3.3. Design of a Conventional H Synthesis
The conventional H
controller is constructed using smooth H
optimization in the frequency domain [
44]. The magnitudes of the closed-loop sensitivity transfer function, that is,
, and the complementary sensitivity transfer function, that is,
, are sculpted by complex weights. Note that
is the closed-loop transfer function between reference input
and output
, whereas
is the closed-loop transfer function between reference
and
. Note that
and
, where
is the loop transfer function. For a typical H
controller, the H
norm, which symbolizes the peak value of weighted frequency domain sensitivity
or its complementary sensitivity
, is reduced. Observe that
and
contain the constraints (peak specs) for modifying the controller parameters, where
and
are shaping weights of the closed-loop and complementary sensitivity transfer functions.
Since sensitivity represents the effectiveness of the closed-loop, it should ideally be very low. The peak requirements show a margin of robustness and avoid high-frequency noise amplification. The H
controller performs better than a traditional PID controller. The shaping weights and constraints are defined using the MIXSYN MATLAB tool. Based on the defined weights and constraints, the robust MIXSYN tool of MATLAB optimized a state-space structure of the H
controller to obtain robust performance. However, the order of this H
controller is equal to the order of the process model plus the order of the weights used to shape closed loops
and
, which is a significant shortcoming of the H
controller. This might not be a big deal in typical applications with plenty of computational resources, but in an industrial process setting, it is often a problem. The state-space results for the design of the H
controller are presented in
Section 4.
3.4. Design of the Genetic Algorithm for a PID Controller
The GA is a stochastic search technique that can be utilized to optimize challenging situations, including the case of both linear and nonlinear systems of equations. Instead of employing deterministic principles, the GA uses random transition rules and manages a population of alternative solutions termed individuals. Through an iterative process, a new generation of individuals is built by altering the chromosomes of the individuals of the current generation, as we can see in [
45]. The fitness of every individual is evaluated using the objective function. The GA performs mainly three operations: mutation, crossover, and selection. Suppose that a GA is used to minimize an objective function
f. Then, individual
is said to be a better solution than individual
if and only if
. This is the fundamental criterion in the selection operation of a GA. This optimization has high efficiency, but with a high computational cost as well.
In Algorithm 1, we can see the basic steps of a GA [
46]. We used this simple approach to carry out our computational experiments. Regarding the GA input,
f is the function to optimize,
n is the number of individuals in each generation (population size), and
iter defines the number of iterations—that is, the number of generations to be built. Just before ending, the GA returns the best solution found
bestSol, and the value obtained
bestFit, by evaluating the best solution in the objective function
f. The best fitness results for a GA-PID design for the continuous MEC process are given in
Section 4.
Algorithm1: General structure of a GA |
|
3.5. Design of the Proposed Fixed Order and Structured H Synthesis
Due to monolithic design and practical restrictions, conventional H
controllers have seen slow industry adoption. The norms of conventional H
controllers are further constrained by complex structure and design specifications, including response time and control bandwidth [
47]. High-order complex weighting filters may be utilized to improve the results. By employing non-smooth H
optimization in the frequency domain, the suggested robust optimization adjusts the essential control elements, such as PIDs. However, this complicates the structure of the ordinary H
controller. The constraints of the conventional H
controller are all overcome by fixed-structure H
controllers. The fixed-structure-based controllers have more practical importance and execute well in terms of response time and quality of the solution [
47]. The fixed and non-tunable elements, tunable control block, and standard formulation of the proposed fixed-structure H
synthesis are all represented in the following paragraph.
Figure 6 displays the standard form of H
synthesis, which consists of two key components: (i) the block
, which contains all the non-tunable (fixed) components of the whole control system, such as an LTI model of the continuous MEC process; and (ii) the second block, which contains the proposed synthesis required structure and fixed-order control elements. In the case of sophisticated multi input multi output (MIMO) systems [
31], all these elements are tunable. When it comes to SISO systems, this block contains just one configurable control element. The customizable diagonal block of tunable control components allows for decoupling complex MIMO processes, where each control element
has a known structure, which is presumed to be an LTI, and is established from
In
Figure 6, the tunable elements are placed in block/controller
, and the remaining elements are placed in block/process
. The error signal
is aggregated in
, with
being the current measurement and
is the reference input, whereas the external input, including the external disturbance
, measurement noise
, and reference inputs, are merged into
. The partition of the standard representation is given by
where
is the signal control. The closed loop objective function formulation from
to
is represented in linear fraction transformation form [
47] as
Optimizing the parameters of the tunable control elements, such as tunable PIDs, is another difficult task. Scalars such as
,
,
, and
are used as parameters for the transfer function of the PID controller, which is found by taking the Laplace transform with its variable
s, and obtained as
Parameters , , , and are tuned by non-smooth H optimization. For the controller structure to be effective and practical, the derivative control term is made up properly using coefficient as the time constant of the first-order filter.
Next, we detail how to optimize the suggested fixed-structure H
synthesis. The controller settings are adjusted in the frequency domain to adhere to standard design criteria. In the case of a SISO system, we must minimize the H
norm, which consists solely of the maximum values of the closed-loop transfer functions
and
across the entire frequency range. The standard form of the objective function
is provided as
whereas from the constraints stated in (
8), we can formulate the criteria for resilient design, such as external and internal disturbance rejection, noise elimination, high stability margins, improved control bandwidth, and enhanced transient specifications.
The appropriate complex weighting transfer functions and were selected to provide the desired forms of the closed-loop transfer functions and , respectively. The intended forms of and follow the design specifications stated as , for , and , for . The proposed fixed-structure H synthesis is independent of the complex weight order.
The structure of conventional H synthesis is bound by the order of the complex shaping weights. In the case of a conventional H controller, a more complex structure arises by using a high order shaping weights to enhance the robustness. For the LTI model of the MEC process in a continuous reactor, is a parametrically optimized and tuned robust PID controller. In MATLAB, the generalized form of the prescribed H synthesis problem is coded as follows:
ltiblock.pid(‘C’,‘pid’);
feedback(1,G(s)*C(s));
feedback(G(s)*C(s),1);
blkdiag(W*S(s),W*T(s));
where tf(1/M wb, 1 wb*A) and .
The generalized state-space structure (
formulation) has two outputs, two inputs, nine states, and a block
, which is a robust PID controller to be tuned. The structured controller must optimize all its parameters by a fast and robust algorithm [
41]. The suggested optimization starts with randomly chosen values of the structured controller parameter and concludes with required resilient parameter values that adhere to the predetermined limitations. To obtain the resilient optimal control parameters through the non-smooth H
optimization, the generalized
formulation is transformed into the
form—that is, the standard form of the objective function stated in (
8). The HINFSTRUCT tool in
MATLAB helped this optimization. The HINFSTRUCT tool employs the fast optimization and robust algorithm [
41] and adjusts the parameters by minimizing the closed-loop objective function gain between the system’s inputs and outputs.
Figure 7 depicts the schematic diagram for the system’s basic feedback loop design, and
Section 4 compares the simulation results of the structured H
synthesis with those of traditional and intelligent controllers.