## An extract on #comolake

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"). 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. The process output is the building 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. 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.' " 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.'" The advanced type of automation that revolutionized manufacturing, aircraft, communications and other industries, is feedback control, which is usually continuous and 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. The theoretical basis of closed loop automation is control theory.

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 poles of its transfer function must have negative-real values, i.e. the real part of each pole must be less than zero. Practically speaking, stability requires that the transfer function complex poles reside 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 Cartesian coordinates where the x {\displaystyle x} axis is the real axis and the discrete Z-transform is in circular coordinates where the {\displaystyle \rho } axis is the real axis. When the appropriate conditions above are satisfied a system is said to be asymptotically stable; the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations. Permanent oscillations occur when a pole has a real part exactly equal to zero (in the continuous time case) or a modulus equal to one (in the discrete time case). If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable; in this case the system transfer function has non-repeated poles at the complex plane origin (i.e. their real and complex component is zero in the continuous time case). Oscillations are present when poles with real part equal to zero have an imaginary part not equal to zero. If a system in question has an impulse response of x [ n ] = 0.5 n u [ n ] {\displaystyle \ x[n]=0.5^{n}u[n]} then the Z-transform (see this example), is given by X ( z ) = 1 1 0.5 z 1 {\displaystyle \ X(z)={\frac {1}{1-0.5z^{-1}}}} which has a pole in z = 0.5 {\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 x [ n ] = 1.5 n u [ n ] {\displaystyle \ x[n]=1.5^{n}u[n]} then the Z-transform is X ( z ) = 1 1 1.5 z 1 {\displaystyle \ X(z)={\frac {1}{1-1.5z^{-1}}}} which has a pole at z = 1.5 {\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 root locus, Bode plots or the Nyquist plots. 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.