As of this writing, Wolfram|Alpha can only analyze linear state space models, which are expressible as a system of vector differential equations: Wolfram|Alpha uses the standard convention, in which input variables are denoted by u, output variables by y, and state variables by x. Some of these variables are the “input, others are the “output, and some are just internal to the system, called “state” variables. In a state space model representation, variables are related to each other using ordinary differential equations. Wolfram|Alpha is equipped to handle both continuous- and discrete-time systems modeled using this framework. These models provide a general framework to mathematically describe the input-output behavior of systems and are commonly used to model a complex process, whether a natural phenomenon, a control system, or something else.
In addition to this repertoire of standard control systems, which we will be expanding in the near future, Wolfram|Alpha can also understand and analyze custom systems, specified using either state space or transfer function models. Type “ PID controller, and Wolfram|Alpha gives you a custom “calculator” that allows you to vary the strength of the proportional, integral, and derivative “gains” and shows how these variations affect the properties of the controller: Usually in the form of a programmable microchip, a PID controller can electronically control something based on the value (or a proportion of the value), the total accumulation ( integral), and the rate of change ( derivative) of some quantity. Typing “ control system: integrator” into Wolfram|Alpha gives a list of properties, graphs, and an interactive response plot where you can see what happens to the output for several common inputs:Īnother standard type of control system Wolfram|Alpha knows about is the PID controller. Put another way, gas is consumed at a rate proportional to how far the gas pedal is depressed. For example, the amount of gas your car has used since you last filled up is basically an integral of the position of the gas pedal over that time. An integrator is a system that accumulates the values of some input over time. We have now incorporated these capabilities into Wolfram|Alpha with the ability to understand some common systems by name.Ī simple example is the integrator. In Version 8 of Mathematica, we added lots of functions that analyze, design, and simulate a variety of systems, including control systems. Most control systems are like this: sensors feed information back into the system, which is then used to decide on an action. By design, the thermostat creates a kind of closed loop called a “negative feedback loop, which tends to stabilize the temperature around a desired value. There is a closed loop of causation formed between the home’s temperature and the state of the furnace. With a thermostat installed, the reverse is also true: the state of the furnace depends on the temperature of the house (it comes on if the temperature is too low, and turns off if the temperature is too high). The temperature of your home depends on many factors, especially how long and how recently the home’s furnace was on. Using various kinds of sensors and actuators, these systems automatically control most common appliances, industrial processes, and even your body’s own biological processes! A control system is any system or device that controls or regulates the behavior of another system. What do your alarm clock, thermostat, coffeemaker, car radio, anti-lock brakes-and almost every other electrical and mechanical device you encounter in your daily life-all have in common? They are all examples of “control systems, one of the most ubiquitous yet unseen modern technologies.
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