Control theory

• 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 cells
  are simulated by a computer as large MIMO control systems.

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

${\displaystyle x}$

${\displaystyle \rho }$

When the appropriate conditions above are satisfied a system is said to be

If a system in question has an impulse response of

${\displaystyle \ x[n]=0.5^{n}u[n]}$

then the Z-transform (see this example), is given by

${\displaystyle \ X(z)={\frac {1}{1-0.5z^{-1}}}}$

which has a pole in ${\displaystyle z=0.5}$ (zero imaginary part). This system is BIBO (asymptotically) stable since the pole is inside the unit circle.

However, if the impulse response was

${\displaystyle \ x[n]=1.5^{n}u[n]}$

then the Z-transform is

${\displaystyle \ X(z)={\frac {1}{1-1.5z^{-1}}}}$

which has a pole at ${\displaystyle z=1.5}$ 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 ${\displaystyle Re[\lambda ]<-{\overline {\lambda }}}$, where ${\displaystyle {\overline {\lambda }}}$ is a fixed value strictly greater than zero, instead of simply asking that ${\displaystyle Re[\lambda ]<0}$.

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 ${\displaystyle m{\ddot {x}}(t)=-Kx(t)-\mathrm {B} {\dot {x}}(t)}$. 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

Processes in industries like

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 industry
  in 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-organized
  system dissipates energy.

• 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 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 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 control
  methods.
• 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.

See also

Examples of control systems

Topics in control theory

Other related topics

References

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. {{cite journal}}: Cite journal requires |journal= (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

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