Robust stability design of PID controllers for arbitrary-order transfer functions with uncertain time delay

Abstract

In this paper a graphical technique is introduced for finding all proportional integral derivative (PID) controllers that satisfy the robust stability constraint of a given singleinput-single-output (SISO) linear time-invariant (LTI) system of any order system with time delay. This problem can be solved by finding all PID controllers that simultaneously stabilize the closed-loop characteristic polynomial and satisfy constraints defined by a set of related complex polynomials. A key advantage of this procedure is that it is only depends on the frequency response of the system and does not require the plant transfer function coefficients. The ability to include the time delay in the nominal model of the system will often allow for designs with reduced conservativeness in plant uncertainty and an increase in size of the set of all PID controllers that robustly stabilize the system.

A unified approach for robust stability design of PID controllers

“…Consequently, all PID controllers that satisfy (8) must lie at the intersection of all controllers that satisfy (10) for all [ , ) T θ π ∈ 0 2 [26].…”

“…Setting ω = 0 in (14), we (15) and (16) the continuous-time robust stability boundaries, which are equivalent to those in [26], are found.…”

“…In [23], [24], [25], [26], and [27] and the authors of these papers developed techniques for finding all achievable PID controllers that simultaneously stabilize the closed-loop continuous-time system and satisfy an H ∞ sensitivity, H ∞ complementary sensitivity, weighted sensitivity, robust stability, or robust performance constraint, respectively. In [28], [29], and [30], this method was extended to a unified approach for continuous-time and discrete-time H ∞ sensitivity, H ∞ complementary sensitivity, or weighted sensitivity design of PID controllers.…”

Stability domains of the delay and PID coefficients for general time-delay systems

“…However, the effect of the small delay perturbations is investigated on the stability. An uncertainty delay range is considered by (Emami & Watkins, 2009) to present a graphical method for computing all robust stabilizing controllers. Again, the delay is assumed a fixed value, the mean of the uncertainty range.…”

Robust performance design of PID controllers with inverse multiplicative uncertainty

“…In [23], [24], [25], [26], and [27] the authors of this papers developed techniques for finding all achievable PID controllers that simultaneously stabilize the closed-loop continuous-time system and satisfy an H ∞ sensitivity, H ∞ complementary sensitivity, weighted sensitivity, robust stability, or robust performance constraint, respectively. In [28], [29], [30], and [31], this method was extended to a unified approach for continuous-time and discrete-time H ∞ sensitivity, H ∞ complementary sensitivity, weighted sensitivity, or robust stability design of PID controllers, respectively.…”