P&id analisis y diseño

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PID Control System Control System Analysis and Design


ith its three-term functionality offering treatment of both transient and steady-state responses, proportional-integral-derivative (PID) control provides a generic and efficient solution to realworld controlproblems [1]–[4]. The wide application of PID control has stimulated and sustained research and development to “get the best out of PID’’ [5], and “the search is on to find the next key technology or methodology for PID tuning” [6]. This article presents remedies for problems involving the integral and derivative terms. PID design objectives, methods, and future directions are discussed.Subsequently, a computerized, simulation-based approach is presented, together with illustrative design results for first-order, higher order, and nonlinear plants. Finally, we discuss differences between academic research and industrial practice, so as to motivate new research directions in PID control.

STANDARD STRUCTURES OF PID CONTROLLERS Parallel Structure and Three-Term Functionality
The transferfunction of a PID controller is often expressed in the ideal form





GPID (s) =

1 U(s) = KP 1 + + TD s , E(s) TI s


where GPD (s) and GPI (s) are the factored PD and PI parts of the PID controller, respectively, and α= 1± √ 1 − 4TD /TI > 0. 2

where U(s) is the control signal acting on the error signal E(s), KP isthe proportional gain, TI is the integral time constant, TD is the derivative time constant, and s is the argument of the Laplace transform. The control signal can also be expressed in three terms as 1 U(s) = KP E(s) + KI E(s) + KD sE(s) s = UP (s) + UI (s) + UD (s),

THE INTEGRAL TERM Destabilizing Effect of the Integral Term


Referring to (1) for TI = 0 and TD = 0, it can be seenthat adding an integral term to a pure proportional term increases the gain by a factor of

where KI = KP /TI is the integral gain and KD = KP TD is the derivative gain. The three-term functionalities include: 1) The proportional term provides an overall control action proportional to the error signal through the allpass gain factor. 2) The integral term reduces steady-state errors throughlow-frequency compensation. 3) The derivative term improves transient response through high-frequency compensation. A PID controller can be considered as an extreme form of a phase lead-lag compensator with one pole at the origin and the other at infinity. Similarly, its cousins, the PI and the PD controllers, can also be regarded as extreme forms of phaselag and phase-lead compensators, respectively.However, the message that the derivative term improves transient response and stability is often wrongly expounded. Practitioners have found that the derivative term can degrade stability when there exists a transport delay [4], [7]. Frustration in tuning KD has thus made many practitioners switch off the derivative term. This matter has now reached a point that requires clarification, as discussedin this article. For optimum performance, KP, KI (or TI ), and KD (or TD) must be tuned jointly, although the individual effects of these three parameters on the closedloop performance of stable plants are summarized in Table 1.


1 = jωTI


1 > 1, for all ω, 2 ω2 TI


and simultaneously increases the phase-lag since 1+ 1 jωTI = tan−1 −1 ωTI < 0, for all ω. (6)

Hence, bothgain margin (GM) and phase margin (PM) are reduced, and the closed-loop system becomes more oscillatory and potentially unstable.

Integrator Windup and Remedies
If the actuator that realizes the control action has saturated range limits, and the saturations are neglected in a linear control design, the integrator may suffer from windup; this causes low-frequency oscillations and leads to...