# Algorithms for U-Model-Based Dynamic Inversion (UM-Dynamic Inversion) for Continuous Time Control Systems

Authors: Ruobing Li; Quanmin Zhu; Janice Kiely; Weicun Zhang
Journal: Complexity
 (2020)

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## Abstract

To setup a universal proper user toolbox from previous individual research publications, this study generalises the algorithms for the U-model dynamic inversion based on the realisation of U-model from polynomial and state-space described continuous-time (CT) systems and presents the corresponding U-control system design in a systematic procedure. Then, it selects four CT dynamic plants plus a wind energy conversion system for simulation case studies in Matlab/Simulink to test/demonstrate the proposed U-model-based design procedure and dynamic inversion algorithms. This work can be treated as a U-control system design user manual in some sense.


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## Body

## 1. Introduction

Linear control system design approaches can be divided into state-space model-based [1] and polynomial model-based, which have been well studied and are widely used. However, in an actual production process, nonlinear control is ubiquitous and more difficult as the superposition principle no longer holds, in contrast to linear systems; therefore, how to design a standard-compliant nonlinear control system to match desired performance properly is a hot issue. For nonlinear systems, a variety of analysis and design approaches already exist, and the most commonly used method to design a nonlinear control system is still linearization. However, the linearization method has certain limitations, and most linear control methods cannot be applied to the design of nonlinear systems directly. For example, compared with the linear polynomial model, nonlinear polynomial models, such as the Nonlinear Autoregressive Moving Average with eXogenous inputs (NARMAX) [2] model, have been used widely in applications and academic research publications [3]; however, there is no systematic routine to convert it into an equivalent state-space model.Generally, there are three methodologies for the nonlinear plant-model-based control system design, two widely used and one less-attended. The first approach is using linear expressions to describe the nonlinear state-space models by feedback linearization approach [4, 5] and then designing this linear-expression corresponding control systems by linear state-space approaches, which has been well studied in [1]. However, this case-by-case method requires certain skills in selecting the appropriate coordinate system and solving the equations requires extraeffort. Furthermore, this state-space linearization approach cannot be directly applied for nonlinear polynomial models. The second method is to use a time-varying linear model to fit the polynomial model, for example, state-dependent parameter (SDP) transformation [6, 7] method can use specified (desired) poles to transform a nonlinear closed-loop control system model to a linear transfer function expression. In summary, it is clear that these approaches for designing a nonlinear control system are trying to convert the original nonlinear system into a quasi-linear domain system firstly and then to choose an opportune linear control approach for the system. In the model structure, the other variables in the system can also determine this quasi-linear SDP transformation’s parameters [8]. The nonlinear polynomial model can be converted into a time-varying linear state-space model by the second method; however, there are obstacles for using this method because this design and transform procedure is not unique, that is, personal and subjective for selection of SDP models.The third approach is the U-model-based design, which is relatively new and less-attended. The U-model is defined as a polynomial or state space function, with time-varying parameters, representing a class of smooth and analytic systems. Zhu et al. [9] proposed the use of a Newton–Raphson iterative algorithm for the root solving of the controller output function, which provided a basic procedure for the design of controllers in the U-model. Zhu and Guo [10] formally proposed the concept of U-model and established the U-model-based control, U-control in short, and system design framework, which provides a general routine to convert smooth nonlinear plant models into U-model. The U-model-based design method can be recognised as converting nonlinear models to time-varying parameter models associated with controller output u, that is, linear control-oriented model structure [10–13].Regarding the research status of U-model-based control, the discrete-time systems has been studied with more attention, especially the representative approaches including pole placement control design method [13, 14], U-Smith predictor with input time delay [15], adaptive U-control of total nonlinear dynamic systems [16], U-neural network enhanced control [17], and underactuated coupled nonlinear adaptive control synthesis, using U-model for multivariable unmanned marine robotics [18]. However, the majority of U-control approaches have assumed that the plant model can be detected without errors or inaccuracies. Therefore, designing an adaptive U-model controller when the plant model is inaccurate, especially the robustness control, will be a hotspot and difficult study area for intensive research. At the same time, there is very little research on U-model-based control system design for continuous-time systems so far [19]. Consequently, the main purpose of this study is to provide a pack of dynamic inversion routines for U-model-based continuous-time control systems.Compared with methods 1 (linear model approximation) and 2 (time-varying linear model approximation), aforementioned, the main contributions of this U-design method are as follows:(1)
In dealing with nonlinearity, the U-model-based design method does not require linearization of the nonlinear models in advance. Instead, this nonlinear plant model-based system is designed directly using linear design methods.(2)
In methodology, using those well-studied linear methods to design nonlinear control systems greatly reduces the complexity of the design procedure.(3)
In design, once the closed loop system output is specified, the only remaining work is to calculate the output of the U model controller.(4)
U-model-based design procedure is more general and effective for designing a linearly behaved control system, which provides new insight and solutions to design the controller.(5)
U-control can be applied together with the other well-developed control system design methods, such as pole placement control, sliding mode control, general predictive control, adaptive, and Smith predictive control [20, 21].(6)
It should be noted that unless the plant model is accurately known, U-model dynamic inversion is very sensitive to internal uncertainties, so the whole control system performance.Accordingly, the main contributions of this study are as follows:(1)
Generalise dynamic inversion algorithms for continuous-time U-model(2)
Generalise U-model-based design procedure for continuous-time dynamic plants in forms of linear/nonlinear and polynomial/state space(3)
Showcases for bench tests and illustration of applications(4)
An industrial backgrounded study: U-control of a wind energy conversion systemFor the rest of the study, Section2 generalises U-polynomial and U-state space model sets and its associated step-by-step U-control design procedure. Section 3 generalises the dynamic inversion algorithms. Section 4 presents a series of computational case studies to test/demonstrate the analytical results numerically and provides an effective procedure for testing designed U-control systems with computational experiments. Section 5 presents an industrial backgrounded case study from modelling, dynamic inversion, and U-control system design to simulation. Section 6 concludes the study.


## 2. U-Model and U-Control System Design

### 2.1. Polynomial U-Model: Single Layer Realisation

#### 2.1.1. Basic Polynomial U-Model

Consider a general continuous-time U-model [19] for Single-Input and Single-Output (SISO) polynomial dynamic systems with a triplet of yt,ut,λt, yt∈ℝ, and ut∈ℝ for the output, input, and parameter, respectively, at time t∈ℝ+:(1)My=∑j=0JλjYM−1,UN−1,ΘNuj,M≥N,where My and Nu are the Mth and Nth order derivatives of the plant output y and the plant input u, respectively, the time-varying parameter λj∗∈ℝ+, associated with the input Nuj, absorbs all the other terms in YM−1=m−1y,m−2y,…y∈ℝM, UN−1=n−1u,n−2u,…u∈ℝN, and coefficients Θ.Here is an example for understanding, consider a classical NAMAX polynomial model:(2)y¨=y−1y˙+1+y2u+1+y˙2u2+y+y+y˙2u3.Its U-model realisation can be determined with(3)y¨=λ0+λ1u+λ2u2+λ3u3,λ0=y−1y˙+y,λ1=1+y2,λ2=1+y˙2,λ3=y+y˙2.Inspection of (2) and (3), the U-realisation is straightforward generally.It should be remarked that the U-polynomial is the same as its presented classical polynomials in the model properties, but oriented expression for control system design [14].


#### 2.1.2. Extended U-Model (Rational Models)

Rational model is totally nonlinear [3], and its polynomial expression is a ratio of two polynomials, that is,(4)My=fpnψn∗Θnfpdψd∗Θd,where fpn is the input-output mapping function of the polynomial in the numerator and fpd is the input-output mapping function of the polynomial in the denominator. Its U-realisation is accordingly described by(5)My=∑j=0nλnjYM−1,UN−1,ΘfnjNu∑j=0dλdjYM−1,UN−1,ΘfdjNu,M≥N,where fnj and fdj are vector functions of the control vector Nu in numerator and denominator, respectively, and λnj and λdj are the associated parameters vectors absorbing all the other terms in the model. Here is a simple example of the rational model:(6)y˙=0.1y3+sinu+0.5u31+cosy+u2.Its U-model realisation can be determined with(7)y˙=λn0+λn1sinu+λn2u2+λn3u3λd0+λd1u+λd2u2,where(8)λn0=0.1y3,λn1=1,λn2=0,λn3=0.5,λd0=1+cosy,λd1=0,λd2=1.


### 2.2. State Space U-Model: Multilayer Realisation

For a general SISO CT state space model,(9)X˙=FX,u,y=HX,where u,y∈ℝ, X∈ℝn, and F∈ℝn are smooth mapping to represent the input to the state and H∈ℝ is a smooth mapping to drive the states to the outputs. In this study, assume that there is no unstable zero dynamics (i.e., the model reversible) and that the state X can be obtained through measurement or observation.Expand state-space model (9) into a multilayer polynormal expression as follows:(10)x˙1=F1x1,x2,…,xn,x˙2=F2x1,x2,…,xn,⋮x˙n=Fnx1,x2,…,xn,u,y=Hx1,x2,…,xn.Convert state-space model (10) into a multilayer U-model expression as follows:(11)x˙1=∑i=0nλ1if1ix2,x˙2=∑i=0nλ2if2ix3,⋮x˙n=∑i=0nλnifniu,y=∑i=0nhix1,x2,…,xn.For each line of (11), λ∗∗ and f∗∗ are time-varying parameter absorbing all the other variables and the U-basis function, respectively.For illustration, consider a nonlinear SISO system state-space model of(12)x˙1=x1+0.5sinx2,x˙2=−x1+u,y=x1.Using the absorbing rule to convert (12) into multilayer U-model as(13)x˙1=λ10+λ11f11x2,x˙2=λ20+λ21f21u,y=x1,where(14)λ10=x1,λ11=0.5,f11x2=sinx2,λ20=−x1,λ21=1,f21u=u.


### 2.3. U-Control System Design

Here is a systematic summary of the U-control framework.Figure1 shows the classical control system framework, where GP is the plant model, which could be linear or nonlinear dynamics and can be described by polynomial and state space models [14]. Let G (not shown in the figure) be the closed-loop performance function, specified with ad hoc applications in advance by designers and/or customers, r is the reference, which is the desired output of the control system, e is the difference (error) between the output y, and the reference r. Gc is the designed controller.Figure 1
Classical control system framework.The main principles of this kind of control system design framework are to generate a suitable control input signalu to drive the system output trajectory y following a set of specified closed loop performances (both transient and steady state).Figure2 shows the U-control system framework [3, 5], in which Gc1 is a linear invariant controller, can be designed by Gc1=G/1−G while GP−1GP=1, where GP−1 is the controlled plant’s dynamic inverse. U-control framework is applicable to both linear and nonlinear structures as long as dynamic inverse GP−1 exist.Figure 2
U-control system framework.To explain the control system design procedure, consider a CT SISO linear closed-loop feedback control system framework with a set of (F,Gc1,Gip):(15)∑=F,Gc1,Gip,where Gip=GP−1GP.In general, the U-control system design procedure has two separate steps:(1)
Assume the plant modelGP stable and bounded, and its inversse GP−1 exists. From Figure 2, the controller which is shown in the dashed line block has two parts: the invariant controller Gc1 and plant’s dynamic inverter GP−1. To facilitate the design of GP−1, the convert plant model GP (1) into its U-model. Alternatively, (15) can be expressed as(16)∑=F,UGc1,GP−1,Gp,whereGc1,GP−1 is defined as U-controller. Determine GP−1 to work out the controller output u, which is the control input to the plant GP. The eventual goal is to make plant output equal to the invariant (IV) controller output: v=y, that is, to reach GP−1GP=1 under the proper dynamic inversion.(2)
Design invariant controllerGc1. Figure 3 shows the U-control structure while GP−1GP=1 achieved, that is,(17)∑=F,Gc1.Figure 3
U-model-based simplified control system.This is a type of linear control systems. Therefore, the desired closed-loop transfer functionG can be expressed as G=Gc1/1+Gc1, where G can be effectively designed with two significant factors shaping linear system response, damping ratio ζ, and undamped natural frequency ωn. Therefore, the invariant controller Gc1 can be obtained by inversing G into Gc1=G/1−G while GP−1GP=1.As the invariant controllerGc1 design is independent of plant GP, the U-control allows once-off design for all stable-nonminimum phase plants, except designing the inverter GP−1 of the considered plant.


## 2.1. Polynomial U-Model: Single Layer Realisation

### 2.1.1. Basic Polynomial U-Model

Consider a general continuous-time U-model [19] for Single-Input and Single-Output (SISO) polynomial dynamic systems with a triplet of yt,ut,λt, yt∈ℝ, and ut∈ℝ for the output, input, and parameter, respectively, at time t∈ℝ+:(1)My=∑j=0JλjYM−1,UN−1,ΘNuj,M≥N,where My and Nu are the Mth and Nth order derivatives of the plant output y and the plant input u, respectively, the time-varying parameter λj∗∈ℝ+, associated with the input Nuj, absorbs all the other terms in YM−1=m−1y,m−2y,…y∈ℝM, UN−1=n−1u,n−2u,…u∈ℝN, and coefficients Θ.Here is an example for understanding, consider a classical NAMAX polynomial model:(2)y¨=y−1y˙+1+y2u+1+y˙2u2+y+y+y˙2u3.Its U-model realisation can be determined with(3)y¨=λ0+λ1u+λ2u2+λ3u3,λ0=y−1y˙+y,λ1=1+y2,λ2=1+y˙2,λ3=y+y˙2.Inspection of (2) and (3), the U-realisation is straightforward generally.It should be remarked that the U-polynomial is the same as its presented classical polynomials in the model properties, but oriented expression for control system design [14].


### 2.1.2. Extended U-Model (Rational Models)

Rational model is totally nonlinear [3], and its polynomial expression is a ratio of two polynomials, that is,(4)My=fpnψn∗Θnfpdψd∗Θd,where fpn is the input-output mapping function of the polynomial in the numerator and fpd is the input-output mapping function of the polynomial in the denominator. Its U-realisation is accordingly described by(5)My=∑j=0nλnjYM−1,UN−1,ΘfnjNu∑j=0dλdjYM−1,UN−1,ΘfdjNu,M≥N,where fnj and fdj are vector functions of the control vector Nu in numerator and denominator, respectively, and λnj and λdj are the associated parameters vectors absorbing all the other terms in the model. Here is a simple example of the rational model:(6)y˙=0.1y3+sinu+0.5u31+cosy+u2.Its U-model realisation can be determined with(7)y˙=λn0+λn1sinu+λn2u2+λn3u3λd0+λd1u+λd2u2,where(8)λn0=0.1y3,λn1=1,λn2=0,λn3=0.5,λd0=1+cosy,λd1=0,λd2=1.


## 2.1.1. Basic Polynomial U-Model

Consider a general continuous-time U-model [19] for Single-Input and Single-Output (SISO) polynomial dynamic systems with a triplet of yt,ut,λt, yt∈ℝ, and ut∈ℝ for the output, input, and parameter, respectively, at time t∈ℝ+:(1)My=∑j=0JλjYM−1,UN−1,ΘNuj,M≥N,where My and Nu are the Mth and Nth order derivatives of the plant output y and the plant input u, respectively, the time-varying parameter λj∗∈ℝ+, associated with the input Nuj, absorbs all the other terms in YM−1=m−1y,m−2y,…y∈ℝM, UN−1=n−1u,n−2u,…u∈ℝN, and coefficients Θ.Here is an example for understanding, consider a classical NAMAX polynomial model:(2)y¨=y−1y˙+1+y2u+1+y˙2u2+y+y+y˙2u3.Its U-model realisation can be determined with(3)y¨=λ0+λ1u+λ2u2+λ3u3,λ0=y−1y˙+y,λ1=1+y2,λ2=1+y˙2,λ3=y+y˙2.Inspection of (2) and (3), the U-realisation is straightforward generally.It should be remarked that the U-polynomial is the same as its presented classical polynomials in the model properties, but oriented expression for control system design [14].


## 2.1.2. Extended U-Model (Rational Models)

Rational model is totally nonlinear [3], and its polynomial expression is a ratio of two polynomials, that is,(4)My=fpnψn∗Θnfpdψd∗Θd,where fpn is the input-output mapping function of the polynomial in the numerator and fpd is the input-output mapping function of the polynomial in the denominator. Its U-realisation is accordingly described by(5)My=∑j=0nλnjYM−1,UN−1,ΘfnjNu∑j=0dλdjYM−1,UN−1,ΘfdjNu,M≥N,where fnj and fdj are vector functions of the control vector Nu in numerator and denominator, respectively, and λnj and λdj are the associated parameters vectors absorbing all the other terms in the model. Here is a simple example of the rational model:(6)y˙=0.1y3+sinu+0.5u31+cosy+u2.Its U-model realisation can be determined with(7)y˙=λn0+λn1sinu+λn2u2+λn3u3λd0+λd1u+λd2u2,where(8)λn0=0.1y3,λn1=1,λn2=0,λn3=0.5,λd0=1+cosy,λd1=0,λd2=1.


## 2.2. State Space U-Model: Multilayer Realisation

For a general SISO CT state space model,(9)X˙=FX,u,y=HX,where u,y∈ℝ, X∈ℝn, and F∈ℝn are smooth mapping to represent the input to the state and H∈ℝ is a smooth mapping to drive the states to the outputs. In this study, assume that there is no unstable zero dynamics (i.e., the model reversible) and that the state X can be obtained through measurement or observation.Expand state-space model (9) into a multilayer polynormal expression as follows:(10)x˙1=F1x1,x2,…,xn,x˙2=F2x1,x2,…,xn,⋮x˙n=Fnx1,x2,…,xn,u,y=Hx1,x2,…,xn.Convert state-space model (10) into a multilayer U-model expression as follows:(11)x˙1=∑i=0nλ1if1ix2,x˙2=∑i=0nλ2if2ix3,⋮x˙n=∑i=0nλnifniu,y=∑i=0nhix1,x2,…,xn.For each line of (11), λ∗∗ and f∗∗ are time-varying parameter absorbing all the other variables and the U-basis function, respectively.For illustration, consider a nonlinear SISO system state-space model of(12)x˙1=x1+0.5sinx2,x˙2=−x1+u,y=x1.Using the absorbing rule to convert (12) into multilayer U-model as(13)x˙1=λ10+λ11f11x2,x˙2=λ20+λ21f21u,y=x1,where(14)λ10=x1,λ11=0.5,f11x2=sinx2,λ20=−x1,λ21=1,f21u=u.


## 2.3. U-Control System Design

Here is a systematic summary of the U-control framework.Figure1 shows the classical control system framework, where GP is the plant model, which could be linear or nonlinear dynamics and can be described by polynomial and state space models [14]. Let G (not shown in the figure) be the closed-loop performance function, specified with ad hoc applications in advance by designers and/or customers, r is the reference, which is the desired output of the control system, e is the difference (error) between the output y, and the reference r. Gc is the designed controller.Figure 1
Classical control system framework.The main principles of this kind of control system design framework are to generate a suitable control input signalu to drive the system output trajectory y following a set of specified closed loop performances (both transient and steady state).Figure2 shows the U-control system framework [3, 5], in which Gc1 is a linear invariant controller, can be designed by Gc1=G/1−G while GP−1GP=1, where GP−1 is the controlled plant’s dynamic inverse. U-control framework is applicable to both linear and nonlinear structures as long as dynamic inverse GP−1 exist.Figure 2
U-control system framework.To explain the control system design procedure, consider a CT SISO linear closed-loop feedback control system framework with a set of (F,Gc1,Gip):(15)∑=F,Gc1,Gip,where Gip=GP−1GP.In general, the U-control system design procedure has two separate steps:(1)
Assume the plant modelGP stable and bounded, and its inversse GP−1 exists. From Figure 2, the controller which is shown in the dashed line block has two parts: the invariant controller Gc1 and plant’s dynamic inverter GP−1. To facilitate the design of GP−1, the convert plant model GP (1) into its U-model. Alternatively, (15) can be expressed as(16)∑=F,UGc1,GP−1,Gp,whereGc1,GP−1 is defined as U-controller. Determine GP−1 to work out the controller output u, which is the control input to the plant GP. The eventual goal is to make plant output equal to the invariant (IV) controller output: v=y, that is, to reach GP−1GP=1 under the proper dynamic inversion.(2)
Design invariant controllerGc1. Figure 3 shows the U-control structure while GP−1GP=1 achieved, that is,(17)∑=F,Gc1.Figure 3
U-model-based simplified control system.This is a type of linear control systems. Therefore, the desired closed-loop transfer functionG can be expressed as G=Gc1/1+Gc1, where G can be effectively designed with two significant factors shaping linear system response, damping ratio ζ, and undamped natural frequency ωn. Therefore, the invariant controller Gc1 can be obtained by inversing G into Gc1=G/1−G while GP−1GP=1.As the invariant controllerGc1 design is independent of plant GP, the U-control allows once-off design for all stable-nonminimum phase plants, except designing the inverter GP−1 of the considered plant.


## 3. UM-Dynamic Inversion

Nonlinear dynamic inversion (NDI) is a generic control technique in nature, that is, improving control performance through control system design. Currently, NDI has been a challenging research issue and practical significance in mechanical motion control systems, such as turbines, robots, and flying vehicles [22]. Basic NDI calculation procedure is differentiating the plant output equation results N times to find the direct relationship between the input u and Nth order derivative of the output y under the Lie derivative formulation [22]. However, NDI is very sensitive and unstable in case of model inaccuracies and mismatch. In order to combat the uncertainty of the plant and improve the system robustness, Incremental Nonlinear Dynamic Inversion (INDI) [22] and adaptive INDI [23] have been introduced in another complicated formulation.Different from the computational complexity of basic NDI under the Lie derivative expression, this study converts the plant model into U-model realisation in a systematic concise formulation, which is generically applicable to both polynomial and state space equations. This also establishes a foundation for future development of robust UM-dynamic inversion.The U-model-based dynamic inversion (UM-dynamic inversion) algorithm is to obtain the inputu by solving the root from (1), that is,(18)Gp−1⇔Nu∈My−∑j=0JλjYM−1,UM−1,ΘNuj=0.For the solution which exists, the systems must be Bounded Input and Bounded Output (BIBO) stable and no unstable zero dynamic (nonminimum phase).


### 3.1. Algorithms for Polynomial Models

#### 3.1.1. Linear Plant

Use Laplace transform (S operator, jy=djy/dtj⇔sjY, ∫⋯∫ydt⇔1/sjY) to expresses a set of general linear dynamic plants as(19)Y=∑j=0j=NβjsN−j∑j=0j=MαjsM−ju=β0sN+β1sN−1+…+βN−1s+βNα0sM+α1sM−1+⋯+αM−1s+αMU,M≥N,where Y and U are Laplace transform of the output and input, respectively, and M and N are the orders (highest power) of the denominator and numerator functions, respectively.Accordingly, its U-realisation is given as(20)My=∑j=01λjYM−1,UN−1,ΘNuj,where(21)λoYM−1,UN−1,Θ=−∑j=1j−M−1αjSM−jY+∑j=1j=N−1βjSN−jUα0,λ1YM−1,UN−1,Θ=β0α0.As the drives are sensitive to noise signals in applications, convert the operations into integral implementations by multiplying1/sN on both sides of (20), and this gives(22)M−Ny=∑j=01λjYM−1,UN−1,ΘusN,where(23)λoYM−1,UN−1,Θ=−∑j=1j=M−1αjsM−j/sNY+∑j=1j=N−1βjsN−j/sNUα0,λ1YM−1,UN−1,Θ=β0α0,Therefore, the alternative U-model is(24)M−Ny=∑j=01λjYM−1,UN−1,Θuj.


#### 3.1.2. Nonlinear Plant

For UM-dynamic inversion of (18), replace all the derivatives with integrals through division by the output derivative order, which is formulated as(25)jy=My1sM−j,where jy=djy/dtjHere is the practical implementation of (18):(26)Gp−1⇔u∈y−∑j=0JλjYM−1,UM−1,ΘNujsM=0.To illustrate the conversion to U-model from a nonlinear polynomial, consider an example of(27)y¨=1−y2y˙−u+1+yu˙+1+y˙2u˙2+u˙3.In U-realisation, its derivative-based operation becomes(28)y¨=λ0+λ1u˙+λ2u˙2+λ3u˙3,where(29)λ0=1−y2y˙−u,λ1=1+y2+y˙2,λ2=1+y˙2,λ3=1.To convert it into integration operation, the corresponding U-realisation has the form of(30)y=λ0s2+λ1su+λ2su2+λ3su3.


### 3.2. Algorithms for State Space Models

#### 3.2.1. Linear Plant

For a general SISO linear CT state-space system model, it has(31)x˙=Ax+Bu,y=Cx+Du,where u,y∈ℝ, x∈ℝn, let A=010⋯⋯⋱⋱⋯−α1⋯−αn, B=00⋯1T, C=β1⋯βn, and D=0. Expanding (14) gives rise to(32)x˙1=x2,x˙2=x3,⋮x˙n=∑j=1nαjxj+u,y=∑j=1nβjxj,where x is the state vector and u and y are the input and output of the model, respectively.For taking up such UM-dynamic inversion, first use a systematic approach [24] to convert the linear state space model into input/output transfer function by(33)YsUs=CsIn−A−1B,where Ys and Us are the Laplace transforms of the output and input, respectively, and In is an identity matrix of dimension n.Then, use the linear polynomial dynamic inversion procedure presented in Section3.1.1.


#### 3.2.2. Nonlinear Plant

For the model of (10), here is the step by step procedure for UM-dynamic inversion.(1)
Generate direct mapping between the outputy and the input u, by differentiating the state variables in the output equation till it is directly related to u in the state equation(2)
Use the procedure for nonlinear polynomial dynamic inversion in Section3.1.2 to determine the solutionsIt should be noted that the above computations require the full state variable necessarily available/measurable.To illustrate the realisation, consider a nonlinear state space model of(34)x˙1=x1−x2−x32,x˙2=−x1+x22−x3,x˙3=−x12−x2+x3+u,y=x1.Differentiatingy twice against x1 with the output equation gives(35)y˙=x˙1=x1−x2−x32,y¨=x˙1−x˙2−2x3x12−x2+x3+u.As the second line of (35) directly relates the output and the input, it can be used for the dynamic inversion. The corresponding polynomial U-model is given by(36)y¨=λ0+λ1u,λ0=x˙1−x˙2−2x3x12−x2+x3,λ1=−2x3.Convert to integral expression as(37)y=λ0s2+λ1s2u.


### 3.3. Iterative Algorithms

There are systematic routines to find roots for the 1st order (linear) and 2nd order (nonlinear) polynomials. However, it is difficult analytically to determine the roots for the 3rd and up order polynomials. Commonly iterative root-solving algorithms are considered.Newton–Raphson algorithm [25] is a classical iterative algorithm, which can be used to determine the roots of polynomial U-models, that is, the solution of UM-dynamic inversion. In formulation, this algorithm is given by(38)NuK+1=NuK−My−∑j=0JλjYM−1,UM−1,ΘNujdMy−∑j=0JλjYM−1,UM−1,ΘNuj/dNu.In addition, to test U-control systems in Simulink/Matlab simulation, Matlab functions can be used to find the roots directly.


## 3.1. Algorithms for Polynomial Models

### 3.1.1. Linear Plant

Use Laplace transform (S operator, jy=djy/dtj⇔sjY, ∫⋯∫ydt⇔1/sjY) to expresses a set of general linear dynamic plants as(19)Y=∑j=0j=NβjsN−j∑j=0j=MαjsM−ju=β0sN+β1sN−1+…+βN−1s+βNα0sM+α1sM−1+⋯+αM−1s+αMU,M≥N,where Y and U are Laplace transform of the output and input, respectively, and M and N are the orders (highest power) of the denominator and numerator functions, respectively.Accordingly, its U-realisation is given as(20)My=∑j=01λjYM−1,UN−1,ΘNuj,where(21)λoYM−1,UN−1,Θ=−∑j=1j−M−1αjSM−jY+∑j=1j=N−1βjSN−jUα0,λ1YM−1,UN−1,Θ=β0α0.As the drives are sensitive to noise signals in applications, convert the operations into integral implementations by multiplying1/sN on both sides of (20), and this gives(22)M−Ny=∑j=01λjYM−1,UN−1,ΘusN,where(23)λoYM−1,UN−1,Θ=−∑j=1j=M−1αjsM−j/sNY+∑j=1j=N−1βjsN−j/sNUα0,λ1YM−1,UN−1,Θ=β0α0,Therefore, the alternative U-model is(24)M−Ny=∑j=01λjYM−1,UN−1,Θuj.


### 3.1.2. Nonlinear Plant

For UM-dynamic inversion of (18), replace all the derivatives with integrals through division by the output derivative order, which is formulated as(25)jy=My1sM−j,where jy=djy/dtjHere is the practical implementation of (18):(26)Gp−1⇔u∈y−∑j=0JλjYM−1,UM−1,ΘNujsM=0.To illustrate the conversion to U-model from a nonlinear polynomial, consider an example of(27)y¨=1−y2y˙−u+1+yu˙+1+y˙2u˙2+u˙3.In U-realisation, its derivative-based operation becomes(28)y¨=λ0+λ1u˙+λ2u˙2+λ3u˙3,where(29)λ0=1−y2y˙−u,λ1=1+y2+y˙2,λ2=1+y˙2,λ3=1.To convert it into integration operation, the corresponding U-realisation has the form of(30)y=λ0s2+λ1su+λ2su2+λ3su3.


## 3.1.1. Linear Plant

Use Laplace transform (S operator, jy=djy/dtj⇔sjY, ∫⋯∫ydt⇔1/sjY) to expresses a set of general linear dynamic plants as(19)Y=∑j=0j=NβjsN−j∑j=0j=MαjsM−ju=β0sN+β1sN−1+…+βN−1s+βNα0sM+α1sM−1+⋯+αM−1s+αMU,M≥N,where Y and U are Laplace transform of the output and input, respectively, and M and N are the orders (highest power) of the denominator and numerator functions, respectively.Accordingly, its U-realisation is given as(20)My=∑j=01λjYM−1,UN−1,ΘNuj,where(21)λoYM−1,UN−1,Θ=−∑j=1j−M−1αjSM−jY+∑j=1j=N−1βjSN−jUα0,λ1YM−1,UN−1,Θ=β0α0.As the drives are sensitive to noise signals in applications, convert the operations into integral implementations by multiplying1/sN on both sides of (20), and this gives(22)M−Ny=∑j=01λjYM−1,UN−1,ΘusN,where(23)λoYM−1,UN−1,Θ=−∑j=1j=M−1αjsM−j/sNY+∑j=1j=N−1βjsN−j/sNUα0,λ1YM−1,UN−1,Θ=β0α0,Therefore, the alternative U-model is(24)M−Ny=∑j=01λjYM−1,UN−1,Θuj.


## 3.1.2. Nonlinear Plant

For UM-dynamic inversion of (18), replace all the derivatives with integrals through division by the output derivative order, which is formulated as(25)jy=My1sM−j,where jy=djy/dtjHere is the practical implementation of (18):(26)Gp−1⇔u∈y−∑j=0JλjYM−1,UM−1,ΘNujsM=0.To illustrate the conversion to U-model from a nonlinear polynomial, consider an example of(27)y¨=1−y2y˙−u+1+yu˙+1+y˙2u˙2+u˙3.In U-realisation, its derivative-based operation becomes(28)y¨=λ0+λ1u˙+λ2u˙2+λ3u˙3,where(29)λ0=1−y2y˙−u,λ1=1+y2+y˙2,λ2=1+y˙2,λ3=1.To convert it into integration operation, the corresponding U-realisation has the form of(30)y=λ0s2+λ1su+λ2su2+λ3su3.


## 3.2. Algorithms for State Space Models

### 3.2.1. Linear Plant

For a general SISO linear CT state-space system model, it has(31)x˙=Ax+Bu,y=Cx+Du,where u,y∈ℝ, x∈ℝn, let A=010⋯⋯⋱⋱⋯−α1⋯−αn, B=00⋯1T, C=β1⋯βn, and D=0. Expanding (14) gives rise to(32)x˙1=x2,x˙2=x3,⋮x˙n=∑j=1nαjxj+u,y=∑j=1nβjxj,where x is the state vector and u and y are the input and output of the model, respectively.For taking up such UM-dynamic inversion, first use a systematic approach [24] to convert the linear state space model into input/output transfer function by(33)YsUs=CsIn−A−1B,where Ys and Us are the Laplace transforms of the output and input, respectively, and In is an identity matrix of dimension n.Then, use the linear polynomial dynamic inversion procedure presented in Section3.1.1.


### 3.2.2. Nonlinear Plant

For the model of (10), here is the step by step procedure for UM-dynamic inversion.(1)
Generate direct mapping between the outputy and the input u, by differentiating the state variables in the output equation till it is directly related to u in the state equation(2)
Use the procedure for nonlinear polynomial dynamic inversion in Section3.1.2 to determine the solutionsIt should be noted that the above computations require the full state variable necessarily available/measurable.To illustrate the realisation, consider a nonlinear state space model of(34)x˙1=x1−x2−x32,x˙2=−x1+x22−x3,x˙3=−x12−x2+x3+u,y=x1.Differentiatingy twice against x1 with the output equation gives(35)y˙=x˙1=x1−x2−x32,y¨=x˙1−x˙2−2x3x12−x2+x3+u.As the second line of (35) directly relates the output and the input, it can be used for the dynamic inversion. The corresponding polynomial U-model is given by(36)y¨=λ0+λ1u,λ0=x˙1−x˙2−2x3x12−x2+x3,λ1=−2x3.Convert to integral expression as(37)y=λ0s2+λ1s2u.


## 3.2.1. Linear Plant

For a general SISO linear CT state-space system model, it has(31)x˙=Ax+Bu,y=Cx+Du,where u,y∈ℝ, x∈ℝn, let A=010⋯⋯⋱⋱⋯−α1⋯−αn, B=00⋯1T, C=β1⋯βn, and D=0. Expanding (14) gives rise to(32)x˙1=x2,x˙2=x3,⋮x˙n=∑j=1nαjxj+u,y=∑j=1nβjxj,where x is the state vector and u and y are the input and output of the model, respectively.For taking up such UM-dynamic inversion, first use a systematic approach [24] to convert the linear state space model into input/output transfer function by(33)YsUs=CsIn−A−1B,where Ys and Us are the Laplace transforms of the output and input, respectively, and In is an identity matrix of dimension n.Then, use the linear polynomial dynamic inversion procedure presented in Section3.1.1.


## 3.2.2. Nonlinear Plant

For the model of (10), here is the step by step procedure for UM-dynamic inversion.(1)
Generate direct mapping between the outputy and the input u, by differentiating the state variables in the output equation till it is directly related to u in the state equation(2)
Use the procedure for nonlinear polynomial dynamic inversion in Section3.1.2 to determine the solutionsIt should be noted that the above computations require the full state variable necessarily available/measurable.To illustrate the realisation, consider a nonlinear state space model of(34)x˙1=x1−x2−x32,x˙2=−x1+x22−x3,x˙3=−x12−x2+x3+u,y=x1.Differentiatingy twice against x1 with the output equation gives(35)y˙=x˙1=x1−x2−x32,y¨=x˙1−x˙2−2x3x12−x2+x3+u.As the second line of (35) directly relates the output and the input, it can be used for the dynamic inversion. The corresponding polynomial U-model is given by(36)y¨=λ0+λ1u,λ0=x˙1−x˙2−2x3x12−x2+x3,λ1=−2x3.Convert to integral expression as(37)y=λ0s2+λ1s2u.


## 3.3. Iterative Algorithms

There are systematic routines to find roots for the 1st order (linear) and 2nd order (nonlinear) polynomials. However, it is difficult analytically to determine the roots for the 3rd and up order polynomials. Commonly iterative root-solving algorithms are considered.Newton–Raphson algorithm [25] is a classical iterative algorithm, which can be used to determine the roots of polynomial U-models, that is, the solution of UM-dynamic inversion. In formulation, this algorithm is given by(38)NuK+1=NuK−My−∑j=0JλjYM−1,UM−1,ΘNujdMy−∑j=0JλjYM−1,UM−1,ΘNuj/dNu.In addition, to test U-control systems in Simulink/Matlab simulation, Matlab functions can be used to find the roots directly.


## 4. Simulation Demonstrations

This simulation demonstration selected four plant models to tests the UM-dynamic inversion and their associated U-control systems with the following bullet points:(i)
To demonstrate the generality and effectiveness of UM dynamic inversion(ii)
To demonstrate the principle of model-independent design in U-control, supported by the dynamic inversions(iii)
To demonstrate a once-off design with the linear invariant controller in accordance with a closed-loop performance specification irrespective of the plant model structures(iv)
To validate the applicability, conciseness, and efficiency of the U-control and UM-dynamic inversion, particularly in designing nonlinear control systems


### 4.1. U-Control System Design

With reference to the previous introduction in Section2, for these simulations, design a unique U-control system with a desired system output response in terms of damping ratio ζ=0.7, undamped natural frequency ωn=1, and zero steady state error to a step reference input [15]. Accordingly, this closed-loop transfer function was specified as(39)YSRS=GS=1s2+1.4s+1.The invariant controllerGc1 was determined by taking the inverse of (39):(40)Gc1=G1−G=1s2+1.4s.


### 4.2. Case 1: Linear Polynomial and State Space Models

#### 4.2.1. Plant 1: Polynomial Model

(41)GPs=3s2+6s+4s2+2s+1.The corresponding U-model was(42)y=λ0s2+λ1u,λ0=−2y˙−y+6u˙+4u,λ1=3.


#### 4.2.2. Plant 2: State Space Model

(43)x˙1x˙2=01.5−0.5−2x1x2+01,y=10x1x2+0.The corresponding U-model was(44)y¨=λ0+λ1u,λ0=1.5−0.5x1−2x2,λ1=1.5.


### 4.3. Case 2: Nonlinear Polynomial and State Space Models

#### 4.3.1. Plant 3: Polynomial Model

(45)y˙=u˙3+u˙2−u˙−0.5y+sinu.The corresponding U-model was(46)y˙=λ0+λ1u˙+λ2u˙2+λ3u˙3,λ0=−0.5y+sinu,λ1=−1,λ2=1,λ3=1.


#### 4.3.2. Plant 4: State Space Model

(47)x˙1=x2,x˙2=sinx1+u,y=x1.The corresponding U-model model was(48)y¨=λ0+λ1u,λ0=sinx1,λ1=1.


### 4.4. Simulation

Figures4 and 5 show the U-control system design simulation structures. Figure 6 and 7 show the simulation results. All of these have demonstrated that the purposes outlined at the beginning of the section have been achieved.Figure 4
(a) U-control of plant 1. (b) U-control of plant 2.
(a)(b)Figure 5
(a) U-control of plant 3. (b) U-control of plant 4.
(a)(b)Figure 6
(a) Plant outputs and reference for plant1 and plant2. (b) Controller outputs for plant1 and plant2.
(a)(b)Figure 7
(a) Plant outputs and reference for plant 3 and plant 4. (b) Controller outputs for plant 3 and plant 4.
(a)(b)


## 4.1. U-Control System Design

With reference to the previous introduction in Section2, for these simulations, design a unique U-control system with a desired system output response in terms of damping ratio ζ=0.7, undamped natural frequency ωn=1, and zero steady state error to a step reference input [15]. Accordingly, this closed-loop transfer function was specified as(39)YSRS=GS=1s2+1.4s+1.The invariant controllerGc1 was determined by taking the inverse of (39):(40)Gc1=G1−G=1s2+1.4s.


## 4.2. Case 1: Linear Polynomial and State Space Models

### 4.2.1. Plant 1: Polynomial Model

(41)GPs=3s2+6s+4s2+2s+1.The corresponding U-model was(42)y=λ0s2+λ1u,λ0=−2y˙−y+6u˙+4u,λ1=3.


### 4.2.2. Plant 2: State Space Model

(43)x˙1x˙2=01.5−0.5−2x1x2+01,y=10x1x2+0.The corresponding U-model was(44)y¨=λ0+λ1u,λ0=1.5−0.5x1−2x2,λ1=1.5.


## 4.2.1. Plant 1: Polynomial Model

(41)GPs=3s2+6s+4s2+2s+1.The corresponding U-model was(42)y=λ0s2+λ1u,λ0=−2y˙−y+6u˙+4u,λ1=3.


## 4.2.2. Plant 2: State Space Model

(43)x˙1x˙2=01.5−0.5−2x1x2+01,y=10x1x2+0.The corresponding U-model was(44)y¨=λ0+λ1u,λ0=1.5−0.5x1−2x2,λ1=1.5.


## 4.3. Case 2: Nonlinear Polynomial and State Space Models

### 4.3.1. Plant 3: Polynomial Model

(45)y˙=u˙3+u˙2−u˙−0.5y+sinu.The corresponding U-model was(46)y˙=λ0+λ1u˙+λ2u˙2+λ3u˙3,λ0=−0.5y+sinu,λ1=−1,λ2=1,λ3=1.


### 4.3.2. Plant 4: State Space Model

(47)x˙1=x2,x˙2=sinx1+u,y=x1.The corresponding U-model model was(48)y¨=λ0+λ1u,λ0=sinx1,λ1=1.


## 4.3.1. Plant 3: Polynomial Model

(45)y˙=u˙3+u˙2−u˙−0.5y+sinu.The corresponding U-model was(46)y˙=λ0+λ1u˙+λ2u˙2+λ3u˙3,λ0=−0.5y+sinu,λ1=−1,λ2=1,λ3=1.


## 4.3.2. Plant 4: State Space Model

(47)x˙1=x2,x˙2=sinx1+u,y=x1.The corresponding U-model model was(48)y¨=λ0+λ1u,λ0=sinx1,λ1=1.


## 4.4. Simulation

Figures4 and 5 show the U-control system design simulation structures. Figure 6 and 7 show the simulation results. All of these have demonstrated that the purposes outlined at the beginning of the section have been achieved.Figure 4
(a) U-control of plant 1. (b) U-control of plant 2.
(a)(b)Figure 5
(a) U-control of plant 3. (b) U-control of plant 4.
(a)(b)Figure 6
(a) Plant outputs and reference for plant1 and plant2. (b) Controller outputs for plant1 and plant2.
(a)(b)Figure 7
(a) Plant outputs and reference for plant 3 and plant 4. (b) Controller outputs for plant 3 and plant 4.
(a)(b)


## 5. Test of U-Control of a Wind Energy Conversion System

### 5.1. Brief Review of Wind Energy Conversion Control Systems

Wind power is a clean natural resource to supplement the other power resources from fossil fuels, coal, solar, and so on. This rich power source is widely distributed, renewable, has no greenhouse gas emissions, and uses little land [26]. In conversion of wind power, in which air flows drive wind turbines to generate electrical power, the need of effective control strategies for cost reduction and power acquisition performance is commonly recognised. Particularly, such control system design is very critical for Variable Speed Wind Turbines (VSWT). The other unavoidable issue in wind energy conversion is that the turbine performs according to linear dynamics, but the power conversion is nonlinear from the multiplication of two dynamic variables (the wind power is obtained as a product of torque/input and rotor angular speed). Designing such control systems is challenging in formulation and implementation. Precup et al. [27] gives a collection of the up to date research on advanced control and optimization paradigms for wind energy systems. Regarding U-control of the wind energy conversion systems, Zhu et al. [28] presents the 1st U-model-based control system formulisation and design for wind energy conversion systems. In contrast, this study removes the demand on solving Diophantine equation for pole placement assignment and directly uses closed loop inversion to design the invariant controller. Furthermore, this study is a continuous time control and gives emphasis on illustration of the general platform for industrial applications using Simulink block diagram connections, rather than Matlab-coded programs.


### 5.2. Plant Models [29]

Modelling of the wind turbine has played a significant role in understanding of the behaviour of the wind turbine over its region of operation because it allows for the development of comprehensive control systems that aid in optimal operation of a wind turbine. Such mathematical models are the foundation to quantify control performance of the energy systems. Furthermore, these models are essential reference for the design of the turbines and minimise generation costs leading to cost reduction in wind energy, consequently making it an economically viable alternative source of energy [30]. This section characterises these wind turbine section models into an integrated nonlinear dynamic plant operational model to describe input/output relationships.


#### 5.2.1. Drive Train in Lumped Mass Model (as Shown in Figure8)

Here is the nomenclature list.ωr: rotor angular speed aerodynamic torqueJr and Jg: rotor and generator inertias, respectively.Kr and Kg: rotor and generator external damping, respectivelyJt: integrated inertiaKt: integrated dampingTem: converted electromagnetic torqueTa: rotor torque, from external wind power in practical systemsTg: generator torque, regulate the system operation to generate powerFigure 8
Schematic diagram of the drive train.The rotor speedωr is driven by the rotor torque Ta and the low-speed torque Tls. The generator speed ωg is driven by the high-speed torque Ths and the electromagnetic torque Tem. It should be noted that using the gear box can change the generator speed. The dynamics of the rotor and the generator can be described by Newton’s law in forms of(49)Jrω˙r=Ta−Krωr−Tls,Jgω˙g=Ths−Kgωg−Tem.Define the gearbox rationg as(50)ng=ωgωr=TlsThs.Invoking (50), the generator dynamic in (49) can be rewritten as(51)ng2Jgω˙r=Tls−ng2Kgωr−ngTem.Thus, the drive train model can be described by combining (49) with (51) as(52)Jtω˙r=Ta−Ktωr−Tg,where(53)Jt=Jr+ng2Jg,Kt=Kr+ng2Kg,Tg=ngTem.


#### 5.2.2. Power OutputPg

Pg, the power output from the generator, is given by(54)Pg=Tgωr.


#### 5.2.3. Wind Speed TorqueTa

In system (1), external wind is the source of the driven force. The wind speed torque is given by(55)Ta=12ρπR3Cqλ,βv^−ξ2,where ρ, v, and R are the air density, wind speed, and rotor radius, respectively, β is the blade pitch angle, λ=Rωr/v is the tip-speed ratio; v^ is the estimation of the effective wind speed v, which can be measured via anemometer, and ξ denotes the measurement noise; Cqλ,β, the efficiency for the wind turbine power conversion, is given by(56)Cqλ,β=0.22λ116m−0.4β−5exp−12.5m,with 1/m=1/λ+0.08β−0.035/β3+1


#### 5.2.4. Energy Conversion Input-Output Model for Control System Design

From the above physical principle models, the energy conversion input-output model for control system design can be expressed as(57)P˙gTg−T˙gPg=1JtTaTg2−KtJtPgTg−1JtTg3,where the control input is the generator torque Tg and the plant output is the power output Pg.


#### 5.2.5. Power Conversion in Low Speed Region

For the drive train (49) to collect the maximum quantity of energy embedded in the low speed wind region, it requires(58)Pd=npPamax,with(59)Pamax=12ρπR2Cpmaxv3,where Cpmax is the maximum power coefficient and np is the ratio between the desired generator power Pd and the maximised available power Pamax.


### 5.3. U-Control System

The U-control system was the same as designed in Section4. The U-realisation of the input-output plant model was derived as(60)P˙g=λ0+λ1T˙g,λ0=1/JtTaTg2−Kt/JtPgTg−1/JtTg3Tg,λ1=PgTg.


### 5.4. Simulation Results

The selected generator, equipped with three blade, horizontal axis, and up wind variable speed wind turbine, generates 1.5 MW electrical output, made by WINDEY Co. This category of generators has been used worldwide [27]. The major parameters are listed in Table 1.Table 1
Wind turbine characteristics.
Rated power1.5 MWRotor radiusR = 38.5 mRotor inertiaJr = 4456761 kg·m2Generator inertiaJg = 123 kg·m2Rotor friction coefficientKr = 45.52 N·m/rad/sGenerator friction coefficientKg = 0.4 N·m/rad/sGearbox rationg = 104.494In consequence, the parameters of the turbine dynamic model are determined with(61)Jt=Jr+ng2Jg=5.7998×106,Kt=Kr+ng2Kg=4.4131×103,KtJt=7.609×10−4.For U-model (60), its time varying parameters are assigned with(62)P˙g=λ0+λ1T˙g,λ0=15.7998×106TaTg−7.609×10−4Pg−15.7998×106Pg2,λ1=PgTg.The rest of the simulation conditions/parameters include the desired powerPd, wind torque Ta from (55) and (56), the specified wind speed v with a mean of 9 m/s and turbulence intensity of 10%, sensor noise represented by a uniformly distributed random sequence of [−0.3, 0.3], np=0.8 for the ratio between the desired generator power Pd and the maximum available power Pamax, Cpmax=0.4382 for the maximum power ratio, and ρ=1.12 for the air density.Figure9 shows the constructed U-control system in Simulink block diagrams. Figure 10 shows the simulation results which are the same as those obtained from [27]. However, the differences are (1) this study is continuous time control system against the discrete time control systems [27]; (2) this study uses concise closed loop inversion to determine the invariant controller against solving the complicated Diophantine equation; and (3) this study uses the Simulink block diagram to build up the control system against using Matlab functions to develop coded programs, which block diagram-based simulation is much more engineering meaningful, transparent, and cost-effective than coded programming. Inspection of the generated plots: Figure 10(a) shows the simulated wind speed profile, Figure 10(b) shows that the generated output power properly follows the specified power trajectory and is similar to wind speed profile even though with wind measurement noise, Figure 10(c) shows the variation of the rotor torque following the wind force, and Figure 10(d) shows the tracking error amplitude converged to zero mean and small variance, and this indicates that the U-control system has converted into the maximum available power from the wind power.Figure 9
U-control of the wind energy conversion system.Figure 10
(a) Effective wind speedvt. (b) Desired power Pdt and generator output power Pgt. (c) Wind torque Tat. (d) Tracking control error between Pdt and Pgt.
(a)(b)(c)(d)


## 5.1. Brief Review of Wind Energy Conversion Control Systems

Wind power is a clean natural resource to supplement the other power resources from fossil fuels, coal, solar, and so on. This rich power source is widely distributed, renewable, has no greenhouse gas emissions, and uses little land [26]. In conversion of wind power, in which air flows drive wind turbines to generate electrical power, the need of effective control strategies for cost reduction and power acquisition performance is commonly recognised. Particularly, such control system design is very critical for Variable Speed Wind Turbines (VSWT). The other unavoidable issue in wind energy conversion is that the turbine performs according to linear dynamics, but the power conversion is nonlinear from the multiplication of two dynamic variables (the wind power is obtained as a product of torque/input and rotor angular speed). Designing such control systems is challenging in formulation and implementation. Precup et al. [27] gives a collection of the up to date research on advanced control and optimization paradigms for wind energy systems. Regarding U-control of the wind energy conversion systems, Zhu et al. [28] presents the 1st U-model-based control system formulisation and design for wind energy conversion systems. In contrast, this study removes the demand on solving Diophantine equation for pole placement assignment and directly uses closed loop inversion to design the invariant controller. Furthermore, this study is a continuous time control and gives emphasis on illustration of the general platform for industrial applications using Simulink block diagram connections, rather than Matlab-coded programs.


## 5.2. Plant Models [29]

Modelling of the wind turbine has played a significant role in understanding of the behaviour of the wind turbine over its region of operation because it allows for the development of comprehensive control systems that aid in optimal operation of a wind turbine. Such mathematical models are the foundation to quantify control performance of the energy systems. Furthermore, these models are essential reference for the design of the turbines and minimise generation costs leading to cost reduction in wind energy, consequently making it an economically viable alternative source of energy [30]. This section characterises these wind turbine section models into an integrated nonlinear dynamic plant operational model to describe input/output relationships.


### 5.2.1. Drive Train in Lumped Mass Model (as Shown in Figure8)

Here is the nomenclature list.ωr: rotor angular speed aerodynamic torqueJr and Jg: rotor and generator inertias, respectively.Kr and Kg: rotor and generator external damping, respectivelyJt: integrated inertiaKt: integrated dampingTem: converted electromagnetic torqueTa: rotor torque, from external wind power in practical systemsTg: generator torque, regulate the system operation to generate powerFigure 8
Schematic diagram of the drive train.The rotor speedωr is driven by the rotor torque Ta and the low-speed torque Tls. The generator speed ωg is driven by the high-speed torque Ths and the electromagnetic torque Tem. It should be noted that using the gear box can change the generator speed. The dynamics of the rotor and the generator can be described by Newton’s law in forms of(49)Jrω˙r=Ta−Krωr−Tls,Jgω˙g=Ths−Kgωg−Tem.Define the gearbox rationg as(50)ng=ωgωr=TlsThs.Invoking (50), the generator dynamic in (49) can be rewritten as(51)ng2Jgω˙r=Tls−ng2Kgωr−ngTem.Thus, the drive train model can be described by combining (49) with (51) as(52)Jtω˙r=Ta−Ktωr−Tg,where(53)Jt=Jr+ng2Jg,Kt=Kr+ng2Kg,Tg=ngTem.


### 5.2.2. Power OutputPg

Pg, the power output from the generator, is given by(54)Pg=Tgωr.


### 5.2.3. Wind Speed TorqueTa

In system (1), external wind is the source of the driven force. The wind speed torque is given by(55)Ta=12ρπR3Cqλ,βv^−ξ2,where ρ, v, and R are the air density, wind speed, and rotor radius, respectively, β is the blade pitch angle, λ=Rωr/v is the tip-speed ratio; v^ is the estimation of the effective wind speed v, which can be measured via anemometer, and ξ denotes the measurement noise; Cqλ,β, the efficiency for the wind turbine power conversion, is given by(56)Cqλ,β=0.22λ116m−0.4β−5exp−12.5m,with 1/m=1/λ+0.08β−0.035/β3+1


### 5.2.4. Energy Conversion Input-Output Model for Control System Design

From the above physical principle models, the energy conversion input-output model for control system design can be expressed as(57)P˙gTg−T˙gPg=1JtTaTg2−KtJtPgTg−1JtTg3,where the control input is the generator torque Tg and the plant output is the power output Pg.


### 5.2.5. Power Conversion in Low Speed Region

For the drive train (49) to collect the maximum quantity of energy embedded in the low speed wind region, it requires(58)Pd=npPamax,with(59)Pamax=12ρπR2Cpmaxv3,where Cpmax is the maximum power coefficient and np is the ratio between the desired generator power Pd and the maximised available power Pamax.


## 5.2.1. Drive Train in Lumped Mass Model (as Shown in Figure8)

Here is the nomenclature list.ωr: rotor angular speed aerodynamic torqueJr and Jg: rotor and generator inertias, respectively.Kr and Kg: rotor and generator external damping, respectivelyJt: integrated inertiaKt: integrated dampingTem: converted electromagnetic torqueTa: rotor torque, from external wind power in practical systemsTg: generator torque, regulate the system operation to generate powerFigure 8
Schematic diagram of the drive train.The rotor speedωr is driven by the rotor torque Ta and the low-speed torque Tls. The generator speed ωg is driven by the high-speed torque Ths and the electromagnetic torque Tem. It should be noted that using the gear box can change the generator speed. The dynamics of the rotor and the generator can be described by Newton’s law in forms of(49)Jrω˙r=Ta−Krωr−Tls,Jgω˙g=Ths−Kgωg−Tem.Define the gearbox rationg as(50)ng=ωgωr=TlsThs.Invoking (50), the generator dynamic in (49) can be rewritten as(51)ng2Jgω˙r=Tls−ng2Kgωr−ngTem.Thus, the drive train model can be described by combining (49) with (51) as(52)Jtω˙r=Ta−Ktωr−Tg,where(53)Jt=Jr+ng2Jg,Kt=Kr+ng2Kg,Tg=ngTem.


## 5.2.2. Power OutputPg

Pg, the power output from the generator, is given by(54)Pg=Tgωr.


## 5.2.3. Wind Speed TorqueTa

In system (1), external wind is the source of the driven force. The wind speed torque is given by(55)Ta=12ρπR3Cqλ,βv^−ξ2,where ρ, v, and R are the air density, wind speed, and rotor radius, respectively, β is the blade pitch angle, λ=Rωr/v is the tip-speed ratio; v^ is the estimation of the effective wind speed v, which can be measured via anemometer, and ξ denotes the measurement noise; Cqλ,β, the efficiency for the wind turbine power conversion, is given by(56)Cqλ,β=0.22λ116m−0.4β−5exp−12.5m,with 1/m=1/λ+0.08β−0.035/β3+1


## 5.2.4. Energy Conversion Input-Output Model for Control System Design

From the above physical principle models, the energy conversion input-output model for control system design can be expressed as(57)P˙gTg−T˙gPg=1JtTaTg2−KtJtPgTg−1JtTg3,where the control input is the generator torque Tg and the plant output is the power output Pg.


## 5.2.5. Power Conversion in Low Speed Region

For the drive train (49) to collect the maximum quantity of energy embedded in the low speed wind region, it requires(58)Pd=npPamax,with(59)Pamax=12ρπR2Cpmaxv3,where Cpmax is the maximum power coefficient and np is the ratio between the desired generator power Pd and the maximised available power Pamax.


## 5.3. U-Control System

The U-control system was the same as designed in Section4. The U-realisation of the input-output plant model was derived as(60)P˙g=λ0+λ1T˙g,λ0=1/JtTaTg2−Kt/JtPgTg−1/JtTg3Tg,λ1=PgTg.


## 5.4. Simulation Results

The selected generator, equipped with three blade, horizontal axis, and up wind variable speed wind turbine, generates 1.5 MW electrical output, made by WINDEY Co. This category of generators has been used worldwide [27]. The major parameters are listed in Table 1.Table 1
Wind turbine characteristics.
Rated power1.5 MWRotor radiusR = 38.5 mRotor inertiaJr = 4456761 kg·m2Generator inertiaJg = 123 kg·m2Rotor friction coefficientKr = 45.52 N·m/rad/sGenerator friction coefficientKg = 0.4 N·m/rad/sGearbox rationg = 104.494In consequence, the parameters of the turbine dynamic model are determined with(61)Jt=Jr+ng2Jg=5.7998×106,Kt=Kr+ng2Kg=4.4131×103,KtJt=7.609×10−4.For U-model (60), its time varying parameters are assigned with(62)P˙g=λ0+λ1T˙g,λ0=15.7998×106TaTg−7.609×10−4Pg−15.7998×106Pg2,λ1=PgTg.The rest of the simulation conditions/parameters include the desired powerPd, wind torque Ta from (55) and (56), the specified wind speed v with a mean of 9 m/s and turbulence intensity of 10%, sensor noise represented by a uniformly distributed random sequence of [−0.3, 0.3], np=0.8 for the ratio between the desired generator power Pd and the maximum available power Pamax, Cpmax=0.4382 for the maximum power ratio, and ρ=1.12 for the air density.Figure9 shows the constructed U-control system in Simulink block diagrams. Figure 10 shows the simulation results which are the same as those obtained from [27]. However, the differences are (1) this study is continuous time control system against the discrete time control systems [27]; (2) this study uses concise closed loop inversion to determine the invariant controller against solving the complicated Diophantine equation; and (3) this study uses the Simulink block diagram to build up the control system against using Matlab functions to develop coded programs, which block diagram-based simulation is much more engineering meaningful, transparent, and cost-effective than coded programming. Inspection of the generated plots: Figure 10(a) shows the simulated wind speed profile, Figure 10(b) shows that the generated output power properly follows the specified power trajectory and is similar to wind speed profile even though with wind measurement noise, Figure 10(c) shows the variation of the rotor torque following the wind force, and Figure 10(d) shows the tracking error amplitude converged to zero mean and small variance, and this indicates that the U-control system has converted into the maximum available power from the wind power.Figure 9
U-control of the wind energy conversion system.Figure 10
(a) Effective wind speedvt. (b) Desired power Pdt and generator output power Pgt. (c) Wind torque Tat. (d) Tracking control error between Pdt and Pgt.
(a)(b)(c)(d)


## 6. Conclusions

In U-control system design/operation, the condition ofGp−1Gp=1 is the backbone. Therefore, this requires an accurate model of Gp and an effective routine/algorithm for the UM-dynamic inversion Gp−1. This study provides a generalised methodology, with the assumption of an accurate model of Gp, a set of algorithms for the UM-dynamic inversion Gp−1. Simulated bench examples have demonstrated the analytical results and provided an effective procedure in testing designed control systems with computational experiments.The remaining challenging issues with UM-dynamic inversion are robust UM-dynamic inversion dealing with uncertainties in modelGp and data driven dynamic inversion (DD-dynamic inversion) for an unknown model of Gp. These solutions, no doubt will significantly make U-control realistically feasible and supplementary to the other exiting approaches for a wide range of applications.



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*Source: 3640210-2020-07-16.xml*