An ℋ_∞ Optimal Robust Pole Placement with Fixed Transparent Controller Structure on the Basis of Nonnegativity of Even Spectral Polynomials

Abstract

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 ℋ_∞ are for a more reliable and effective formulation of objective functions for optimization presented in the form of a spectral polynomial with positivity conditions. The method enables robust low-order controller design by using plant simplification with partial-fraction decomposition, where the simplification remainder is added to the performance weight. The controller structure is assembled of well-known parts such as disturbance rejection, and reference tracking. The approach also allows the possibility of multiobjective optimization of robust criteria, application of mixed sensitivity problem, and other closed-loop limitation criteria, where the common criteria function can be composed from different unrelated criteria. Optimization and controller design are performed with iterative evolution algorithm.

1. Introduction

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 ‎[1], but few of them can be presented as quasiconvex or convex problems in a well-known LMI framework [2–5]. However, convex problems become difficult to solve if some constraints are imposed on the controller coefficient or on the feedback gain matrix in the state space approach. Linear convex problem with an imposed structure of the feedback loop or of the controller is called Structured Linear Control problem (SLC) ‎[6] and includes a large class of various problems, such as fixed-order feedback, norm bounded gain controller, and multiobjective optimization problems. All these problems are classified as NP-hard, with many excellent solving procedures [7–10] and different separate optimization aspects such as optimal solution, stable controller, robust stability, robust performance, and others [11–13]. Many of these optimization conditions in previous citations are separate and unrelated to each method. Proper solutions of solving algorithms depend on an intersection of conditions, where only suboptimality is guaranteed, with no further information about the existence of the solution [6, 13]. Our approach proposes an optimal solution with combined weighted objective function, where each single weight is selected by the importance of optimization criteria. The proper solution of the problem can be proved before the optimization starts.

Robust control paradigms, such as ℋ_∞, ℋ₂, and 𝜇-synthesis, enable design of robust controllers with consideration of the uncertainty of nominal plants [14–16]. Plant uncertainty for LTI systems is usually formulated as unstructured or structured uncertainty, where different uncertainty models ‎[16] and different descriptions of the uncertainty parameter space ‎[17] require different approaches. In this paper, we will deal only with unstructured uncertainty for different models. Despite the excellence and quality of recent approaches, the developed methods also have certain limitations. The most notable limitation of the synthesis is the influence of the weighting function structures on the controller order and on hidden performances of the feedback system. Mostly, the design of robust controller is a compromise between the controller order and the precision of the plant uncertainty description ‎[15], which requires many iterations and changes by weight selection and plant-order reduction. Also the performance of the closed-loop system is conditioned by the optimization procedure and the selected objective function, where the final closed-loop performance is fuzzy related to the designer’s initial dynamic criteria. Possible solutions of the presented problem can be dealt with by choosing a fixed characteristic polynomial and introducing parametric solutions. Fixed characteristic polynomial sets the stability and dynamic performance of the controlled system. Parametric solution in a polynomial equation is closely related to Y-K parameterization and coprime factorization [18, 19], with a crucial difference: the parametric solution does not influence the characteristic polynomial. Closed-loop stability and performance do not depend on the optimization procedure. In other words, the design procedure of the robust controller has two steps. The first step of the method is selecting a fixed characteristic polynomial with a set of parametric solutions. The second part of the design procedure is ensuring robust stability and optimal closed-loop performance with no back impact on the first step. The chosen characteristic polynomial remains unchanged during the whole design procedure, which means that the position of closed-loop poles is fixed.

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 ‖⋅‖_∞ will be presented. The objective function of the optimization procedure can be composed from different unrelated criteria, like robust stability, strong stabilization, reference tracking, disturbances rejection, and so forth, with prior defined region of suboptimal solution on the basis of Šiljak’s stability criterion [20, 21]. The condition of robustness is tested with metric ℋ_∞, similarly as in papers [22–24], where conditions of robustness are formed in even spectral polynomials with a positivity condition. Such formulation of robust criteria allows for testing the robustness on the basis of polynomial coefficients and positivity conditions, with optimization algorithm (GA) and differential evolution (DE) ‎[25].

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 3. Sections 4 and 5 describe the influence of parametric solutions on the norm ‖⋅‖_∞ and the assessment of robust criteria with a spectral polynomial nonnegativity test. Section 5 concludes with a formulation of cost functions for multiobjective optimization with DE. Section 6 describes synthesis examples to show the effectiveness of the proposed approach. This is followed by a conclusion.

2. Pole Placement and Parametric Solutions

Let us consider a given feedback system, with nominal plant 𝑃₀(𝑠) and controller 𝐾(𝑠) in direct branch (‎Figure 1).

Figure 1 

Negative feedback configuration.

Plant and controller configuration is shown in Figure 1: where 𝑃₀(𝑠) and 𝐾(𝑠) are as the following: 𝑃₀(𝑠)=𝐵(𝑠)𝐴(𝑠)(𝑠)=𝐿(𝑠),𝐾𝑅(𝑠). (2.1)

Polynomials 𝐴(𝑠), 𝐵(𝑠), 𝑅(𝑠), and 𝐿(𝑠) can be written as the following: 𝐴(𝑠)=𝑎_𝑛𝑠^𝑛+𝑎_𝑛−1𝑠^𝑛−1+⋯+𝑎₁𝑠+𝑎₀,𝐵(𝑠)=𝑏_𝑚𝑠^𝑚+𝑏_𝑚−1𝑠^𝑚−1+⋯+𝑏₁𝑠+𝑏₀,𝑅(𝑠)=𝑟_𝑗𝑠^𝑗+𝑟_𝑗−1𝑠^𝑗−1+⋯+𝑟₁𝑠+𝑟₀,𝐿(𝑠)=𝑙_𝑘𝑠^𝑘+𝑙_𝑘−1𝑠^𝑘−1+⋯+𝑙₁𝑠+𝑙₀. (2.2)

Complementary sensitivity and sensitivity functions are 𝑇(𝑠)=𝐵(𝑠)𝐿(𝑠)(𝐴(𝑠)𝑅(𝑠)+𝐵(𝑠)𝐿(𝑠))⁻¹=𝐵(𝑠)𝐿(𝑠)𝐶⁻¹(𝑠),𝑆(𝑠)=𝐴(𝑠)𝑅(𝑠)(𝐴(𝑠)𝑅(𝑠)+𝐵(𝑠)𝐿(𝑠))⁻¹=𝐴(𝑠)𝑅(𝑠)𝐶⁻¹(𝑠), (2.3) where 𝐶(𝑠) is a closed-loop characteristic polynomial, 𝐶(𝑠)=𝑐_𝑖𝑠^𝑖+𝑐_𝑖−1𝑠^𝑖−1+⋯+𝑐₁𝑠+𝑐₀. (2.4)

Controller coefficients 𝑟_{𝑗,𝑗−1,…0} and 𝑙_{𝑘,𝑘−1,…0} are determined by solving polynomial equation (2.5): 𝐴(𝑠)𝑅(𝑠)+𝐵(𝑠)𝐿(𝑠)=𝐶(𝑠). (2.5)

Pole placement synthesis of the controller is based on the choice of the characteristic polynomial (2.4), ‎[26]. The choice of 𝐶(𝑠) for high-order systems can be laborious, but with the use of different standard forms such as Manabe, Bessele, Kessler, Binomial function ‎[27], Lipatov criterion ‎[28], simple approximation of dominant dynamic/poles, and optimization procedure (ISE, IAE, etc.) in which a polynomial of any order can be chosen. Standard polynomial synthesis is presented as a 2DOF system structure. Such synthesis enables separate design of tracking the reference signal and elimination of external interferences. For simplification of control structure we can also assume that 2DOF configuration can be presented as 1DOF, where prefilter design is eliminated with an additional performance weight in the optimization procedure, similar to the ℋ_∞ loop-shaping method ‎[16].

The Diophantine equation (2.5) is solvable if and only if any greatest common divisor of 𝐴(𝑠) and 𝐵(𝑠) is a factor of 𝐶(𝑠) [26]. If polynomials 𝐴(𝑠) and 𝐵(𝑠) are coprime, then (2.5) can have no solutions, exactly one solution, or a family of parametric solutions. Only the last statement is used in our approach. Equation (2.5) has a family of parametric solutions if the following holds true: (𝑠)(𝑠),(𝑠)(𝑠)(𝑠).deg𝑅≥deg𝐴deg𝐶=deg𝑅+deg𝐴 (2.6)

The number of parametric solutions is equal to −(),̃̃∑∑𝑤𝑟̃𝑝=𝑗+𝑘+2𝑖+1𝑗+𝑘,𝑗=𝑧∧𝑘=_̃𝑗∈𝑟_𝑗,𝑟₀∧𝑙𝑤̃∈𝑙_𝑗,𝑙₀, (2.7) where 𝑝 is the number of free parameters selected in controller polynomials 𝑅(𝑠) and 𝐿(𝑠). Indexes 𝑗̃ and 𝑘 denote the position of selected free parameters and refer to the power of the Laplace operator for each polynomial, 𝑅(𝑠) and 𝐿(𝑠).

Matrix form of the polynomial equation (2.5) with condition (2.6) is 𝐶_𝑦=𝑅_𝑦𝑆_𝑦+𝑃𝑆_𝑦. (2.8)

Solution of (2.8) is as follow: 𝑅_𝑦=𝐶_𝑦−𝑃𝑆_𝑦⋅𝑆_𝑦⁻¹, (2.9) where matrixes 𝐶_𝑦 and the Sylvester matrix 𝑆_𝑦 [23] contain a known coefficient of polynomials 𝐶(𝑠), 𝐴(𝑠), and 𝐵(𝑠). Matrix 𝑅_𝑦 contains the unknown controller parameters of polynomials 𝑅(𝑠) and 𝐿(𝑠), where 𝑆_𝑦 is a modified 𝑆_𝑦 matrix, with the same structure and coefficients. Matrix 𝑃 contains a set of chosen free parameters 𝑝. Equation (2.9) is solvable only if matrix 𝑆_𝑦 is not singular. The arrangement of the coefficients of polynomials 𝐴(𝑠) and 𝐵(𝑠) in the matrix 𝑆_𝑦 depends on the position of the selected free parameters in the polynomials 𝑅(𝑠) and 𝐿(𝑠). The position of free parameters 𝑝 can be selected as any coefficient of the polynomial 𝑅(𝑠) or 𝐿(𝑠) under condition (2.6), with respect to the singularity of matrix 𝑆_𝑦. Hence, it follows that the combination of free parameters in 𝑅(𝑠), 𝐿(𝑠) cannot be selected randomly. Each incorrect selection of free parameter combination in 𝑅(𝑠), 𝐿(𝑠) lowers matrix rank 𝑆_𝑦 at least by one, which means that the system of linear equation is linearly dependent and (2.9) has no solution.

As we have mentioned in the introduction, the polynomial approach is similar to the well-known Y-K parameterization, where Bezout’s identity and 𝑄 stable transfer function must be solved and selected. Verification of closed-loop poles’ position and closed-loop dynamics is thus somewhat blurred in the proposed approach. The next section will discuss transparent controller parameterization with introduced parametric solutions for ensuring good closed-loop performance in the sense of tracking and disturbance rejection.

3. Controller Parametrization with Parametric Solution

After stabilization of the closed-loop system with the selected closed-loop polynomial 𝐶(𝑠), it is also important to ensure proper behaviour of the controlled system in terms of reference tracking 𝑣, disturbance rejection 𝑤₁, 𝑤₂, robustness and time transient performances, and so forth (‎Figure 2). The issue of robustness will be discussed in the next section.

Figure 2 

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

Good reference tracking of closed-loop system 𝑒=0 can be ensured if the sensitivity function |𝑆(𝜔)| is small at frequency 𝜔₀ of the reference signal 𝑣: lim_𝜔→𝜔0|𝑆(𝜔)|≈0. (3.1)

The system provides good tracking on the set of reference signals with the frequency span 𝐵_𝜔=[𝜔_𝑙,𝜔_ℎ], if |𝑆(𝐵_𝜔)|=0 [14] is true. From (2.3) we see that the tracking performance of the closed-loop system depends on the polynomial root location of 𝑅(𝑠) and 𝐴(𝑠). With this assumption, controller 𝐾(𝑠) can be parameterised with different polynomial structures, with a known effect on the sensitivity frequency response |𝑆(𝜔)|. The most often used structure in controller parameterization with a known effect on low frequency tracking performance is root 𝑠=0 in polynomial 𝑅(𝑠). Also ramp tracking performance can be achieved with double roots at 𝑠=0 in 𝑅(𝑠), and so forth. Guided systems often require tracking harmonic reference (HR) signals with frequency 𝜔_𝑟. Good performance can be achieved with Noch structure or with an added imaginary root (𝑠²+𝜔²_𝑟) in 𝑅(𝑠), |𝑆(𝜔_𝑟)|=0.

The same effect as with tracking performance can be achieved also for input disturbance 𝑤₁, where the following holds true: 𝑤₁𝑦=𝑃₀(𝑠)𝑆(𝑠). (3.2)

And for output disturbance 𝑤₂, if 𝑒𝑣=𝑤₂𝑦=𝐴(𝑠)𝑅(𝑠)𝐶(𝑠)=𝑆(𝑠). (3.3)

From (3.2) and (3.3), we see that disturbance rejection (DR) is also related to sensitivity 𝑆(𝑠), (3.1), [16, 18], meaning that for input disturbance 𝑤₁ rejection is more effective with passive plants 𝑃₀(𝑠).

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 ‎[16]. It often occurs that despite the PIP condition being fulfilled, synthesis leads to an unstable controller although all performance conditions are fulfilled ‎[29]. Also many characteristics in the controller structure such as integral action, differential part (PID, PI structure), or complex zero can on one hand highly improve the closed-loop performance and on the other hand highly impair it (speed versus accuracy D-I part, overshot versus rise time, waterbed effect, etc.). In many cases, a compromise can be reached between performance requirements with optimal selection of controller structure and coefficients values. Let us take limited sensors accuracy for example. It is evident that a compromise can be achieved with a close approximation of the known structure. Integral action is approximated with stable zero (𝑠+𝛿), ℝ={𝛿∣0<𝛿≪1} in 𝑅(𝑠), where perfect accuracy |𝑆(0)|=0 is lowered for 𝛿. With proper selection of 𝛿: |𝑆(𝛿)|<0≪1 sufficient damping level of |𝑆(𝛿)| can be guaranteed so that it does not exceed the limited sensor accuracy. A similar approach can be used for another property, double-integrator or complex zero (𝑠²+𝜀𝑠+𝜔²_𝑟), ℝ={𝜀∣0<𝜀≪1}. This approach offers a compromise and an optimal setting of parameters 𝛿, 𝜀, which slightly improves the overall closed-loop performance. With an added fixed structure in controller 𝐾(𝑠) a set of parametric solutions of the polynomial equation (2.5), (2.8) is provided.

Parameterized controller with an added known structure is 𝐾(𝑠)=𝐾_𝑎(𝑠)𝐾^′(𝑠),𝐾(𝑠)=𝐿^′(𝑠)𝑅^′(𝑠)𝑅_𝑎(𝑠), (3.4) where 𝐾_𝑎(𝑠), 𝑅_𝑎(𝑠) are known structures and 𝐾^′(𝑠),  𝑅^′(𝑠),  𝐿^′(𝑠) are the solution of polynomial equation (2.8). The number of parametric solutions with parameterized controller (3.4) and condition (2.6) is 𝑝=deg𝑅−deg𝐴+1,deg𝑅=𝑝+deg𝐴−1. (3.5)

The degree of the characteristic polynomial 𝐶(𝑠) is deg𝐶=𝑝+2deg𝐴−1. (3.6)

Sensitivity function with controller (3.4) is 𝑆(𝑠)=𝐴(𝑠)𝑅^′(𝑠)𝑅_𝑎(𝑠)𝐴(𝑠)𝑅^′(𝑠)𝑅_𝑎(𝑠)+𝐵(𝑠)𝐿^′(𝑠), (3.7) where |𝑆(𝜔)|=𝑆^′(𝜔)𝑅_𝑎(𝜔). (3.8)

Minimizing the tracking error 𝑒≈0 or effective disturbance rejection of exogenous signals 𝑤₁, 𝑤₂ can be achieved with an added structure 𝑅_𝑎(𝑠) if the following is true: lim_𝜔→𝜔0𝑅_𝑎(𝜔)≈0. (3.9)

Sensitivity function for step reference signals and disturbances with an added proxy structure of integral action 𝑅_𝑎𝑖(𝑠)=(𝑠+𝛿), ℝ={𝛿∣0<𝛿≪1} is as the following: lim_𝜔→0|𝑆(𝜔)|=lim_𝜔→0𝑆^′(𝜔)𝜔²+𝛿²,lim_𝜔→0|𝑆(𝜔=𝑎)|⁰𝑟₀𝛿𝑐₀,0<𝛿≪1,lim_𝜔→0|𝑆(𝜔)|≈𝛿⇒|𝑆(0,𝛿)|≈0. (3.10)

The value of parameter 𝛿 determines closed-loop tracking accuracy, disturbance rejection capability, and closed-loop dynamic with characteristic polynomial 𝐶(𝑠). Parameter 𝛿 can also be used as a limited arbitrary parametric solution in the further optimization technique, with an admissible interval [0,𝛿_max]∈{𝛿∈ℝ∣0<𝛿≤𝛿_max≪1}. Considering (3.6), the degree of the characteristic polynomial with two parametric solutions {1,𝛿} is equal to deg𝐶=1+2deg𝐴.

The same properties can be introduced for ramp reference signals and output disturbances, 𝑅_𝑎𝑖𝑖(𝑠)=(𝑠²+2𝛿𝑠+𝛿²), ℝ={𝛿∣0<𝛿≪1}, where deg𝐶=2+2deg𝐴, as the following: lim_𝜔→0|𝑆(𝜔)|=lim_𝜔→0𝑆^′(𝜔)𝜔⁴+𝜔²𝛿²2𝛿²−1+𝛿⁴,lim_𝜔→0|𝑆(𝜔=𝑎)|⁰𝑟₀𝛿²𝑐₀,0<𝛿≪1,lim_𝜔→0|𝑆(𝜔)|≈𝛿²⇒𝑆0,𝛿²≈0. (3.11)

For tracking harmonic reference signals and harmonic disturbance (HD) rejection, a structure with stable complex roots 𝑅_𝑎𝜔(𝑠)=(𝑠²+𝜀𝑠+𝜔²_𝑟)ℝ={𝜀∣0<𝜀≪1} is proposed. 𝜔_𝑟 is a well-dumped frequency of sensitivity function |𝑆(𝜔)| with region 𝜀=𝜔_𝑟/Q. Q is the “quality factor” of HR tracking, HD rejection and mean frequency gap width of |𝑆(𝜔)|, with symmetric centre 𝜔_𝑟in logarithmic scale. The quality factor is also defined with bandwidth (BW), where 𝜔_𝑙 is a low frequency and 𝜔_ℎ a high frequency of the symmetric gap with a damping level −3 dB from the maximum, 𝜔𝑄=^𝑟𝜔_𝑙−𝜔_ℎ=𝜔_𝑟.𝐵𝑊 (3.12)

Selected parameter 𝜀 is equal to 𝜀=𝐵𝑤. Sensitivity function for harmonic property with 𝜔_𝑟 is lim_{𝜔→𝜔𝑟}|𝑆(𝜔)|=lim_{𝜔→𝜔𝑟}𝑆^′(𝜔)−𝜔+𝜔_𝑟²+𝜔²𝜀²,lim_{𝜔→𝜔𝑟}|𝑆(𝜔)|=𝑆^′𝜔_𝑟𝜔_𝑟|𝑆(𝜀,0,𝜀)|≈0,∀𝜀≪1∧∀𝜔_𝑟<1. (3.13)

Tracking and rejecting performance is slightly dependent on frequency 𝜔_𝑟 and on the value of parameter 𝜀 in respect to polynomial 𝐶(𝑠). The tracking/rejecting performance can also be determined with a damping band between [0,𝜔_𝑟], if zeros of 𝑅_𝑎(𝑠) are closer to origin than zeros of 𝐶(𝑠). Damping band (DB) is equal to DB=𝑎₀𝑟₀𝜔_𝑟𝑐₀,𝜔_𝑟, (3.14) where the damping value of |𝑆(DB)| in dB is equal to 𝜁(dB)𝑎=20log⁰𝑟₀𝜔_𝑟𝑐₀. (3.15)

The same property can be used for step and low frequency signals. If inequality 𝑎₀𝑟₀𝜔_𝑟/𝑐₀≪1 is true, then good tracking/rejection performance with damping (3.15) can be ensured for signals with frequency smaller than 𝜔_𝑟. The degree of the chosen characteristic polynomial 𝐶(𝑠) is equal to deg𝐶=2+2deg𝐴.

Proper selection of coefficients 𝛿, 𝜔_𝑟, 𝜀 according to optimal performance and robust criteria is the main issue in the optimization procedure. Objective functions of inequality of parameters 0<𝛿<𝛿_max, 0<𝜔_𝑟<𝜔_𝑟max, and 0<𝜀<𝜀_max, will compose the main objective function with robust criteria, where the limiting values 𝛿_max, 𝜔_𝑟max, 𝜀_max determine the admissible property of the system in accordance with the tracking/rejecting closed-loop performance. Controller parameterization 𝐾(𝑠) can also be determined with fixed selection of 𝛿, 𝜔_𝑟, 𝜀, with an added auxiliary set of free parameters 𝑝 in controller structure 𝐾^′(𝑠). The degree of controller 𝐾(𝑠) with fixed structure 𝑅_𝑎𝑖(𝑠) and a set of parameters 𝑝 is deg𝑅=𝑝+1+deg𝐴, (3.16) and with fixed structure 𝑅_𝑎𝑖(𝑠), 𝑅_𝑎𝜔(𝑠), it is deg𝑅=𝑝+2+deg𝐴. (3.17)

Auxiliary parameters 𝑝 allow a wider range of optimization space, but on the other hand raise the degree of the controller 𝐾(𝑠). The possibility of simplification of the controller order for high-order plants with an emphasized low-order dominant dynamic will be discussed in the next section.

4. Robust Stability Assessment

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 ℋ_∞ performance are proposed for models with nonstructured uncertainty. The closed-loop system with different types of non-structured uncertainty is robustly stable if the following holds true [15, 16]: 𝑊₁𝑆_∞<1,𝑊₂𝑈_∞<1,𝑊₃𝑇_∞<1. (4.1)(4.2)(4.3)

𝑆(𝑠), 𝑈(𝑠), 𝑇(𝑠) are nominal sensitivity, controller output, and complementary sensitivity transfer functions, respectively. The selected stable and proper weights 𝑊₁⁻¹(𝑗𝜔), 𝑊₂⁻¹(𝑗𝜔), 𝑊₃⁻¹(𝑗𝜔) represent the upper limit of the frequency band of closed-loop transfer functions 𝑆(𝑗𝜔), 𝑈(𝑗𝜔), 𝑇(𝑗𝜔) [16]. The main goal of the proposed synthesis is optimized robust criteria (4.1)–(4.3) with a set of parametric solutions in controller coefficients 𝐾(𝑠)=𝑓𝑟𝑧̃,𝑙𝑤̃, (4.4) where min_𝐾𝑟𝑧̃_,𝑙𝑤̃_{∈𝑅ℋ∞}𝑊₁𝑆_∞=𝛾_min<1,min_𝐾𝑟𝑧̃_,𝑙𝑤̃_{∈𝑅ℋ∞}𝑊₂𝑈_∞=𝛾_min<1,min_𝐾𝑟𝑧̃_,𝑙𝑤̃_{∈𝑅ℋ∞}𝑊₃𝑇_∞=𝛾_min<1. (4.5)

The ℋ_∞ extended mixed sensitivity problem: 𝑇_𝑧𝑤=𝑊₁𝑆𝑊₂𝑈𝑊₃𝑇_∞. (4.6)

with an optimal solution: min_{𝐾∈𝑅ℋ∞}𝑇_𝑧𝑤𝑟𝑧̃,𝑙𝑤̃_∞=𝛾_min. (4.7)

Pole placement with controller parameterization which involves free parameters 𝑝 or 𝜔_𝑟, 𝜀, and metrics norm ‖⋅‖_∞ can be used for loop shaping with known partial controller structure and its effect on the closed-loop performance discussed in the previous section.

4.1. Partial-Fractional Decomposition of 𝑃₀(𝑠) and Robust Criteria

As we have mentioned in Section 3, it is sometimes recommended in practice to use or implement a simpler controller structure. Low-order controller structure is more adequate for further closed-loop analysis and more appropriate for real-time computing on mid and low-range embedded systems or on efficient complex systems with high dynamic. Much research and many articles have been devoted to the question of model order simplification with different approaches such as balanced truncation, Hankle singular value decomposition, and others ‎[30]. Most approaches assess the deviation between the simplified and the given model in frequency or time domain on the basis of optimal value of objective function, with no assessment of the difference in mathematical form such as the parametric model error in state space or transfer function. Advanced approach to robust optimal controller design for a simplified model is proposed under the condition that all closed-loop performance requirements for the simplified model are fulfilled also for the primary model. This can be ensured if we know the difference between both and can take it into account in controller design without back evaluation after each single calculation step. The proposed approach of model simplification with partial fraction decomposition is efficient for systems with emphasized low-order dynamics, where the remainder transfer function of order reduction is added to performance weight (4.5), with a minor loss in the robustness criterion in general. For example, partial fraction decomposition of the model can be used for many thermodynamic and electromechanical systems.

Partial fraction decomposition of the plant 𝑃₀(𝑠) is as follows: 𝑃₀(𝑠)=𝑃_low(𝑠)+𝑄_rem(𝑠),𝐵(𝑠)𝐴(𝑠)=𝐵_low(𝑠)𝐴_low(𝑠)+𝐵_rem(𝑠)𝐴_rem(𝑠), (4.8) where 𝐵(𝑠)Π^𝑛−𝑞_𝑙=1𝑠±𝑝_𝑙Π^𝑞/2_𝑐=1𝑠²±𝑑_1𝑐𝑠+𝑑₀=𝐵_low(𝑠)Π^{𝑛low−𝑞low}_𝑙=1𝑠±𝑝_𝑙Π^𝑞low/2_𝑐=1𝑠²±𝑑_1𝑐𝑠+𝑑₀+𝐵_rem(𝑠)Π^{𝑛−𝑛low−𝑞low}_𝑙=1𝑠+𝑝_𝑙Π^{(𝑞−𝑞low)/2}_𝑐=1𝑠²+𝑑_1𝑐𝑠+𝑑₀. (4.9)

𝑃_low(𝑠) is the reduced model, 𝑄_rem(𝑠) is the remainder function of reduction 𝑃₀(𝑠), 𝑛 is the order of 𝐴(𝑠), 𝑞 is the number of complex roots of 𝐴(𝑠), 𝑛_low is the order of 𝐴_low(𝑠), and 𝑞_low is the number of complex roots of 𝐴_low(𝑠). Function 𝑃_low(𝑠) contains dominant and unstable poles of 𝑃₀(𝑠). For good fitting at low frequency between 𝑃₀(𝑠) and 𝑃_low(𝑠) the following must be true: 𝐵_low(0)≈𝐵(0). (4.10)

Remainder 𝑄_rem(𝑠) must be a stable and proper function. The stability property of 𝑄_rem(𝑠) is used for further transformation of robust criterion (4.1)–(4.3) with reconstructed weights 𝑊^′(𝑠). New weights 𝑊^′(𝑠) are formed for each robust criterion separately. Before we introduce robust stability criteria for 𝑃_low(𝑠), it is necessary to consider the stability property for 𝑃₀(𝑠) with controller 𝑃_low(𝑠)⇒𝐾_low(𝑠).

The synthesized controller 𝐾_low(𝑠) for plant 𝑃_low(𝑠) stabilizes 𝑃₀(𝑠) if (4.8), (4.10), 𝑄_rem(𝑠)∈ℛℋ_∞, ‖𝑄_rem‖_∞<1, and ‖𝑄_rem‖_∞<‖𝑃_low‖_∞ is true. 1+𝑃_low+𝑄_rem𝐾_low>0,1+𝑃_low𝐾_low1+1+𝑃_low𝐾_low⁻¹𝑄_rem𝐾_low𝑄>0,_rem𝐾_low∞<1+𝑃_low𝐾_low∞,∀𝑄_rem∧∀𝑃_low,𝑄_rem∞≤𝑃_low∞. (4.11)

All three properties are achieved with decomposition of 𝑃₀(𝑠) to dominant and unstable poles for 𝑃_low(𝑠) and remain stable poles for 𝑄_rem(𝑠).

Robust stability property (4.1) considered by small gain theorem for reduced model 𝑃_low(𝑠) and inverse uncertainty model 𝑃=𝑃₀(1+𝑊₁)⁻¹ is 1+𝐾𝑃_low1+𝑆_low𝑊₁+𝐾𝑄_{rem𝑊′1(𝐾)}𝑆>0,_low𝑊^′₁(𝐾)_∞<1. (4.12)

𝑆_low(𝑠) is the sensitivity closed-loop function of the reduced model 𝑃_low(𝑠) with synthesized controller 𝐾(𝑠) and reconstructed weight 𝑊^′₁(𝑠).

Robust stability property (4.2) for reduced model 𝑃_low(𝑠) and additive uncertainty model 𝑃=𝑃₀+𝑊₂ is 𝑈_low𝑄_rem+𝑊_2∞𝑈<1,_low𝑊^′_2∞<1, (4.13) where 𝑈_low(𝑠) is the controller output transfer function with 𝐾_low(𝑠).

Robust stability property (4.3) for reduced model 𝑃_low(𝑠) and multiplicative uncertainty model 𝑃=𝑃₀(1+𝑊₃) is 𝑈_low𝑊₃𝑃₀+𝑄_rem∞𝑈<1,_low𝑊^′_3∞<1. (4.14)

5. Optimization of Robust Criteria with Even Spectral Polynomials

Transformation of robust criteria (4.1)–(4.3) in even polynomial form is applied in many optimization techniques, particularly if objective functions ensure convex property ‎[3]. Many efficient robust approaches use zero-order optimization method ‎[31], where the selection of objective function is of key importance.

The norm ‖⋅‖_∞ for 𝑃₀(𝑠) can be defined with function Φ(𝑠): Φ(𝑠)=𝛾²𝐼−𝑃₀(𝑠)𝑃₀(−𝑠). (5.1)

Φ(𝑠) is a continuous function for all 𝜔∈ℝ∪{∞} and has no imaginary axis zero if ‖𝑃₀‖_∞<𝛾. Proper condition of suboptimal robustness (4.1)–(4.3) is fulfilled, if ‖𝑋‖_∞<1, where 𝛾=1. From (5.1), inequality ‖𝑃₀‖_∞<1 can be derived: 𝐴(𝑠)𝐴(−𝑠)−𝐵(𝑠)𝐵(−𝑠)>0,𝐴(𝑗𝜔)²−𝐵(𝑗𝜔)²(𝜔)𝐴𝜔=Π>0,₂−𝐵𝜔²=Π(𝜔)>0. (5.2)

Spectral polynomial Π(𝜔)=∑^𝑣_𝑤=0𝑞_2𝑤𝜔^2𝑤>0 with real coefficients 𝑞_2𝑤 is an even function, where ‖𝑃₀‖_∞<1 is true, if polynomial Π(𝜔) has no real roots. Polynomial Π(𝜔) must be a strictly positive function: 𝑞_2𝑤>0forall𝜔. Nonnegativity of even polynomials can be tested with algebraic Šiljak’s stability test ‎[21] if the condition 𝑞₀∧𝑞_𝑣>0forall𝜔 is fulfilled. Šiljak’s test is used for testing the existence of proper solutions before the optimization method starts. A more detailed discussion of the whole algorithm and the use of Šiljak’s test are presented in ‎[32].

From condition (5.2), spectral polynomial Π(𝜔) can be derived for each robust criterion (4.1)–(4.3) separately.

Robust condition (4.1) with weight 𝑊₁(𝑠) or 𝑊^′₁(𝑠) is 𝐴(𝑠)𝑅(𝑠)𝑤_𝑏1(𝑠)𝐶(𝑠)𝑤_𝑎1(𝑠)_∞<1, (5.3) where 𝑊₁(𝑠)∧𝑊^′₁(𝑠)=𝑤_𝑏1(𝑠)𝑤⁻¹_𝑎1(𝑠)=(𝑤_𝑏1𝑔𝑠^𝑔+⋯+𝑤_𝑏10)(𝑤_𝑎1𝑔𝑠^𝑔+⋯+𝑤_𝑎10)⁻¹ and the derived spectral polynomialΠ_𝑆(𝜔) is Π_𝑠𝑠,𝑝=𝐶(𝑠)𝐶(−𝑠)𝑤_𝑎1(𝑠)𝑤_𝑎1(−𝑠)−𝐴(𝑠)𝐴(−𝑠)𝑅𝑠,𝑝𝑅−𝑠,𝑝𝑤_𝑏1(𝑠)𝑤_𝑏1(),Π−𝑠_𝑠𝜔,𝑝=𝐶𝑤_𝑎1(𝜔)−𝐴𝑅𝑤_𝑏1𝜔,𝑝>0. (5.4)

Robust condition (4.2) with weight 𝑊₂(𝑠), 𝑊^′₂(𝑠)=𝑤_𝑏2(𝑠)𝑤⁻¹_𝑎2(𝑠) and spectral polynomial Π_𝑈(𝜔) is 𝐴(𝑠)𝐿(𝑠)𝑤_𝑏2(𝑠)𝐶(𝑠)𝑤_𝑎2(𝑠)_∞<1,Π_𝑈𝜔,𝑝=𝐶𝑤_𝑎2(𝜔)−𝐴𝐿𝑤_𝑏2𝜔,𝑝>0. (5.5)(5.6)

Robust condition (4.3) with weight 𝑊₃(𝑠), 𝑊^′₃(𝑠)=𝑤_𝑏3(𝑠)𝑤⁻¹_𝑎3(𝑠) and spectral polynomial Π_𝑇(𝜔) is 𝐵(𝑠)𝐿(𝑠)𝑤_𝑏3(𝑠)𝐶(𝑠)𝑤_𝑎3(𝑠)_∞<1,Π_𝑇𝜔,𝑝=𝐶𝑤_𝑎3(𝜔)−𝐵𝐿𝑤_𝑏3𝜔,𝑝>0. (5.7)(5.8)

With simple algebra we can determine the necessary edge conditions of controller parameters for ensuring nonnegativity of the spectral polynomials (5.4), (5.6), and (5.8). Edge conditions for criteria (5.3), (5.5), and (5.7) are as follows: 0<𝑟₀<𝑐₀𝑤_𝑎10𝑎₀𝑤_𝑏10⁻¹∀Π_𝑆𝜔,𝑝,0≤𝑙₀≤𝑐₀𝑤_𝑎20𝑎₀𝑤_𝑏20⁻¹∀Π_𝑈𝜔,𝑝,0≤𝑙₀≤𝑐₀𝑤_𝑎30𝑏₀𝑤_𝑏30⁻¹∀Π_𝑇𝜔,𝑝. (5.9)

Edge conditions (5.9) determine the solvability of the given problem, with the chosen controller structure. The central issue of optimization procedure is finding a global minimum of maximized nonnegative spectral polynomial Π(𝜔). The objective function 𝑓_obj of DE optimization procedure is composed of spectral polynomial and auxiliary condition 𝑓_od, like the strong stabilization property, and of additional closed-loop time performance criteria. The strong stabilization condition is ensured by testing the root location of 𝑅(𝑠) in LHP with Lipatov criterion ‎[32]. The spectral polynomial is a quasiconvex function and can be considered as a convex function with the selection of 𝜔≤0 or 𝜔≥0. With bounded optimization space 𝜔≤0 or 𝜔≥0, the optimization procedure can be treated as a convex problem with no loss of generality. Objective function for single criterion is 𝑓_obj=_𝑆𝜔,𝑝∨_𝑈𝜔,𝑝∨_𝑇𝜔,𝑝∧𝑓_ad. (5.10) Objective function for the mixed sensitivity problem is 𝑓_obj=_𝑆𝜔,𝑝∧_𝑈𝜔,𝑝∧_𝑇𝜔,𝑝∧𝑓_ad. (5.11)

6. Design Example

Example 6.1. 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-𝑧, longitudinal movement-𝑥, 𝑦, and orientation (yaw)-𝜙 (Figure 3).

Figure 3 

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 𝑃(𝑠) with coupled dynamic 𝐻(𝑠) is 𝑥(𝑠)𝑦(𝑠)𝑧(𝑠)𝜙(𝑠)=𝑃_𝑥(𝑠)𝐻_𝑥𝑦(𝑠)0𝐻_yaw𝑥(𝑠)𝐻_𝑦𝑥(𝑠)𝑃_𝑦(𝑠)0𝐻_yaw𝑦(𝑠)00𝑃_alt(𝑠)𝐻_yaw𝑎(𝑠)𝐻_𝑥yaw(𝑠)𝐻_𝑦yaw(𝑠)0𝑃_yaw(𝑠)×𝑢_𝑥(𝑠)𝑢_𝑦(𝑠)𝑢_𝑧(𝑠)𝑢_𝜙(𝑠), (6.1) where individual transfer function is 𝑃_𝑥(𝑠)=𝑃_𝑦(𝑠)=46345×(𝑠+0.837)𝑠⁴+7.049×𝑠³+13.73×𝑠²,𝑃+4.484×𝑠_alt(𝑠)=46345×𝑠+0.837𝑠³+21.05×𝑠²,𝑃+0.965×𝑠+0.01_yaw(𝑠)=85.74×𝑠+26.46𝑠².+0.312×𝑠 (6.2) The coupled dynamic transfer function is 𝐻_𝑥𝑦(𝑠)=𝐻_𝑥𝑦(𝑠)=2.1×𝑠²+0.039×𝑠+0.0000983𝑠²,𝐻+0.2×𝑠+0.0075_yaw𝑥(𝑠)=𝐻_yaw𝑦(𝑠)=4.5×𝑠²+10.12×𝑠+0.0011𝑠²,𝐻+15.01×𝑠+0.135_yaw𝑎(𝑠)=3×𝑠⁴+190.1×𝑠³+4.355×𝑠²+10.11×𝑠+0.0001𝑠⁴+64.01×𝑠³+724.6×𝑠²,𝐻+1207×𝑠+12_𝑥yaw(𝑠)=𝐻_𝑥yaw(𝑠)=1.085⋅10⁻⁴×𝑠²+924.4×𝑠+279.1𝑠².+0.421×𝑠+0.034 (6.3)

For control design robust-mixed sensitivity problem (4.6) is used with weights as the following: 𝑊_𝑀𝑥(𝑠)=𝑊_𝑀𝑦(𝑠)=0.015×𝑠⁴+0.1479×𝑠³+0.538×𝑠²+0.855×𝑠+0.5022𝑠⁴+4.713×𝑠³+7.099×𝑠²,𝑊+3.651×𝑠+0.5602_𝑀alt(𝑠)=0.112×𝑠³+5.615×𝑠²+0.7537×𝑠+7.045⋅10⁻⁵𝑠³+13.06×𝑠²,𝑊+0.6501×𝑠+0.003406_𝑀yaw(𝑠)=0.00126×𝑠²+0.0974×𝑠+0.02929𝑠²,𝑊+0.4286×𝑠+0.03703_𝑖𝑥(𝑠)=𝑊_𝑖𝑦(𝑠)=𝑠²+0.2×𝑠+0.00752.1𝑠²,𝑊+0.0819×𝑠+0.000195_𝑖alt(𝑠)=𝑠⁴+64.01×𝑠³+724.6×𝑠²+1207×𝑠+123×𝑠⁴+570.3×𝑠³+13.01×𝑠²,𝑊+30.3×𝑠+0.0003_𝑖yaw(𝑠)=𝑠²+0.4286×𝑠+0.037030.00126×𝑠²,+0.0974×𝑠+0.02929 (6.4) where weights𝑊_𝑀𝑥(𝑠), 𝑊_𝑀𝑦(𝑠),𝑊_𝑀alt(𝑠), and 𝑊_𝑀yaw(𝑠) represent the multiplicative uncertainty of the nominal plant ‎[15]. Uncertainty weights were determined on the basis of system parameter variation, like different load weight (battery, camera), different propeller characteristic, and so forth. Weight 𝑊_𝑀(𝑠) describes the deviation of the plant at a lower frequency of nominal characteristic. Weight 𝑊_𝑖(𝑠) is used to decuple a feedback MIMO plant on four single-axis SISO systems. The influences of other inputs for each axis are represented as output disturbances 𝑤₂ (Figure 2) with a frequency characteristic 𝑊_𝑖(𝑠). Weight 𝑊_𝑖(𝑠) is selected from coupled dynamic transfer functions 𝐻(𝑠) for each axis separately and represents the lowest upper boundary of the sensitivity function for single-axis SISO feedback system. Additional simplification of controller structure and design procedure is carried out with partial-fractional decomposition on stable higher dynamic transfer function 𝑄_rem(𝑠) and the remaining lower dynamic, unstable transfer function 𝑃_low(𝑠). The new weight 𝑊^′_𝑀(𝑠) and robust criteria are transformed with expression (4.14). Partial-fractional decomposition is as follows: 𝑃_𝑥(𝑠)=𝑃_𝑦(𝑠)=𝑃_low𝑥/𝑦(𝑠)+𝑄_rem𝑥/𝑦(𝑠)=2872×𝑠+3519𝑠²−+0.484×𝑠2872×𝑠+2.1⋅10⁴𝑠²,𝑃+6.643×𝑠+11.03_alt(𝑠)=𝑃_lowalt(𝑠)+𝑄_remalt(𝑠)=1297×𝑠+1.811𝑠²+0.0459×𝑠+4.7⋅10⁻⁴−1297.23,𝑃𝑠+21.03_yaw(𝑠)=𝑃_lowyaw(𝑠)+𝑄_remyaw(𝑠)=84.81𝑠+0.93.𝑠+0.312 (6.5)

Mixed sensitivity problem (4.6) with stability criterion for multiplicative uncertainty with modified weights 𝑊^′_𝑀(𝑠) and robust performance criterion with weights 𝑊_𝑖(𝑠) is 𝑇_𝑥=𝑇_𝑦=𝑊_{𝑖𝑥/𝑦}𝑆_𝑥/𝑦𝑊^′_{𝑀𝑥/𝑦}𝑈_𝑥/𝑦∞,𝑇_alt=𝑊_𝑖alt𝑆_alt𝑊^′_𝑀alt𝑈_alt∞,𝑇_yaw=𝑊_yaw𝑆_yaw𝑊^′_yaw𝑈_yaw∞. (6.6) Time performance requirements of the controlled system are as the following: (i)overshot 𝑀_pr(%): 𝑥/𝑦-axes <20%, 𝑧-axis <30%, 𝜙-axis <30%;(ii)settling time 𝑡_𝑠(s): 𝑥/𝑦-axis <20s, 𝑧-axis <15s, 𝜙-axis <6s;(iii)steady state error for all four axes: 𝑒(%)<1%;(iv)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;(v)strong stabilization system.

According to control requirements, the controller structure for each axis can be determined. The selected controller structure for longitudinal movement 𝐾_𝑥(𝑠) and 𝐾_𝑦(𝑠) is 𝐾_𝑥𝑠,𝑟₀̃,𝑟₁̃,𝑙₀̃=𝐾_𝑦𝑠,𝑟₀̃,𝑟₁̃,𝑙₀̃=𝑠+𝑙₀̃𝑠+𝑟₁̃⋅𝑙₂𝑠²+𝑙₁𝑠+𝑙₀𝑟₂𝑠²+𝑟₁𝑠+𝑟₀̃, (6.7) where free parameters 𝑟₀̃, 𝑟₁̃, 𝑙₀̃ enable good ability of optimization criteria with weights 𝑊_𝑖(𝑠), 𝑊_𝑀(𝑠). According to strong stabilization and overshot requirements, the acceptable interval limits for the free parameter are determined as follows: 0<𝑟₀̃<5, 50<𝑟₁̃<750, 10<𝑙₀̃<200. Selected characteristic polynomial coefficients (Manabe polynomial form and settling time 𝑡_𝑠(s)<10s, ‎[27]) and polynomial order requirements (2.6), (2.7) with controller structure 𝐾_𝑥/𝑦(𝑠) and plant 𝑃_low𝑥/𝑦(𝑠) are as follows: 𝐶_𝑥(𝑠)=𝐶_𝑦(𝑠)=𝑠⁵+690×𝑠⁴+2397×𝑠³+2950×𝑠²+1366×𝑠+310. (6.8)

Altitude controller 𝐾_alt(𝑠) with added Noch characteristic at a damping frequency 𝜔_noch=0.23rad/s and three free parameters is 𝐾_alt𝑠,𝑟₀̃,𝑟₁̃,𝑙₀̃=𝜔²_noch𝑠²+𝑟₁̃𝑠+𝜔²_noch⋅𝑙₀̃𝑠⁵+𝑙₄𝑠⁴+𝑙₃𝑠³+𝑙₂𝑠²+𝑙₁𝑠+𝑙₀𝑟₅𝑠⁵+𝑟₄𝑠⁴+𝑟₃𝑠³+𝑟₂𝑠²+𝑟₁𝑠+𝑟₀̃. (6.9) Noch characteristic is chosen from the property of weight 𝑊_𝑖alt(𝑠) with a frequency gap at ≈0.23rad/s.

The acceptable interval limit for free parameter 𝑟₀̃ is 0<𝑟₀̃<4 for ensuring good low-frequency tracking property and disturbance rejection. Admissible interval for parameter 𝑟₁̃ is 0<𝑟₁̃<0.01 for ensuring good damping at frequency 𝜔_noch. The admissible interval for parameter 𝑙₀̃ is 0<𝑙₀̃<0.01 for ensuring optimization property at higher frequency. Selected characteristic polynomial (Manabe form) according to controller order 𝐾_alt(𝑠) and plant 𝑃_lowalt(𝑠) is: 𝐶_alt(𝑠)=𝑠⁹+70.4×𝑠⁸+2478×𝑠⁷+4.04⋅10⁴×𝑠⁶+9.44⋅10⁴×𝑠⁵+1.24⋅10⁵×𝑠⁴+6.5⋅10⁴×𝑠³+1.98⋅10⁴×𝑠²+53.59×𝑠+0.037. (6.10)

Selected yaw controller structure 𝐾_yaw(𝑠) with free parameters 𝑟₀̃, 𝑟₁̃ is 𝐾_yaw𝑠,𝑟₀̃,𝑟₁̃=𝑙₂𝑠²+𝑙₁𝑠+𝑙₀𝑟₂𝑠²+𝑟₁̃𝑠+𝑟₀̃. (6.11) Admissible interval for parameter 𝑟₀̃ is 0<𝑟₀̃<1, and for 𝑟₀̃ it is  1<𝑟₁̃<10 to ensure strong stabilization system and good low-frequency closed-loop system ability. The characteristic polynomial is 𝐶_yaw(𝑠)=𝑠³+2.75×𝑠²+3.92×𝑠+2.26. (6.12)

The results of the optimization procedure with DE algorithm are described in ‎[32]. Longitudinal movement controller 𝐾_𝑥/𝑦(𝑠) is as follows: 𝐾_𝑥(𝑠)=𝐾_𝑦(𝑠)=𝑠+93.8⋅𝑠+6862.2⋅10⁻⁴×𝑠²+0.00127×𝑠+0.0009𝑠².+2.98×𝑠+2.464 (6.13) The values of norms are ‖𝑊_{𝑖𝑥/𝑦}𝑆_{𝑖𝑥/𝑦}‖_∞=0.58, ‖𝑊^′_{𝑀𝑥/𝑦}𝑈_{𝑖𝑥/𝑦}‖_∞=0.896, ‖𝑊_{𝑀𝑥/𝑦}𝑇_{𝑖𝑥/𝑦}‖_∞=0.896.

Altitude controller 𝐾_alt(𝑠) is as follows: 𝐾_alt(𝑠)=0.0531𝑠²⋅+0.001×𝑠+0.0531376.1×𝑠⁵+1308×𝑠⁴+1466×𝑠³+941.6×𝑠²+275.5×𝑠+0.384𝑠⁵+10.35×𝑠⁴+2475×𝑠³+1.37⋅10⁴×𝑠².+2610×𝑠+3.6 (6.14) The values of norms are ‖𝑊_𝑖alt𝑆_𝑖alt‖_∞=0.54, ‖𝑊^′_𝑀alt𝑈_𝑖alt‖_∞=0.95, ‖𝑊_𝑀alt𝑇_𝑖alt‖_∞=0.959.

Yaw controller 𝐾_yaw(𝑠) is as follows 𝐾_yaw(𝑠)=0.0145×𝑠²+0.0361×𝑠+0.0267𝑠².+1.524×𝑠+0.852 (6.15) The values of norms are ‖𝑊_𝑖yaw𝑆_𝑖yaw‖_∞=0.51, ‖𝑊^′_𝑀yaw𝑈_𝑖yaw‖_∞=0.75, ‖𝑊_𝑀yaw𝑇_𝑖yaw‖_∞=0.78.

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

Longitudinal movement control corresponds to 𝐾_𝑥/𝑦(𝑠) on step reference signal with low-frequency output disturbance 𝜔=0.08 rad/s, Figure 4.

Figure 4 

Longitudinal 𝑥/𝑦 movement control with controller 𝐾𝑥/𝑦(𝑠) and output disturbance with frequency 𝜔=0.081 rad/s.

Altitude control with controller 𝐾_alt(𝑠) and output disturbance signal, 𝜔=0.22 rad/s, is presented in Figure 5.

Figure 5 

Altitude control with controller 𝐾alt(𝑠) and output periodic disturbance.

Orientation-yaw control with controller 𝐾_yaw(𝑠) and output disturbance signal, 𝜔=0.12 rad/s, is presented in Figure 6.

Figure 6 

Orientation (yaw) control with controller 𝐾yaw(𝑠) and output periodic disturbance.

A trajectory following with controllers 𝐾_𝑥(𝑠), 𝐾_𝑦(𝑠), 𝐾_alt(𝑠), is presented in Figure 7.

Figure 7 

Quadcopter trajectory following.

The quadcopter guidance controllers 𝐾_𝑥(𝑠), 𝐾_𝑦(𝑠), 𝐾_alt(𝑠), and 𝐾_yaw(𝑠) have satisfied the whole set of control requirements (see Figures 4 and 5). Controllers are robustly stable for multiplicative uncertainty models and allow the desired tracking performance for step and periodic reference signals at frequency (0–0.15 rad/s) for 𝑥, 𝑦, and 𝜙 axes and (0–0.27 rad/s) for altitude control 𝑧-axis. The feedback system also allows the desired output disturbance rejection for signal at frequency (0–0.18 rad/s) for 𝑥, 𝑦, 𝜙-axes and (0–0.35 rad/s) for 𝑧-axis.

Example 6.2. The second design example is a stabilization of the single-axis magnetic ball suspension system.

The ball (1) in Figure 8 can move only vertically in the magnetic field of the electromagnet (2.3). The position of the ball is detected by the optical sensor. Nominal model of the plant ‎[33] is 𝑋(𝑠)𝑈(𝑠)=𝑃_sus(𝑠)=28.9𝑠⁴+147.4×𝑠³+454.7×𝑠²−85.49⋅10³×𝑠−656.6⋅10³. (6.16)

Figure 8 

Single-axis magnetic system.

Used weights are 𝑊_𝑆(𝑠)=2.88⋅10⁻³×𝑠²+0.161×𝑠+0.8𝑠²+0.16⋅10⁻³×𝑠+6.4⋅10⁻⁹,𝑊_𝐴(𝑠)=1.1⋅10⁻⁵×𝑠²+0.011×𝑠+0.0011𝑠³+15×𝑠²,+3×𝑠+2 (6.17) where weights 𝑊_𝑆(𝑠) and 𝑊_𝐴(𝑠) represent desired sensitivity characteristic of closed-loop system and additive uncertainty of the nominal system, respectively. Uncertainty weights describe nonlinear characteristic of the magnetic field and the system parameter deviation, especially different ball weight.

Simplification of controller structure with partial-fractional decomposition on 𝑃_lowsus(𝑠) and 𝑄_susrem(𝑠) (4.8) is as the following: 𝑃_sus(𝑠)=𝑃_suslow(𝑠)+𝑄_susrem(𝑠)=0.06×𝑠+33.32𝑠³+7.371×𝑠²+−577.2×𝑠−46900.001274.𝑠+139.34 (6.18)

The new weights 𝑊^′_𝐴(𝑠) and 𝑊^′_𝑆(𝑠) are transformed with expressions (4.12), (4.13). Mixed sensitivity problem (4.6) with modified weights 𝑊^′_𝑀(𝑠),𝑊^′_𝑆(𝑠) are 𝑇_mix=𝑊^′_𝑆𝑆_suslow𝑊^′_𝑀𝑈_suslow∞. (6.19) Performance requirements of the controlled system are as the following.(i)overshot:𝑀_pr(%)<10%.(ii)settling time: 𝑡_𝑠(s)<0.8s.(iii)the tracking error for step response and periodic reference signal over frequency band (0–4 rad/s): 𝑒(%)<3%.(iv)robust stabilized system and good output low frequency disturbance rejection (>−40 dB) over frequency band (0–5 rad/s). (v)strong stabilization system.

Selected controller structure 𝐾(𝑠,𝑟₀̃,𝑟₁̃) with prefilter 𝐹(𝑠) designed according to ‎[23], 𝐾𝑠,𝑟₀̃,𝑟₁̃=𝑙₄𝑠⁴+𝑙₃𝑠³+𝑙₂𝑠²+𝑙₁𝑠+𝑙₀𝑟₄𝑠⁴+𝑟₃𝑠³+𝑟₂𝑠²+𝑟₁̃𝑠+𝑟₀̃,𝐹(𝑠)=‖𝑇𝐹‖_∞𝑙₄𝑠⁴+𝑙₃𝑠³+𝑙₂𝑠²+𝑙₁𝑠+𝑙₀, (6.20) where free parameters 𝑟₀̃, 𝑟₁̃ improve the ability of optimization criteria with weights 𝑊^′_𝑀(𝑠),  𝑊^′_𝑆(𝑠). According to strong stabilization and overshot requirements, the acceptable interval limits for the free parameter are determined as the following: 0<𝑟₀̃<1, 150<𝑟₁̃<3000. Selected characteristic polynomial coefficients according to (2.6), (2.7), and ‎[27] are 𝐶(𝑠)=𝑠⁷+285.7×𝑠⁶+4.1⋅10⁴×𝑠⁵+2.9⋅10⁶×𝑠⁴+1.4⋅10⁸×𝑠³+1.8⋅10⁹×𝑠²+1.6⋅10¹⁰×𝑠+5.9⋅10¹⁰. (6.21)

The results of the optimization procedure are 𝐾𝑠,𝑟₀̃,𝑟₁̃=9.6⋅10⁴𝑠⁴+45.8×𝑠³+732.7×𝑠²+5.95⋅10³×𝑠+2.12⋅10⁴𝑠⁴+278×𝑠³+3.9⋅10⁴×𝑠²,𝐹(𝑠)=+783×𝑠+0.052.12⋅10⁴𝑠⁴+45.8×𝑠³+732.7×𝑠²+5.95⋅10³×𝑠+2.12⋅10⁴. (6.22) The values of norms are ‖𝑊^′_𝑆𝑆_suslow‖_∞=0.508, ‖𝑊^′_𝑀𝑈_suslow‖_∞=0.64,‖𝑊_𝑆𝑆_sus‖_∞=0.501, ‖𝑊_𝑀𝑈_sus‖_∞=0.63.

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 𝜔₁=1.5 rad/s and 𝜔₂=4 rad/s and offset value 0.3 mm, Figure 9.

Figure 9 

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

Step reference tracking with output disturbance Figure 10.

Figure 10 

Step reference tracking with output disturbance.

Periodic reference tracking with periodic output disturbance Figure 11.

Figure 11 

Periodic reference tracking with output disturbance.

The closed-loop systems with controller 𝐾(𝑠,𝑟₀̃,𝑟₁̃) and prefilter 𝐹(𝑠) have satisfied the whole set of control requirements (see Figures 10 and 11). Controllers are robustly stable for additive uncertainty model and allow the desired tracking performance for step and periodic reference signals at frequency (0–4 rad/s). The feedback system also allows the desired output disturbance rejection for signal at frequency (0–5 rad/s).

7. Conclusion

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 ℋ₂, where ℋ₂ means optimization of the feedback system energy. In general, the method is useful for all controller designs, where low-order transparent controllers are required and the structure of the controller is predefined.