the term cybernetics in the 1940s.
• John R. Ragazzini introduced digital control and the use of Z-transform in control theory (invented by Laplace) in the 1950s.
• Lev Pontryagin introduced the maximum principle and the bang-bang principle.
• Pierre-Louis Lions developed viscosity solutions into stochastic control and optimal control methods.
• Rudolf Kalman pioneered the state-space approach to systems and control. Introduced the notions of controllability and observability. Developed the Kalman filter for linear estimation.
• Ali H. Nayfeh who was one of the main contributors to nonlinear control theory and published many books on perturbation methods
• Jan C. Willems Introduced the concept of dissipativity, as a generalization of Lyapunov function to input/state/output systems. The construction of the storage function, as the analogue of a Lyapunov function is called, led to the study of the linear matrix inequality (LMI) in control theory. He pioneered the behavioral approach to mathematical systems theory.

Examples of control systems

Topics in control theory

Other related topics

• Levine, William S., ed. (1996). The Control Handbook. New York: CRC Press. ISBN 978-0-8493-8570-4.
• Karl J. Åström; Richard M. Murray (2008). Feedback Systems: An Introduction for Scientists and Engineers (PDF). Princeton University Press. ISBN 978-0-691-13576-2.
• Christopher Kilian (2005). Modern Control Technology. Thompson Delmar Learning. ISBN 978-1-4018-5806-3.
• Vannevar Bush (1929). Operational Circuit Analysis. John Wiley and Sons, Inc.
• Robert F. Stengel (1994). Optimal Control and Estimation. Dover Publications. ISBN 978-0-486-68200-6.
• Franklin; et al. (2002). Feedback Control of Dynamic Systems (4 ed.). New Jersey: Prentice Hall. ISBN 978-0-13-032393-4.
• Joseph L. Hellerstein; Dawn M. Tilbury; Sujay Parekh (2004). Feedback Control of Computing Systems. John Wiley and Sons. ISBN 978-0-471-26637-2.
• Diederich Hinrichsen and Anthony J. Pritchard (2005). Mathematical Systems Theory I – Modelling, State Space Analysis, Stability and Robustness. Springer. ISBN 978-3-540-44125-0.
• Andrei, Neculai (2005). "Modern Control Theory – A historical Perspective" (PDF). Retrieved October 10, 2007. Cite journal requires |journal= (help)
• Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition (PDF). Springer. ISBN 978-0-387-98489-6.
• Goodwin, Graham (2001). Control System Design. Prentice Hall. ISBN 978-0-13-958653-8.
• Christophe Basso (2012). Designing Control Loops for Linear and Switching Power Supplies: A Tutorial Guide. Artech House. ISBN 978-1608075577.
• Boris J. Lurie; Paul J. Enright (2019). Classical Feedback Control with Nonlinear Multi-loop Systems (3 ed.). CRC Press. ISBN 978-1-1385-4114-6.

For Chemical Engineering

• Luyben, William (1989). Process Modeling, Simulation, and Control for Chemical Engineers. McGraw Hill. ISBN 978-0-07-039159-8.

Wikimedia Commons has media related to Control theory.

Wikibooks has a book on the topic of: Control Systems

• Control Tutorials for Matlab, a set of worked-through control examples solved by several different methods.
• Control Tuning and Best Practices
• Advanced control structures, free on-line simulators explaining the control theory
• The Dark Side of Loop Control Theory, a professional seminar taught at APEC in 2012 (Orlando, FL).