Control theory
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Control theory is a field of control engineering and applied mathematics that 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
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
History
Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of the
A notable application of dynamic control was in the area of crewed 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.[9][10] 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
Open-loop and closed-loop (feedback) control
Fundamentally, there are two types of control loop:
In open-loop control, the control action from the controller is independent of the "process output" (or "controlled process variable"). 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 switching on/off of the boiler, but the controlled variable should be the building temperature, but is not because this is open-loop control of the boiler, which does not give closed-loop control of the temperature.
In closed loop control, the control action from the controller is dependent on the process output. In the case of the boiler analogy this would include a thermostat to monitor the building temperature, and thereby feed back a signal to ensure the controller maintains the building at the temperature set on the thermostat. A closed loop controller therefore has a feedback loop which ensures the controller exerts a control action to give a process output the same as the "reference input" or "set point". For this reason, closed loop controllers are also called feedback controllers.[11]
The definition of a closed loop control system according to the
Classical control theory
A closed-loop controller or feedback controller is a control loop which incorporates feedback, in contrast to an open-loop controller or non-feedback controller. A closed-loop controller uses feedback to control
In the case of linear
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
- improved rectification of random fluctuations[15]
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
Linear and nonlinear control theory
The field of control theory can be divided into two branches:
- resonant frequencies, zeros and poles, which give solutions for system response and design techniques for most systems of interest.
- 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]
Analysis techniques - frequency domain and time domain
Mathematical techniques for analyzing and designing control systems fall into two different categories:
- 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 equationsin 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 languageshave made their analysis routine.
In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain
System interfacing - SISO & MIMO
Control systems can be divided into different categories depending on the number of inputs and outputs.
- audio system, in which the control input is the input audio signal and the output is the sound waves from the speaker.
- 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 cellsare simulated by a computer as large MIMO control systems.
Classical SISO System Design
The scope of classical control theory is limited to
Modern MIMO System Design
Modern control theory is carried out in the
Topics in control theory
Stability
The stability of a general dynamical system with no input can be described with Lyapunov stability criteria.
- A linear system is called bounded-input bounded-output (BIBO) stable if its output will stay bounded for any bounded input.
- Stability for nonlinear systems that take an input is input-to-state stability (ISS), which combines Lyapunov stability and a notion similar to BIBO stability.
For simplicity, the following descriptions focus on continuous-time and discrete-time linear systems.
Mathematically, this means that for a causal linear system to be stable all of the
- in the open left half of the complex plane for continuous time, when the Laplace transform is used to obtain the transfer function.
- inside the unit circle for discrete time, when the Z-transform is used.
The difference between the two cases is simply due to the traditional method of plotting continuous time versus discrete time transfer functions. The continuous Laplace transform is in
When the appropriate conditions above are satisfied a system is said to be
If a system in question has an impulse response of
then the Z-transform (see this example), is given by
which has a pole in (zero imaginary part). This system is BIBO (asymptotically) stable since the pole is inside the unit circle.
However, if the impulse response was
then the Z-transform is
which has a pole at and is not BIBO stable since the pole has a modulus strictly greater than one.
Numerous tools exist for the analysis of the poles of a system. These include graphical systems like the
Mechanical changes can make equipment (and control systems) more stable. Sailors add ballast to improve the stability of ships. Cruise ships use antiroll fins that extend transversely from the side of the ship for perhaps 30 feet (10 m) and are continuously rotated about their axes to develop forces that oppose the roll.
Controllability and observability
Controllability and observability are main issues in the analysis of a system before deciding the best control strategy to be applied, or whether it is even possible to control or stabilize the system. Controllability is related to the possibility of forcing the system into a particular state by using an appropriate control signal. If a state is not controllable, then no signal will ever be able to control the state. If a state is not controllable, but its dynamics are stable, then the state is termed stabilizable. Observability instead is related to the possibility of observing, through output measurements, the state of a system. If a state is not observable, the controller will never be able to determine the behavior of an unobservable state and hence cannot use it to stabilize the system. However, similar to the stabilizability condition above, if a state cannot be observed it might still be detectable.
From a geometrical point of view, looking at the states of each variable of the system to be controlled, every "bad" state of these variables must be controllable and observable to ensure a good behavior in the closed-loop system. That is, if one of the
Solutions to problems of an uncontrollable or unobservable system include adding actuators and sensors.
Control specification
Several different control strategies have been devised in the past years. These vary from extremely general ones (PID controller), to others devoted to very particular classes of systems (especially robotics or aircraft cruise control).
A control problem can have several specifications. Stability, of course, is always present. The controller must ensure that the closed-loop system is stable, regardless of the open-loop stability. A poor choice of controller can even worsen the stability of the open-loop system, which must normally be avoided. Sometimes it would be desired to obtain particular dynamics in the closed loop: i.e. that the poles have , where is a fixed value strictly greater than zero, instead of simply asking that .
Another typical specification is the rejection of a step disturbance; including an integrator in the open-loop chain (i.e. directly before the system under control) easily achieves this. Other classes of disturbances need different types of sub-systems to be included.
Other "classical" control theory specifications regard the time-response of the closed-loop system. These include the rise time (the time needed by the control system to reach the desired value after a perturbation), peak overshoot (the highest value reached by the response before reaching the desired value) and others (settling time, quarter-decay). Frequency domain specifications are usually related to robustness (see after).
Modern performance assessments use some variation of integrated tracking error (IAE, ISA, CQI).
Model identification and robustness
A control system must always have some robustness property. A robust controller is such that its properties do not change much if applied to a system slightly different from the mathematical one used for its synthesis. This requirement is important, as no real physical system truly behaves like the series of differential equations used to represent it mathematically. Typically a simpler mathematical model is chosen in order to simplify calculations, otherwise, the true system dynamics can be so complicated that a complete model is impossible.
- System identification
The process of determining the equations that govern the model's dynamics is called system identification. This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically its transfer function or matrix. Such identification from the output, however, cannot take account of unobservable dynamics. Sometimes the model is built directly starting from known physical equations, for example, in the case of a mass-spring-damper system we know that . Even assuming that a "complete" model is used in designing the controller, all the parameters included in these equations (called "nominal parameters") are never known with absolute precision; the control system will have to behave correctly even when connected to a physical system with true parameter values away from nominal.
Some advanced control techniques include an "on-line" identification process (see later). The parameters of the model are calculated ("identified") while the controller itself is running. In this way, if a drastic variation of the parameters ensues, for example, if the robot's arm releases a weight, the controller will adjust itself consequently in order to ensure the correct performance.
- Analysis
Analysis of the robustness of a SISO (single input single output) control system can be performed in the frequency domain, considering the system's transfer function and using
- Constraints
A particular robustness issue is the requirement for a control system to perform properly in the presence of input and state constraints. In the physical world every signal is limited. It could happen that a controller will send control signals that cannot be followed by the physical system, for example, trying to rotate a valve at excessive speed. This can produce undesired behavior of the closed-loop system, or even damage or break actuators or other subsystems. Specific control techniques are available to solve the problem:
System classifications
Linear systems control
For MIMO systems, pole placement can be performed mathematically using a
Nonlinear systems control
Processes in industries like
Decentralized systems control
When the system is controlled by multiple controllers, the problem is one of decentralized control. Decentralization is helpful in many ways, for instance, it helps control systems to operate over a larger geographical area. The agents in decentralized control systems can interact using communication channels and coordinate their actions.
Deterministic and stochastic systems control
A stochastic control problem is one in which the evolution of the state variables is subjected to random shocks from outside the system. A deterministic control problem is not subject to external random shocks.
Main control strategies
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
- 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.[20] 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.[21] 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.
- aerospace industryin the 1950s, and have found particular success in that field.
- 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.
- dynamic system.
- self-organizedsystem dissipates energy.
People in systems and control
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.
- discontinuous automatic control and applied it to automatic aircraft control systems.
- Alexander Lyapunov in the 1890s marks the beginning of stability theory.
- negative feedback amplifiersin 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 in the 1940s.[23]
- Warren E. Dixon, control theorist and a professor
- Kyriakos G. Vamvoudakis, developed synchronous reinforcement learning algorithms to solve optimal control and game theoretic problems
- 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.
- bang-bang principle.
- viscosity solutions into stochastic control and optimal controlmethods.
- state-space approach to systems and control. Introduced the notions of controllability and observability. Developed the Kalman filterfor 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.
See also
- Examples of control systems
- Automation
- Deadbeat controller
- Distributed parameter systems
- Fractional-order control
- H-infinity loop-shaping
- Hierarchical control system
- Model predictive control
- Optimal control
- Process control
- Robust control
- Servomechanism
- State space (controls)
- Vector control
- Topics in control theory
- Coefficient diagram method
- Control reconfiguration
- Feedback
- H infinity
- Hankel singular value
- Krener's theorem
- Lead-lag compensator
- Minor loop feedback
- Multi-loop feedback
- Positive systems
- Radial basis function
- Root locus
- Signal-flow graphs
- Stable polynomial
- State space representation
- Steady state
- Transient response
- Transient state
- Underactuation
- Youla–Kucera parametrization
- Markov chain approximation method
- Other related topics
- Adaptive system
- Automation and remote control
- Bond graph
- Control engineering
- Control–feedback–abort loop
- Controller (control theory)
- Cybernetics
- Intelligent control
- Mathematical system theory
- Negative feedback amplifier
- Outline of management
- People in systems and control
- Perceptual control theory
- Systems theory
References
- ^ Maxwell, J. C. (1868). "On Governors" (PDF). Proceedings of the Royal Society. 100. Archived (PDF) from the original on December 19, 2008.
- .
- ^ GND. "Katalog der Deutschen Nationalbibliothek (Authority control)". portal.dnb.de. Retrieved April 26, 2020.
- JSTOR 112510.
- ISSN 1575-9822.
- ^ Routh, E.J.; Fuller, A.T. (1975). Stability of motion. Taylor & Francis.
- ^ Routh, E.J. (1877). A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion: Particularly Steady Motion. Macmillan and co.
- ^ Hurwitz, A. (1964). "On The Conditions Under Which An Equation Has Only Roots With Negative Real Parts". Selected Papers on Mathematical Trends in Control Theory.
- ^ Flugge-Lotz, Irmgard; Titus, Harold A. (October 1962). "Optimum and Quasi-Optimum Control of Third and Fourth-Order Systems" (PDF). Stanford University Technical Report (134): 8–12. Archived from the original (PDF) on April 27, 2019.
- ISBN 9781849722704.
- ^ "Feedback and control systems" - JJ Di Steffano, AR Stubberud, IJ Williams. Schaums outline series, McGraw-Hill 1967
- ^ Mayr, Otto (1970). The Origins of Feedback Control. Clinton, MA US: The Colonial Press, Inc.
- ^ Mayr, Otto (1969). The Origins of Feedback Control. Clinton, MA US: The Colonial Press, Inc.
- .
- .
- ^ "trim point".
- ISBN 978-0-07-070096-3.
- JSTOR 2589614.
- PMID 26423222.
Here we use tools from control and network theories to offer a mechanistic explanation for how the brain moves between cognitive states drawn from the network organization of white matter microstructure
- .
- S2CID 32067734.
- .
- .
Further reading
- 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; ISBN 978-0-471-26637-2.
- ISBN 978-3-540-44125-0.
- Andrei, Neculai (2005). "Modern Control Theory – A historical Perspective" (PDF). Retrieved October 10, 2007.
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(help) - 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.
External links
- 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