U-Model and U-Control Methodology for Nonlinear Dynamic Systems

Key Laboratory of Knowledge Automation for Industrial Processes of Ministry of Education, School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China University of the West of England, Coldharbour Lane, Bristol BS16 1QY, UK Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan Department of Electrical Engineering, University of Zanjan, Zanjan 45371-38791, Iran School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China


Introduction
In general, there are three frameworks for control system design. e two popular frameworks are (1) the model-based approach and (2) the model-free/data-driven approach. e third is relatively new and that is (3) the model-independent/ U-model-based approach. Here is a brief introduction to the three frameworks.

Model-Based Control System Design.
To show this framework, consider the general cascade feedback control system shown in Figure 1, consisting of the following elements: G p : plant, which could be modelled as a linear transfer function or a nonlinear dynamic equation in either the polynomial or state-space expression; G c : classic controller; and G: closed-loop performance function, specified in advance by designers and/or users.
For a linear plant G p , the controller G c could be designed by means of For a nonlinear plant G p , the controller could be designed as follows: where f( * ) is a function that links the plant and closed-loop performance to determine the control through a certain type of inversion.
Here are some remarks on the control-design framework: (i) e model of the plant G p is requested in advance, where the model sets include the linear/nonlinear polynomial and state-space expressions. (ii) Advantages: there are many mature approaches available for this design framework [1][2][3]. It has been the predominant approach in academic research and industrial applications. (iii) Disadvantage 1: the framework features unnecessary repetition in design. Taking a linear plant model as an example, it unnecessarily repeats the calculation of (G/1 − G) if the plant model changed in (1). (iv) Disadvantage 2: it is difficult to design nonlinear plant-based control systems and it is also difficult to specify the transient responses of nonlinear control systems with this framework. (v) Disadvantage 3: the model structure affects the approach needed for the linear/nonlinear and polynomial/state-space models, which is a common feature of model-based design frameworks.

Model-Free/Data-Driven Control System Design.
ere are various approaches to model-free control system design. A few well-known designs are described below.
(1) PID control by the Ziegler-Nichols approach [4]: this heuristic method of tuning a PID controller G c (see Figure 1) has the following features: No need for a model of the plant G p , even when mild conditions are required for the controlled plants.
Advantage: it is the most commonly and easily used trial-and-error approach. Disadvantages: this approach wastes experimental work to obtain plant models. Almost all engineering plants/processes and input/output measurements are possible to model in principle, although it is sometimes a difficult task. (2) Iterative learning control (ILC) [5]: this framework (see Figure 2) has the following features: No need for a model of the plant G p in design, even when mild conditions are required for the controlled plant.
Requires iterative learning to improve the controller G c with repeated reference stimulation; we finally achieve G c G p � G − 1 p G p � 1.
Advantages: this approach considers every possibility for integrating past control information into the next round of control design. ere is no need for a clear model structure. Disadvantage 1: this approach wastes experimental work to obtain plant models, which is an issue with almost every engineering process. Disadvantage 2: this approach is only available in a repeatable control environment under strict conditions Disadvantage 3: it is challenging to control nonlinear dynamic plants with this approach. (3) Model-free control (MFC) [6]: this framework (see Figure 1) has the following features: No need for a model of the plant G p , even under mild constraints (e.g., an ultralocal model y (v) � F + αu where α is a coefficient and u is the controller output) on the controlled plants. is approach is an enhanced PID controller (u � − (F − y (v) + k p e + k i e + k d _ e/α)) in which F needs to be estimated each time. Advantages: the ultralocal model can be used to approximate complex dynamic plants and improve control performance in this approach. Disadvantages: they are similar to those of PID controllers. [7][8][9][10][11].

Model-Independent Control System Design
is framework (see Figure 3) consists of the following: G c1 , linear invariant controller, and G − 1 p , dynamic inversion of plant.
Some remarks are given on the control framework.
(i) It features model-independent controller design.
(ii) Advantage 1: the parallel design controller and dynamic inversion make the design procedure applicable to linear/nonlinear polynomial/state-space model structures. Transient responses can be specified for nonlinear systems. It is neat in design without waste/repetition if the plant model changes.

Discrete Time U-Model
Set e U-model expresses an explicit input-output relationship U(y(t), u(t − 1), Ψ(t)) at time t with time-varying parameters Ψ(t) to absorb dynamics implicitly. is is a controloriented model and is derived from existing principle models or data-fitting models. is section explains (1) the definition of the U-model and the principles of converting classic models into U-models, (2) the dynamic inversion of U-polynomial models, and (3) the dynamic inversion of Ustate-space models.

U-Models.
Definition: for a single input (u ∈ R single) and single output (y ∈ R) dynamic system , assign a triplet (U, Ψ, f U ), where U � U α | α ∈ u(t − 1) is a vector of appropriate dimension and Ψ � Ψ α | α ∉ u(t − 1) is a dynamic absorbing vector of appropriate dimension that is associated with U. Accordingly, system U-model U− model is defined as a polynomial/rational system, where the polynomial/rational function f U � f U (α) | α ∈ Ψ, U is a mapping f U : u(t − 1) Ψ∪U ⟶ y(t) ∈ R from the input space to the output space.

U-Model Realisation from Classic Polynomials.
Consider a general classical SISO polynomial model in the form of where y(t) ∈ R u(t − 1) ∈ R are the output/input, respectively, at t ∈ Z + and Φ( * ) � ϕ 0 ( * ) · · · ϕ L ( * ) ∈ R L+1 , where Y t− 1 , U t− 1 are expanded from the output and input, respectively, in the proper dimensions. Let where n is the plant dynamic order and Θ � θ 0 · · · θ L ∈ R L+1 is the associated parametric vector. Let the function f P : u ⟶ y be a polynomial mapping of the input space to the output space.
e vector form of the expanded equation (4) is given as follows: where the bases ϕ l ( * ) are the smooth functions in the space expanded from the past inputs/outputs, for example, , and the associated coefficients θ l are real constants.
In other terms, this is a general expression of a nonlinear autoregressive moving average with exogenous input model (NARMAX) [12].
To realise a U-model from this classical polynomial, set up an absorbing rule.
Absorbing rule: let μ: R L+1 ⟶ R M+1 be a map from a polynomial f P to its U-polynomial f U and suppose that its inverse μ − 1 exists; therefore, it has e mapping has some proper algebra properties as [8].
(a) ∀(P( * ), Θ), (Ψ( * ), U(u(t − 1))), Accordingly, with reference to (7), the mapping is (a) injective (one to one), (b) surjective (onto) and bijective as both (a) and (b), and (c) invertible (I is an identity function). In system aspect, the map, except making the structure expression changed, does not change any characteristics of both models, such as output response, stability, dynamics, and statics. e absorbing rule is a formation of Ψ( * ) from the polynomial f P with reference to u(t − 1): first identify a control basis function U(u(t − 1)) and then absorb all the other associated functions as a coefficient that varies with time.
erefore, using the absorbing rule, realising f U mapped from polynomial f P (5) gives the following: is function is expanded from the above nonlinear function f P as a polynomial in terms of u(t − 1). M is the Figure 3: Model-independent control system design.
Complexity number of items associated with input u(t − 1) and the timevarying parameter vector Ψ( * ) � ψ 0 (t) · · · ψ M (t) ∈ R M+1 is a function derived from absorbing the other regression terms and the coefficients. Example 1. Consider the polynomial model as shown below: Absorbing the terms associated with u(t − 1) into the vector Ψ( * ) gives the corresponding U-model realisation as follows: where ψ 0 (t) � 0.2 * sin(y(t − 1)),

U-Model Realisation from Classic Rational Models.
Rational model, also known as total nonlinear model [13], is a ratio of two polynomials as follows: Here f r is a rational function, the ratio of the f pn /numerator polynomial and f pd /denominator polynomial, which are maps of the input space into the output space. e other definitions follow from the polynomial model above. Note that this rational model is totally nonlinear in terms of parameter estimation and control input design [13].
Continuing with the U-polynomial model conversion, formulate the U-rational model expression as follows: To obtain the model inversion for solving the roots, expand the model as follows: Example 2. Consider the rational model as follows: Absorbing the terms associated with u(t − 1) into the vectors Ψ n ( * ), Ψ d ( * ) gives the corresponding U-model realisation as follows:

U Realisation from a Classical State-Space Mode-
where X ∈ R n u ∈ R y ∈ R ⎧ ⎪ ⎨ ⎪ ⎩ denotes the state, the control, and the output at time t ∈ Z + , respectively. F ∈ R n is a smooth mapping to represent the input to the state output, and h ∈ R is a smooth mapping to drive the states to the outputs. In this study, assume that the system relative degree r equals the system order n and has no unstable zero dynamics (i.e., the model reversible) and that the state X can be obtained through measurement or observation.
Convert state-space model (19) into a multilayer Umodel expression as follows: For each line, M j is the number of terms associated with the next line state variable x j+1 (t) and parameter vector functions absorbing the other state variables. In the penultimate line, M n consists of the terms associated with control u(t) and the time-varying vectors ψ n0 (t) · · · ψ nM n (t) ∈ R M n +1 absorb all the states associated with the control vector U n0 (t) · · · U nM n (t)] ∈ R M n +1 . erefore, each line of the state-space equation is a U-polynomial model, consisting of a multilayer U-model expression.
To illustrate the realisation, consider a nonlinear system represented in terms of state-space model: Take realisation of the corresponding multilayer Umodel by using the absorbing rule as below:

Inversion of U-Polynomial Models.
For simplicity, consider the SISO polynomial U-model (28). Newton-Raphson algorithm [14] is a choice to determine the roots of U-models; that is, the roots are the candidates of controller output u(t − 1). Iteratively, the root searching computation gives rise to the following formulation: Here, index k is the iteration handle: generate the (k+1)th results from the kth iteration, k≻0. ere are also various root solving algorithms available [15]. In parallel, these algorithms are also applicable for U-rational model root solving based on (14).
It should be noted that, in simulation studies, MATLAB codes, such as roots, can be used to find accurate roots of the U-model equations.

Inversion of U-State Space Models.
For simplicity, consider the SISO U-state space model (20). Inversion is a multilayer root solving procedure involving a back-stepping routine whenever x 1 (t + 1) is known; each line of the equation iteratively uses the Newton-Raphson algorithm to obtain x 1 (t + 1) x 2 (t + 1) · · · in back-stepping order.

U-Model-Based Control System Design
A Chinese survey paper [16] has covered the major publications till 2012. Later, representative studies include "U-Block model technique" [8], "control of total nonlinear systems" [9], "U-model enhanced Smith predict control for time delayed nonlinear processes" [11], and "U-neural networks enhanced control system design" [10]. is section further expands/formulates the U-control framework with updated results, including newly introduced two parallel dynamic inversions in design, robust analysis, and a step-bystep procedure for U-control implementation.

U-Control Framework.
Let G p be general dynamic in any expression of linear/nonlinear and polynomial/state-space models. Assumingly, the plant has the mostly claimed properties as those claimed in the other representative works [17]. Accordingly (1) e model inverse G − 1 p exists (2) Lipschitz continuity is satisfied, and G p and its inverse G − 1 p are diffeomorphic and globally uniformly Lipschitz in R n ; that is, For simplicity, but not losing generality, take consideration of a SISO (input u ∈ R 1 and output y ∈ R 1 ) U-model based control system, U-control system in short, which is constructed within an autonomous linear feedback control framework with a bracketed triplet of where F lfbc is a linear feedback loop with functions, linear virtual controller G c1 : y ⟶ u, and virtual unit plant is U-control system structure proposes a model-independent control procedure, because the designs of G c1 : y ⟶ u and G ip � 1: u ⟶ y are independent. ese two independent designs are explained below.
For design of the virtual linear controller G c1 : y ⟶ u, referring to Figure 3(a), it gives where G is a specified closed-loop transfer function with proper dynamic/static responses. For design of the virtual unit plant G ip � 1: u ⟶ y, designing/formulating the plant inverse G − 1 P gives Remark 1. Regarding the merit of the design prototype, the established U-control system framework (25) has two Complexity 5 independent inversion designs: (1) linear controller G c1 : y ⟶ u without involving any plant model structures; therefore, it is also named as linear invariant controller [9]; (2) virtual plant unitisation G ip � G − 1 P G p � 1: u ⟶ y applicable to almost all smooth dynamics models (note: hard nonlinear dynamic models could be sorted out along similar route in the subsequent studies). erefore, the two designs are separately independent and connected within a linear feedback control loop.

Remark 2.
Regarding the efficiency of the U-control system design, linear controller G c1 : y ⟶ u is once-off design irrespective of plant model types and parameters. Plant inverter G − 1 P is formable for polynomial and state-space equations in U-model and numerically solvable for the roots to achieve G ip � G − 1 P G p � 1: u ⟶ y. Consequently, the control system design is reduced to the determination of the plant inverse G − 1 P once the linear controller designed. Consequently, the design procedure is that once-off G c1 : y ⟶ u design and G − 1 P ⟹ G ip � G − 1 P G p � 1: u ⟶ y follow-up design to keep the same closed-loop performance while plant model is changed.

Remark 3.
Regarding the inversion involved in control system design, this is a must for any type of control system design. U-control provides concise structure and less computational effort for its two inversions (one is the inversion of specified linear closed-loop transfer function and the other is the inversion of plant U-model). is aspect can be explained through an inverse function Ψ − 1 ; for U-control systems, it is split into two separate functions of G c1 � Ψ − 1 1 (G) (linear dynamic inversion) and G − 1 p � Ψ − 1 2 (G p )(U-model-based root solving). For the other popular control system design approaches, it is at least a function of Ψ − 1 (G c1 , G p ), which is a common formulation in classical linear feedback control system design. It should be noted that it is more complex in designing control systems with nonlinear plant models.

Remark 4.
With regard to the relationship in control system design between the U-control and the other major approaches, U-control is a supplement to the approaches and takes away the need for the plant structures in controller design and clearly specifies the closed-loop dynamic/static performances. It should be noted that taking the transient performance into consideration when designing nonlinear control systems has received significant attention, and analysing their performance through linear system approaches is a key research domain [18]. U-control is therefore a promising procedure.
In some sense, those, using the other approaches, welldesigned control systems could take U-control as a plug-in box to expand to control different types of plants.

Remark 5.
As U-control is fundamentally based on the assumption G − 1 P G p � 1, it is critical to consider the robustness of the resulting control system in the case of uncertainty, which is very common in practical systems. Surely two types of approaches are the candidates by adding additional robust control loop and/or adaptive loop. Figure 4, here we list a step-by-step design procedure.

Design Procedure. With reference to the aforementioned description and the block diagram in
(1) Establish a stable linear feedback control system structured in Figure 4. Assign G for the whole system transfer function in the closed-loop setup. Specify G by means of damping ratio, undamped natural frequency, and steady-state error and/or the other performance indices (such as poles and zeros and frequency response). (2) Let the plant model be a constant unit or the virtual plant G ip � 1: u ⟶ y that has been achieved. determine a linear invariant controller G c1 by taking inverse of the closed-loop transfer function G using (26). Accordingly, the desired system output is equivalently determined by the output v of the controller G c1 .
Locate/connect the blocks in Figure 5.

U-Model-Based Adaptive Control.
is was first studied in recent publications [9,19]. Figure 6 shows a double-looped (feedback control and adaptation) diagram that adds an adaptation role in dealing with uncertainties and disturbance by online updating model parameters. Interested readers can find the details in the aforementioned reference. Compared with the classic adaptive control scheme, adaptive U-control does not request controller design in each updating step; it only updates the plant model, while the controller is fixed. Here only the framework is explained briefly and the detailed expansions will be reported in the future publications.

Robustness Analysis of U-Control.
is section presents the robustness analysis of U-control based on discrete-time H ∞ using linear matrix inequalities (LMI) technique. Consider the state-space equation in terms of multilayer Urealisation (20) with an external disturbance vector W(t) � [w 1 (t) · · · w n (t)] T as Ψ nj (t)U nj (u(t)) + w n (t), To provide the robustness analysis, take one single line state of x n � (t + 1) from state-space equation (28) at first, and then extend the analysis to the other state variables x i (t + 1). Accordingly, take out e control objective is to minimize the effect of the external disturbance w n on the state vector x n . is study takes the discrete-time H ∞ robust control technique into consideration, where the robust control condition is where β is a known constant defining the upper boundary of H ∞ performance index. Equation (30) can be rewritten as or equivalently with From the above formulations, we have Construct the positive-definite Lyapunov function with where Q > 0. Suppose that the gradient of the Lyapunov function (∇V(x n (t))) is satisfied in the following inequality: In order to prove condition of (36), take summation (Σ) of all terms as Since the first term of (37) is positive, the second term is always negative; that is, which is the same as condition (34). en, inequality (36) is a correct assumption. Determine the gradient of the Lyapunov function by By substituting (39) into (36), we have Now, substituting x n (t + 1) from (28) into (40), we have In what follows, for simplicity, shorten the following notations as Figure 4: U-control framework. Complexity en (41) is expressed as Furthermore, it can be expressed in terms of quadratic form and then matrix form of with Now, applying the Schur complement [20] on (46), we get where en condition of (47) can be simplified as Defining a new variable c: � β − 1 in the first inequality of (49), it changes to where the optimal values Q * and c * can be calculated via Matlab LMI toolbox. e optimal value of β is correspondingly given by β * � (c * ) − 1 . Applying Schur complement on the second inequality of (49), we have Q < 0, which yields − cI < 0. en, from (51), it gives where the existence of optimal solutions for Q * and c * means the robustness of the U-model system versus external disturbances. e robustness analysis for the remainder of equations of state-space model (28) can be proved similar to the abovepresented procedure. In addition, Virtual Equivalent System (VES) methods [21][22][23] can also be used for robustness analysis of U-model control systems.

Simulation Examples
is simulation demonstration selected three plant models: SISO Hammerstein model, SISO nonlinear state-space model, and an extended total nonlinear model. In the control system design, it formulated a commonly used pole placement controller for the three examples. e main purposes for designing the simulation tests of the U-control are as follows: (1) To demonstrate the principle of model-independent design in U-control. (2) To demonstrate the once-off design of the invariant controller which specifies the whole closed loop 8 Complexity performance. In analysis, as always the product of the dynamic inverter and plant is a unit constant, therefore, a uniquely specified invariant controller will achieve the same output performance with different dynamic plants in the selected examples. (3) To demonstrate the workability and conciseness/ simplicity of U-control, particularly in the design of nonlinear control systems. (4) To demonstrate that U-control can supplement/ enhance classic pole placement control.
From previous sections, the design is divided into two parallel blocks: (1) designing the linear invariant control G c1 (thus v(t)) by reversing the specified closed-loop transfer function and (2) determining the control input u(t − 1) by reversing the plant U-model equation.
For familiarisation of different notations used in Ucontrol, this simulation section takes in

Design Invariant Control G c1
. In a popular approach, the conventional pole placement control [24] assigns the closedloop characteristic equation in terms of Z transform: Equivalently the poles are located at 0.6603 ± i0.2463 within the unit circle (stable), a typical decayed oscillatory response with damping ratio of 0.7, and unit undamped natural frequency; this is a commonly used dynamic response index set.
Assign the numerator polynomial in the desired closedloop transfer function as where the constant b is determined by steady-state error requirement to a given reference input. Accordingly, in this case study, to make the steady state follow a given step reference input without error, it sets up ereby, the resultant transfer function is specified as It should be noted that when the condition G ip � 1: u ⟶ y is satisfied, it gives where V(z) is the Z transform of the controller G c1 output as shown in Figure 3.
To determine the linear invariant controller G c1 , temporarily, let the plant G p � 1 or G ip � G − 1 p G p � 1: u ⟶ y. en take inverse of the transfer function G to yield (58) e rest of the control system design will formulate the specific plant inverse G − 1 p in form of U-model for each selected example, which will be implemented in each related subsection. [7]. e Hammerstein style model, a static (memoryless) nonlinear block, is cascaded with a linear differential equation (dynamic) and is a good representative of various nonlinear dynamic plants/processes. Its control has been widely studied with model-based approaches [25]. e simulation example selected [7] is as follows:

Hammerstein Model: A SISO Nonlinear Polynomial
where y t , u t , x t are the plant output, input, and intermediate variable for the static nonlinear component output, respectively. As explained above, the first step in U-control system design is generic to determine the linear invariant controller G c1 , that is, independent of the plant model and universally designed (as was done in the beginning of this section). e second step of the design is specifically working out the controller output u t− 1 by inverting the plant model to find its U-model roots. Accordingly, to realise a U-model for the controller output, it uses the absorbing rule to convert the Hammerstein model into the following U-expression: where en replace the output y t with the virtual controller output v t (i.e., the desired output). Subsequently, it determines one of the roots by solving (60) as the controller output. is gives the following formula: Figure 7 illustrates the simulation results. e control of strictfeedback nonlinear systems is a widely studied, challenging topic [26]. Many leading publications have used neural network model-based approaches to approximate the model set as a pointwise linear model set to alternatively design equivalent linear control systems [27]. e simulation example for the state-space model is as follows: x 1t � x 2(t− 1) + 0.1x 1(t− 1) x 2(t− 1) , where y t , u t , x t denote the plant output and input and x(t) is a state vector, respectively. is represents a second-order nonlinear dynamic plant. Again, in the second step of U-control system design, it requires to work out the specific controller output u t− 1 by inverting the plant U-model. e realised multilayer Umodel is expressed as follows: x 1t � ψ 11 x 2(t− 1) , where is is a two-layer U-model structure. Accordingly, using back-stepping routine with initial x 1t � v t works out the controller output u t− 1 by inverting each line of the equations, as specified in step 4 in U-control design procedure. e simulation results are shown in Figure 8. [9]. e control of nonlinear rational systems, which are modelled as ratios of two nonlinear polynomials, is even more challenging. Until a recent analytical U-model-based approach [9], these models were previously taken as examples of complex systems in neurocontrol system design. e difficulty is that rational model sets are subject to total nonlinearity (both in the parameters/identification and in input/control) [13]. e selected simulation example [9] with dynamics (time delay) and transcendental nonlinearities was as follows:

Extended Total Nonlinear Model
where y t , u t are the plant output and input, respectively. Once again, by applying the absorbing rule, it yields the following U-rational model: � 0.5y t− 1 + sin u t− 1 + u t− 1 ⟹, ψ 0 d y(t) � ψ 0n + ψ 1n sin u t− 1 + ψ 2n u t− 1 . (66) With the same linear invariant controller G c1 used as before, replacing the output y t of (66) with the desired output v t of (58) gives the following: Subsequently, the control input u t− 1 is obtained by the following: (68) Figure 9 illustrates the simulation results. Again, the bench test confirms the performance of the U-control.

Conclusion
U-control has been featured in several publications. is tutorial has been presented to summarise and expand on the essential insights, formulations, and simulated case studies. We hope that this self-contained study can achieve the following purposes: (1) Explain/demonstrate the principle of model-independent design in U-control (2) Explain/demonstrate a universal design for multiple plant model structures (3) Explain/demonstrate U-control workability and effectiveness/efficiency, particularly dealing with nonlinear plant control (4) Explain/demonstrate U-control as a supplement to classic control system design frameworks In terms of research techniques, compared with the two most popular control system design frameworks, modelbased and model-free, this model-independent design effectively relieves the complexity involved in inverting the controller and plant together. e problem of inversion is reduced to inverting the plant model only, which means this framework results in an invariant controller that is universally applicable to the classic model sets and features no repetition if the plant model changes. e most critical issue with this design framework is its robustness because it relies Complexity on having G − 1 p G p � 1. Accordingly, robust U-control is a central topic for research and applications. Additional demonstrations of its use in real cases will help to prepare it for wider application.
In research methodology, U-control is simple/concise and uses basic tools such as poles and zeros for analysing/ designing linear system stability, transient responses (damping ratios and undamped natural frequencies), and the small gain theorem for robustness analysis. All of these are fundamental in postgraduate courses. However, Ucontrol effectively combines them to provide solutions for challenging research problems. It is hoped that this technique will be user-friendly for industrial engineers working with ad hoc applications and easy to use for academics developing further enhancements of the method. As future work, U-model and U-control methodology can be integrated with other concepts in modelling and control of nonlinear dynamic systems, such as multiple model approaches [28][29][30][31][32].

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.