Riassunto teoria Process Control

Lezione 1

The determination of the trend of the control variable is carried out through a component called controller. It often happens that the controlled variable is measurable and available to the regulator; then the control action impressed on the process also depends on the trend of the controlled variable and the controller is said to be in closed loop or in feedback (feedback). If the noise is measurable and the control variable, in open or closed loop, depends on it, we say that the controller performs a noise compensation. On the other hand, open-loop controllers are “blind”, in the sense that they have no way of detecting the consequences of the process being in perturbed, rather than nominal, conditions.

Def. The transducers measure the relative physical quantities by means of suitable sensors and then process the information making them compatible with the controller technology. The actuator converts the variables produced by the controller into manipulatable variables, which are related to the process.

Lezione 2

Assumptions needed to adopt a transfer function

• LTI systems: Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to those inputs. Time-invariant systems are systems where the output does not depend on when an input was applied.
• Continuous time systems: A system is continuous-time when its I/O signals are continuous-time.
• SISO systems: Single input single output systems.

Def. Transfer function of a control system is defined as the ratio of the Laplace transform of the output variable to Laplace transform of the input variable assuming all initial conditions to be zero. () = () ∙ ()

Open loop transfer function is strictly proper, so the output depends on the input not directly, but only through the state. ()/(1 + ())

Closed loop transfer function The equation shows a Plus (+) sign in the denominator representing negative feedback. With a positive feedback system, the denominator will have a Minus (−) sign and the equation becomes: 1 - GH. When H = 1 (unity feedback) and G is very large, the transfer/ → 1. function approaches unity as: Also, as the systems steady state gain G decreases, the ()/(1 + ()) expression of decreases much more slowly. In other words, the system is fairly insensitive to variations in the systems gain represented by G, and which is one of the main advantages of a closed-loop system.

If all poles and zeroes of the transfer function are in the left part of the complex plane, we are in the minimum phase system case. In control theory, a LTI system is said to be minimum phase if the system and its inverse are causal and stable. One of the most useful properties of minimum phase system is that they have an impulse response that is the most compact in time as is possible for any given amplitude function. If the bandwidth is wide, the response of the system is shorter.

Lezioni 3-4

There are limitations on the wideness of the bandwidth because of measurements disturbance. The () = −()/(1 + () ) measurement disturbances sensitivity function is the transfer function describing the effect of the measurement disturbance on the controlled variable (output). If I want to get an attenuation of the effect, the magnitude of the sensitivity function has to be lower than 1 (negative in dB). It is impossible to have the attenuation of measurement disturbance for any.

If we have problems with the design of a controller system, we can use the Pre-filter. The role of pre-filter is to enlarge the closed loop bandwidth a posteriori.

Assumptions for the pre-filter

• Asymptotically stable
• Proper system (number of poles equal to the number of zeroes)
• Unitary gain

The pre-filter can be used to:

• Solve design requirement non solvable by the CL controller
• Enlarge the CL bandwidth of controlled system a posteriori.
• HIGH-PASS FILTER
• Smooth the abrupt peaks of the reference signal. LOW-PASS FILTER (consider not to filter too much because the overall bandwidth of the system is reduced)
• Perform static compensation in a non-robust way (typically done when you have a constant reference signal, the so-called set point).

= lim () = 0: () We want that the permanent error this is ensured by at least type 1 (transfer →∞ function with at least 1 pole in the origin), but sometimes the pole in the origin is critic and for this problem we can add a pre-filter, in order to improve a static performance without introducing the pole. lim () = (0) ∙ (0) and we have to possibility: →0 () 1 () =- such that the static gain is (dynamic pre-filter) () (0)1() = =- (static pre-filter) (0)

Lezione 5

The Parallel compensator aim is to enlarge the CL bandwidth, and so to improve the promptness of the system, in another way. The action of this compensator is added to the output of the controller to get modified the input. ()()+()()′ () ()() = The new CL transfer function becomes: , and we want = 1 (unitary 1+()()′ ()). static gain of The assumptions are: ()- strictly proper (number of poles > number of zeroes) ()- must be causal (output depends only on the present and the past inputs) () () The schemes based on and are two degrees of freedom control schemes.

Lezione 6

The measurements disturbances have harmonics in the high frequency. The process disturbances have harmonics more or less in the same range of the reference signal. When I try to push down the magnitude of one of the two sensitivity functions, the other one tends to go up. 1, ≤ ()() () () = = ≅ {|()|, > 1+()() 1+() 1 , ≤ 1 () = ≅ {|()|1+() 1, > If we are