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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.

Linear control system design approaches can be divided into state-space model-based [

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 [

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. [

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 [

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:

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.

In methodology, using those well-studied linear methods to design nonlinear control systems greatly reduces the complexity of the design procedure.

In design, once the closed loop system output is specified, the only remaining work is to calculate the output of the U model controller.

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.

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 [

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:

Generalise dynamic inversion algorithms for continuous-time U-model

Generalise U-model-based design procedure for continuous-time dynamic plants in forms of linear/nonlinear and polynomial/state space

Showcases for bench tests and illustration of applications

An industrial backgrounded study: U-control of a wind energy conversion system

For the rest of the study, Section

Consider a general continuous-time U-model [

Here is an example for understanding, consider a classical NAMAX polynomial model:

Its U-model realisation can be determined with

Inspection of (

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 [

Rational model is totally nonlinear [

Its U-model realisation can be determined with

For a general SISO CT state space model,

Expand state-space model (

Convert state-space model (

For each line of (

For illustration, consider a nonlinear SISO system state-space model of

Using the absorbing rule to convert (

Here is a systematic summary of the U-control framework.

Figure

Classical control system framework.

The main principles of this kind of control system design framework are to generate a suitable control input signal

Figure

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 (

In general, the U-control system design procedure has two separate steps:

Assume the plant model

where

Design invariant controller

U-model-based simplified control system.

This is a type of linear control systems. Therefore, the desired closed-loop transfer function

As the invariant controller

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 [

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 input

For the solution which exists, the systems must be Bounded Input and Bounded Output (BIBO) stable and no unstable zero dynamic (nonminimum phase).

Use Laplace transform (

Accordingly, its U-realisation is given as

As the drives are sensitive to noise signals in applications, convert the operations into integral implementations by multiplying

Therefore, the alternative U-model is

For UM-dynamic inversion of (

Here is the practical implementation of (

To illustrate the conversion to U-model from a nonlinear polynomial, consider an example of

In U-realisation, its derivative-based operation becomes

To convert it into integration operation, the corresponding U-realisation has the form of

For a general SISO linear CT state-space system model, it has

For taking up such UM-dynamic inversion, first use a systematic approach [

Then, use the linear polynomial dynamic inversion procedure presented in Section

For the model of (

Generate direct mapping between the output

Use the procedure for nonlinear polynomial dynamic inversion in Section

It 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

Differentiating

As the second line of (

Convert to integral expression as

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 [

In addition, to test U-control systems in Simulink/Matlab simulation, Matlab functions can be used to find the roots directly.

This simulation demonstration selected four plant models to tests the UM-dynamic inversion and their associated U-control systems with the following bullet points:

To demonstrate the generality and effectiveness of UM dynamic inversion

To demonstrate the principle of model-independent design in U-control, supported by the dynamic inversions

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

To validate the applicability, conciseness, and efficiency of the U-control and UM-dynamic inversion, particularly in designing nonlinear control systems

With reference to the previous introduction in Section

The invariant controller

The corresponding U-model was

The corresponding U-model was

The corresponding U-model was

The corresponding U-model model was

Figures

(a) U-control of plant 1. (b) U-control of plant 2.

(a) U-control of plant 3. (b) U-control of plant 4.

(a) Plant outputs and reference for plant1 and plant2. (b) Controller outputs for plant1 and plant2.

(a) Plant outputs and reference for plant 3 and plant 4. (b) Controller outputs for plant 3 and plant 4.

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 [

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 [

Here is the nomenclature list.

Schematic diagram of the drive train.

The rotor speed

Define the gearbox ratio

Invoking (

Thus, the drive train model can be described by combining (

In system (

From the above physical principle models, the energy conversion input-output model for control system design can be expressed as

For the drive train (

The U-control system was the same as designed in Section

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 [

Wind turbine characteristics.

Rated power | 1.5 MW |

Rotor radius | |

Rotor inertia | _{r} = 4456761 kg·m^{2} |

Generator inertia | ^{2} |

Rotor friction coefficient | _{r} = 45.52 N·m/rad/s |

Generator friction coefficient | |

Gearbox ratio |

In consequence, the parameters of the turbine dynamic model are determined with

For U-model (

The rest of the simulation conditions/parameters include the desired power

Figure

U-control of the wind energy conversion system.

(a) Effective wind speed

In U-control system design/operation, the condition of

The remaining challenging issues with UM-dynamic inversion are robust UM-dynamic inversion dealing with uncertainties in model

The Simulink block diagrams are included within the article to generate the plot data for supporting the findings of this study.

The authors declare that they have no conflicts of interest.