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This paper presents the synthesis of an optimal robust controller with the use of pole placement technique. The presented method includes solving a polynomial equation on the basis of the chosen fixed characteristic polynomial and introduced parametric solutions with a known parametric structure of the controller. Robustness criteria in an unstructured uncertainty description with metrics of norm

Synthesis of closed-loop control system is based on a mathematical model of the plant, which should enable reliable prediction of input/output response, so that it can be used in controller synthesis. Uncertainty of the plant and external disturbance are the central issues of robust closed-loop control. Modern robust control approaches are mostly presented as a set of optimization problems, where the solution and the optimization procedure depend on the mathematical property of the objective function. Many problems are naturally not convex

Robust control paradigms, such as

The main objective of this paper is to present a robust polynomial approach for unstructured uncertainty, where the order of weighting functions does not influence the order of the synthesised controller. The influence of parametric solutions on the value of the norm

This paper is organized as follows. The second section describes the solutions of the parametric polynomial equation in a matrix form. Based on the second section, the property of parametric solutions, introduced controller parameterizations and their influence on controller feasibility and polynomial equation solvability are proposed in Section

Let us consider a given feedback system, with nominal plant

Negative feedback configuration.

Plant and controller configuration is shown in Figure

Polynomials

Complementary sensitivity and sensitivity functions are

Controller coefficients

Pole placement synthesis of the controller is based on the choice of the characteristic polynomial (

The Diophantine equation (

The number of parametric solutions is equal to

Matrix form of the polynomial equation (

Solution of (

As we have mentioned in the introduction, the polynomial approach is similar to the well-known

After stabilization of the closed-loop system with the selected closed-loop polynomial

Closed-loop system with input, output disturbance, and reference signal.

Good reference tracking of closed-loop system

The system provides good tracking on the set of reference signals with the frequency span

The same effect as with tracking performance can be achieved also for input disturbance

And for output disturbance

From (

In some cases, for example, with higher operation safety or real time computation requirements, a strong stabilization system must be ensured. To be able to synthesize a strong stabilization controller the plant must fulfil the parity interlacing property (PIP) condition

Parameterized controller with an added known structure is

The degree of the characteristic polynomial

Sensitivity function with controller (

Minimizing the tracking error

Sensitivity function for step reference signals and disturbances with an added proxy structure of integral action

The value of parameter

The same properties can be introduced for ramp reference signals and output disturbances,

For tracking harmonic reference signals and harmonic disturbance (HD) rejection, a structure with stable complex roots

Selected parameter

Tracking and rejecting performance is slightly dependent on frequency

The same property can be used for step and low frequency signals. If inequality

Proper selection of coefficients

Auxiliary parameters

After the first step of controller design, that is, selection of closed-loop poles and structure of the controller with a set of parametric solutions, we prepare leeway for further optimization of robustness criteria. Robustness criteria with consideration of

The

with an optimal solution:

Pole placement with controller parameterization which involves free parameters

As we have mentioned in Section

Partial fraction decomposition of the plant

Remainder

The synthesized controller

All three properties are achieved with decomposition of

Robust stability property (

Robust stability property (

Robust stability property (

Transformation of robust criteria (

The norm

Spectral polynomial

From condition (

Robust condition (

Robust condition (

Robust condition (

With simple algebra we can determine the necessary edge conditions of controller parameters for ensuring nonnegativity of the spectral polynomials (

Edge conditions (

In first example, the guidance of the autonomous flight systemquadcopter is taken into account. The system has already been internally stabilized around pitch and roll axis with gyro sensors (hovering mode ability). The problem of the presented controller design example is how to ensure proper guidance ability of the system to follow a desired path trajectory in three-dimensional spaces. The system has the following for degree of freedom 4DOF: altitude-

4DOF quadcopter guidance problem.

The measured data of quadcopter’s position and orientation were obtained from gyro, acceleration, and magnetic sensors with additional reconstruction measure filtering with Kalman recursive algorithm. Altitude measurement was performed with a barometric pressure and ultrasonic distance sensor. Nominal model of the plant in matrix form

For control design robust-mixed sensitivity problem (

Mixed sensitivity problem (

overshot

settling time

steady state error for all four axes:

robust-stabilized system and good output low-frequency disturbance rejection over frequency band (0-0.1 rad/s) and good low-frequency reference signal tracking over (0–0.23 rad/s) for altitude control and (0–0.21 rad/s) for longitudinal movement and orientation;

strong stabilization system.

According to control requirements, the controller structure for each axis can be determined. The selected controller structure for longitudinal movement

Altitude controller

The acceptable interval limit for free parameter

Selected yaw controller structure

The results of the optimization procedure with DE algorithm are described in

Altitude controller

Yaw controller

A closed-loop system responses to the step reference signal and output periodic disturbance.

Longitudinal movement control corresponds to

Longitudinal

Altitude control with controller

Altitude control with controller

Orientation-yaw control with controller

Orientation (yaw) control with controller

A trajectory following with controllers

Quadcopter trajectory following.

The quadcopter guidance controllers

The second design example is a stabilization of the single-axis magnetic ball suspension system.

The ball (1) in Figure

Single-axis magnetic system.

Used weights are

Simplification of controller structure with partial-fractional decomposition on

The new weights

overshot:

settling time:

the tracking error for step response and periodic reference signal over frequency band (0–4 rad/s):

robust stabilized system and good output low frequency disturbance rejection (

strong stabilization system.

Selected controller structure

The results of the optimization procedure are

The closed-loop system responses to the step reference signal and output periodic disturbance.

System response on step reference signal with two different output disturbances, with frequency

System response on step reference signal with two different output disturbances.

Step reference tracking with output disturbance Figure

Step reference tracking with output disturbance.

Periodic reference tracking with periodic output disturbance Figure

Periodic reference tracking with output disturbance.

The closed-loop systems with controller

The proposed method with transparent composed controller structure and optimization of robust criteria with free parameters in polynomial equation provides an efficient tool for feedback system design. Controller parameterization, with known characteristics, such as integral action, double-integrator, and Noch property, and their approximation with stable structure significantly improve feedback performance (overshot, settling time, overall feedback dynamic) and the ability of further implementation on a real system (operation safety, simplification of real-time algorithm). Fraction-partial decomposition is also a useful approach for a simplified controller structure in the polynomial approach, where exact feasibility of the controller depends on the plant degree. It is worth emphasizing that fractional decomposition is limited to plants with strongly expressed dominant and unstable poles which have the highest stored energy of the system (e.g., electromechanical system with a dominant mechanical part). The method can be easily used in combination with metric