| Summary: | 博士 === 國立中正大學 === 化學工程研究所 === 91 === The purposes of this disseration are to apply the notion of principal point to
custruct boundaries of equality constraint set and using the result to PID controller
dsign with differnt constraints. The concept is applying developed analytical expressions
and well-known path-following algorithm to find the feasible domain in parameter space.
In this disseration, we use our result to (1) inversion-based PID controller design,
(2) designing a stabilizing PID controller based on minimizing IE subjecting to
constraints on maximum sensitivity and/or complementary sensitivity amounts,
(3) mixed $H_2/H_\infty$ optimal PID controller design, (4) robust stabilization of
interval plants using PID controllers.
Let the open-loop transfer function be specified a form of $L(s)=$ $G_c(s)G_p(s)$ $=k_a e^{-\tau_d s}/s$.
If the process transfer function $G_p(s)$ is a first or second order model with time selay,
the controller derived from dynamic inversion method is the PID type controller. In this
disseration, we present a systematic approach to construct a simple formula for
gain $k_a$ in terms of time-delay $\tau_d$ and overshoot $\kappa$ of closed-loop
system. The proposed approach is also extend to develop digital controller tunning rule.
As shown by \AA str\"om et al. (1998), the problem of designing a stabilizing PI
controller based on IE associated with step load
disturbance while subjecting to constraints on maximum sensitivity and/or complementary
sensitivity amounts to finding the maximum allowable integral gain.
The latter problem is a non-convex optimization problem whose true solution cannot be
obtained with a guarantee by a gradient-based search algorithm. In this disseration,
we present a novel and effective approach to solve such a non-convex
optimization problem base on constructing boundaries of equality constraints set.
Hence, by constructing the boundary of the feasible domain in the controller gain space,
the maximum allowable integral gain
can be obtained. In addition to having the ability to obtain global
optimal solution, our approach can handle sensitivity and
complementary sensitivity constraints simultaneously without using an iterative procedure.
In this disseration, we also propose a new approach to solve the problem of designing
optimal PID controllers that minimize an $H_2$-norm associated with the set-point response
while subjecting to a constraint on the $H_\infty$-norm of disturbance rejection.
The proposed design approach consists of constructing the feasible domain in the
controller gain space and searching over the domain for the optimal
gain values of minimizing the $H_2$-norm objective function. The construction of the
feasible domain that satisfies the requirements of closed-loop stability and
$H_\infty$-norm constraint is achieved through analytically characterizing the domain
boundary with concept of principal points. The constructed feasible domain saves greatly
computational effort from search for the $H_2$-optimal controller gain values that satisfy
both the stability and disturbance rejection requirements.
We also considers the problem of determining robustly stabilizing PID controllers for
interval plants. We attack the problem of robustly stabilization of an interval plant
family by solving the problems of stabilizing 16 specially selected segment plants.
The solution approach is based on using the concept of D-partition technique. Given an
$n$th-degree closed-loop characteristic polynomial $p(s;\bk,\lambda)$, where
$\bk=(k_p, k_i,k_d)$ represents the PID controller gain vector and $\lambda$ is an
uncertain parameter varying in the interval $[0,1]$, the robust $D$-partition boundary
regions in the gain space $\bR^3$ are respectively defined by
$\bS_0=\{\bk\in\bR^3: p(0;\bk,\lambda)=0,\forall \lambda\in[0,1]\}$,
$\bS_\infty=\{\bk\in\bR^3: p(\infty;\bk,\lambda)=0,\ \forall \lambda\in[0,1]\}$ and
$\bS_\omega=\{\bk\in\bR^3: p(\j\omega;\bk)=0,\ \forall \omega\in (0,\infty),\ \lambda\in[0,1]\}$.
Using the Orlando''s formula, we show that the three robust D-partition boundary regions
$\bS_0,\bS_\infty,$ and $\bS_\omega$ are included in the region
$\bS=\{\bk\in\bR^3: \phi(\bk,\lambda)=0,\ \forall \lambda\in[0,1]\}$, where
$\phi(\bk,\lambda)=p(0;\bk,\lambda)\triangle_{n-1}(\bx,\lambda)$ and $\triangle_{n-1}(\bk,\lambda)$
is the $(n-1)$th Hurwitz determinant of the polynomial $p(s;\bk,\lambda)$.
An analytical characterization of the boundary of set $\bS$ is derived using the notion of
principal points.
By tracing the boundary of the set $\bS$, we can construct the stabilizing PID controller
stability region for a segment plant. By taking the intersection of the stabilizing PID
controller parameter domains of 16 specially selected segment plants, we can finally obtain
the entire set of stabilizing PID controllers an interval plant family.
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