Control theory

In engineering and mathematics, control theory deals with the behavior of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system.

An example
Consider an automobile's cruise control, which is a device designed to maintain a constant vehicle speed. The output variable of the system is vehicle speed. The input variable is the engine's torque output, which is regulated by the throttle.

A simple way to implement cruise control is to lock the throttle position when the driver engages cruise control. However, on hilly terrain, the vehicle will slow down going uphill and accelerate going downhill. This type of controller is called an open-loop controller because there is no direct connection between the output of the system and its input.

In a closed-loop control system, a feedback control monitors the vehicle's speed and adjusts the throttle as necessary to maintain the desired speed. This feedback compensates for disturbances to the system, such as changes in slope of the ground or wind speed.

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 centrifugal governor, conducted by the famous physicist J.C. Maxwell in 1868 entitled "On Governors." This described and analyzed the phenomenon of "hunting" in which lags in the system can lead to overcompensation and unstable behavior. This caused a flurry of interest in the topic, which was followed up by Maxwell's classmate, E.J. Routh, who generalized the results of Maxwell for the general class of linear systems. This result is called the Routh-Hurwitz Criterion.

A notable application of dynamic control was in the area of manned flight. The Wright Brothers made their first successful test flights in December 17, 1903 and by 1904 Flyer III 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). Control of the airplane was necessary for safe flight.

By World War II, control theory was an important part of fire control, guidance systems, and cybernetics. The Space Race to the Moon depended on accurate control of the spacecraft. But control theory is not only useful in technological applications, and is meeting an increasing use in field such economics and sociology.

Classical control theory: the closed-loop controller
To avoid the problems 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 a motor) have an effect on the process outputs (e.g. velocity or position of the motor), which is measured with sensors and processed by the controller; the result (the control signal) is used as input to the process, closing the loop.

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


 * disturbance rejection (such as unmeasured friction in a motor)
 * 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

To obtain good performance, closed-loop and open-loop are used simultaneously; open-loop improves set-point (the value desired for the output) tracking.

The most popular closed-loop controller architecture, by far, is the PID controller.

One example is a fuel injection system using an oxygen sensor to control the fuel/air mixture.

The output of the system $$y(t)$$ is fed back to the reference value $$r(t)$$, through the measurement performed by a sensor. The controller C then takes the difference between the reference and the output, the error e, 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 a so-called single-input-single-output (SISO) control system: example where one or more variables can contain more than one value (MIMO, i.e. Multi-Input-Multi-Output - for example when outputs to be controlled are two or more) are frequent. 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 and the plant P are linear and time-invariant (i.e.: elements of their transfer function $$C(s)$$ and $$P(s)$$ do not depend on time), we can analyze the system above by using the Laplace transform on the variables. This gives us the following relations:


 * $$Y(s) = P(s) U(s)\,\!$$
 * $$U(s) = C(s) E(s)\,\!$$
 * $$E(s) = R(s) - Y(s)\,\!$$

Solving for Y(s) in terms of R(s), we obtain:


 * $$Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)} \right) R(s)$$

The term $$\frac{P(s)C(s)}{1 + P(s)C(s)}$$ is referred to as the transfer function of the system. Where the numerator is the forward gain from r to y, and the denominator is one plus the loop gain of the feedback loop. If we can ensure $$P(s)C(s) >> 1$$, i.e. it has very great norm with each value of $$s$$, then $$Y(s)$$ is approximately equal to $$R(s)$$. This means we control the output by simply setting the reference.

Stability
Stability (in control theory) often means that for any bounded input over any amount of time, the output will also be bounded. This is known as BIBO stability (see also Lyapunov stability). If a system is BIBO stable then the output cannot "blow up" if the input remains finite. Mathematically, this means that for a causal linear continuous-time system to be stable all of the poles of its transfer function must OR
 * lie in the closed left half of the complex plane if the Laplace transform is used (i.e. its real part is less than or equal to zero)
 * lie on or inside the unit circle if the Z-transform is used (i.e. its modulus is less than or equal to one)

In the two cases, if respectively the pole has a real part strictly smaller than zero or a modulus strictly smaller than one, we speak of asymptotic stability: the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations, which are instead present if a pole has exactly a real part equal to zero (or a modulus equal to one). If a simply stable system response neither decays nor grows over time, and has no oscillations, it is referred to as marginally stable: in this case it has non-repeated poles along the vertical axis (i.e. their real and complex component is zero). Oscillations are present when poles with real part equal to zero have imaginary part not equal to zero.

Difference between the two cases are not a contradiction. The Laplace transform is in Cartesian coordinates and the Z-transform is in circular coordinates and it can be shown that
 * the negative-real part in the Laplace domain can map onto the interior of the unit circle
 * the positive-real part in the Laplace domain can map onto the exterior of the unit circle

If the system in question has an impulse response of


 * $$x[n] = 0.5^n u[n]$$

and considering the Z-transform (see this example), it yields


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

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

However, if the impulse response was


 * $$x[n] = 1.5^n u[n]$$

then the Z-transform is


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

which has a pole at $$z = 1.5$$ 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 root locus, Bode plots or the Nyquist plots.

Controllability and observability
Controllability and observability are main issues in the analysis of a system before deciding the best control strategy to be applied. 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 force the system to reach a level of controllability. 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 correct the closed-loop behaviour if such a state is not desirable.

From a geometrical point of view, if we look 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 behaviour in the closed-loop system. That is, if one of the eigenvalues of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the close-loop system which therefore will be unstable. Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis.

Solutions to problems of uncontrollable or unobservable system include adding actuators and sensors.

Control specifications
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 (es. 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: this both if the open-loop is stable or not. An inaccurate choice of the controller, indeed, can even worsen the stability properties of the open-loop system. This must normally be avoided. Sometimes it would be desired to obtain particular dynamics in the closed loop: i.e. that the poles have $$Re[\lambda] < -\overline{\lambda}$$, where $$\overline{\lambda}$$ is a fixed value strictly greater than zero, instead of simply ask that $$Re[\lambda]<0$$.

Another typical specification is the rejection of a step disturbance: this can be easily obtained by including an integrator in the open-loop chain (i.e. directly before the system under control). 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 specification is important: no real physical system truly behaves like the series of differential equations used to represent it in mathematical way. Sometimes a simpler mathematical model can be 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 determination of the equations of a model's dynamics is called model 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 spring-damp system we know that $$ \ddot(t) = - K x(t) - \Beta \dot{x}(t)$$. Even assuming that a "complete" model is used, all the parameters included in these equations (called "nominal parameters") are never known with absolute precision: therefore the control system will have to behave correctly even in presence of their true values.

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 control system can be performed in the frequency domain, considering the system's transfer function and using Nyquist and Bode diagrams. Topics include Phase margin and Amplitude margin. For MIMO and, in general, more complicated control systems one must consider the theoretical results devised for each control technique (see next section): i.e., if particular robustness qualities are needed, the engineer must shift his attention to a control technique including them in its properties.

Constraints
A particular robustness issue is the possibility of a control system to work even in presence of constraints. In the practice every signal is physically limited. It could happen that in its true working a controller will try to send signals that cannot be performed by the machinery: for example, trying to rotate a valve at excessive speed. This can provoke a bad behaviour of the closed-loop system, or even break up actuators or other subsystems. Specific control techniques are available to solve the problem: model predictive control (see later), and anti-wind up systems. The latter consists of another control block that is added to a previously synthesized controller, and ensures that the control signal never overcomes the threshold given.

Main control strategies
Every control system must guarantee first the stability of the closed-loop behaviour. For linear systems, this can be obtained by directly placing the poles. Non-linear control systems use specific theories (normally based on 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. Here a summary list of the main control techniques is shown:

PID controllers
Using a so called PID controller is probably the most used control techniques, being the simplest one. "PID" means: Proportional-Integral-Derivative, referring to the three types of sub.-system which can be added before the system under control. 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 tracking error $$e(t)=r(t)- y(t)$$, a PID controller has the general form


 * $$u(t) = K_P e(t) + K_I \int e(t)dt + K_D \dot{e}(t)$$

The desired closed loop dynamics can be easily obtained by adjusting the three parameters $$ K_P$$, $$ K_I$$ and $$ K_D$$. Stability can be ensured using only the proportional term, but the integral term permits the rejection of a step disturbance (often a striking specification in process control). The derivative term is often omitted. PID controllers are the easiest class of control systems: however, they cannot be used in several more complicated cases, especially if MIMO systems are considered.

Direct pole placement
For MIMO systems, pole placement can be performed mathematically using a State space representation of the open-loop system and calculating a feedback matrix assigning poles in the desired positions. In complicated systems this can require large computer-assisted calculation capabilities, and cannot always ensure robust results.

Optimal control
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 we could need to know the jet thrusts to give in order to bring it again in the desired trajectory after a perturbation, consuming as little fuel as possible in the process. Two classes of optimal controls have been widely used in industrial applications, as it has been shown they can ensure closed-loop stability also. These are Model Predictive Control (MPC) and Linear-Quadratic-Gaussian control (LQG). The first is the more successful one, as it can take account of the presence of constraints over the signals present in the systems, which is an important topic in many industrial processes (by the way, the "optimal control" structure in MPC is only a means to achieve such a result, as it does not optimize a true performance index as a closed-loop control system). Together with PID controllers, MPC systems are the most widely used control technique in process control.

See also:
 * Model predictive control
 * H infinity
 * Coefficient diagram method

Adaptive control
Adaptive control uses an on-line identification of the process parameters, obtaining strong robustness properties. Adaptive controls were applied for the first time in the Aircraft industry (later called Aerospace industry) in the 1950s, and have found particular success in that field.

Non-linear control systems
Processes in industries like Robotics and the Aerospace industry typically have strong non-linear dynamics. In control theory it is sometimes possible to linearize such classes of systems and apply linear techniques: but in many cases it can be necessary to devise from scratch theories permitting control of non-linear systems. These normally take advantage of results based on Lyapunov's theory.