System classifications

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Linear systems control
Nonlinear systems control
Decentralized systems controlDeterministic and stochastic systems control
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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 computer-assisted calculation capabilities, and cannot always ensure robustness. Furthermore, all system states are not in general measured and so observers must be included and incorporated in pole placement design.
Processes in industries like robotics and the aerospace industry typically have strong nonlinear 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 nonlinear systems. These, e.g., feedback linearization, backstepping, sliding mode control, trajectory linearization control normally take advantage of results based on Lyapunov's theory. Differential geometry has been widely used as a tool for generalizing well-known linear control concepts to the non-linear case, as well as showing the subtleties that make it a more challenging problem. Control theory has also been used to decipher the neural mechanism that directs cognitive states.
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.
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.
Turning on a triac is easy: Typically digital logic is connected so its VCC is connected to the triac A1 pin. Some digital logic output pin is connected to a small resistor connected to the triac gate pin. When the digital logic pulls the gate pin low (towards VDD), the triac is triggered and turns all the way on.
As long as the triac is on, the A1 and A2 pins act like they are shorted together. Current can flow in both directions through the triac -- whether A2 is higher or lower than the A1 voltage.
Turning the triac off is a little more difficult. First the digital logic output drives the gate pin high (so it is the same voltage as the common A1 pin). But the triac remains on until the current through the triac drops to less than the holding current. Normally triacs are used with 50 Hz or 60 Hz AC, so the triac may stay on as long as 10 ms after the digital logic tries to turn it off.
After the triac is off, A2 acts like is disconnected and isolated from the rest of the triac (as long as the external circuit doesn't drive it outside its voltage rating, typically plus or minus several hundred volts).
Driving a triac is very similar to driving PNP transistor -- "gate" analogous to "base", "A1" analogous to "emitter", "A2" analogous to "collector".
By far, the most common silicon device to drive AC loads connected to mains voltage is the triac. A triac works better than a BJT or a FET transistor as a switch when controlling AC power. A triac can remain off while A2 can swing up and down hundreds of volts (relative to A1 and the gate). It's difficult to keep a BJT or FET transistor from spontaneously turning on in the "reverse" direction.

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