control system

The sole objective of control system is to generate feasible inputsto the plant (e.g. dynamic systems) such that it will operate as itintended to under a wide range of operating conditions.

Examples of control systems: Cruise control, auto pilot, rice cooker.

For a general finite dimensional dynamic system in its ODE form,

	$\\displaystyle\\dot{x}$	$\\displaystyle=f(x(t),t)+g(x(t),u(t),t),$		(1)
	$\\displaystyle y$	$\\displaystyle=h(x(t),t),$

where $x\\in\\mathbb{R}^{n}$ is the state, $y\\in\\mathbb{R}^{l}$ is theoutput and $u\\in\\mathbb{R}^{m}$ is the control input of the system.In the control literature, equation 1 is general referredas the plant, where the function $f:\\mathbb{R}^{n}\\times\\mathbb{R}\\rightarrow\\mathbb{R}^{n}$ governs the system dynamics, $g:\\mathbb{R}^{m}\\times\\mathbb{R}^{n}\\times\\mathbb{R}\\rightarrow\\mathbb{R}^{n}$ determines how the input (control signals)influence the state $x$ via actuators (e.g. gas turbine) and $h:\\mathbb{R}^{n}\\times\\mathbb{R}\\rightarrow\\mathbb{R}^{l}$ determines how the state generates the output signal. If $m$ is equal to $n$ , the plant isfully actuated. If $m<n$ then the plant is under actuatedand otherwise the plant is over actuated. For a plant that is notexplicitly dependent in time $t$ , such system is called aAutonomous system. The maindifferece between a control system and a general dynamic system isthe additional signal $u(t)$ .

For example, to control an airplane, the control system has to controlthe thrust, flaps, aileron and rudder, which theyare the control signals of the system $u$ . Those control inputinfluence the system state $x$ such as speed (with thrust),attitude and orientation (with flaps, aileron and rudder).To physically alter the state of the airplane, actuators such asgas turbines are needed, which are controlled by the control signals $u$ .

The control signal $u$ can be generated in a closed-loop fashionor open-loop fashion. An open-loop control system generates $u$ with the user, or operator supplied reference state $x_{d}$ or output $y_{d}$ only; meanwhile closed-loop control system uses both referenceand feedback signals that are usually measured fromsensors. In the airplane attitude control example, the desiredattitude is usually represented in roll-pitch-yaw anglerepresentation, and these signals are measurable by attachingsensors to the flaps, aileron and rudder. In engineeringpractice, only closed-loop control systems should be used, sinceopen-loop systems are not robust against uncertainties, modelingerrors and measurement errors.

If a closed-loop control system is based on state feedback, suchcontol system is called a state-feedback control system. By thesame token, a output-feedback control system is based on outputfeedback only. Notice that output signals are available for feedbackby definition, however in reality not all the states are mesurable. Ifa state-feedback control system with all the states available forfeedback, it is called a full-state feedback system and otherwiseis call partial-state feedback system, which usually requires a stateobserver (e.g. Kalman filter) to estimate the unavailable states.

To illustrate the simple concept of control systems, we will use asimple example. A truck driver is required to travel 1000 Km in 10hours. To relive the stress on the driver’s heel, he has placed a stickto the gas paddle so the car travels at $\\,\\mathrm{100Km/h}$ . Under perfectconditions, the driver will reach the destination in the allocatedtime. However, a certain section of the road is up-hill, so the truckslowed down by a considerable amount and will not arrive it’sdestination in time. To remedy this problem, the driver ’implemented’a simple solution using the speed-o-meter such that the gas paddle position $p_{set}$ of the truck is now depends on the current speed $v_{current}$ of the truck, $p_{set}=-K(v_{current}-\\,\\mathrm{100Km/h})$ ,where $K$ is just an adjustable parameter. So if the truck is runningtoo slow (e.g. up-hill), $p_{set}$ will be positive (more gas to theengine) hence speed will increase to maintain the desired speed, sovice-versa for the down-hill case.

In this example, we have outlined all the major components of atypical control system:

•
Actuator: engine,
•
Sensor: speed-o-meter,
•
Plant: truck,
•
Control input: gas paddle,
•
Control objective: 1000 Km in 10 hrs,
•
Control law: $p_{set}=-K(v_{current}-100Km/h)$ ,

The science aspect of control systems is the study ofdesign, synthesis and analysis of control systems using mathematicalconcepts, and the engineering aspect of controls systems is toimplement, construct and adjust the control system according toreal-life situation and limitations.