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

Authors: Weicun Zhang; Quanmin Zhu; Saleh Mobayen; Hao Yan; Ji Qiu; Pritesh Narayan
Journal: Complexity
 (2020)

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## Abstract

This study presents the fundamental concepts and technical details of aU-model-based control (U-control for short) system design framework, including U-model realisation from classic model sets, control system design procedures, and simulated showcase examples. Consequently, the framework provides readers with clear understandings and practical skills for further research expansion and applications. In contrast to the classic model-based design and model-free design methodologies, this model-independent design takes two parallel formations: (1) it designs an invariant virtual controller with a specified closed-loop transfer function in a feedback control loop and (2) it determines the real controller output by resolving the inverse of the plant U-model. It should be noted that (1) this U-control provides a universal control system design platform for many existing linear/nonlinear and polynomial/state-space models and (2) it complements many existing design approaches. Simulation studies are used as examples to demonstrate the analytically developed formulations and guideline for potential applications.


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## Body

## 1. Introduction

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


### 1.1. Model-Based Control System Design

To show this framework, consider the general cascade feedback control system shown in Figure1, consisting of the following elements: Gp: plant, which could be modelled as a linear transfer function or a nonlinear dynamic equation in either the polynomial or state-space expression; Gc: classic controller; and G: closed-loop performance function, specified in advance by designers and/or users.Figure 1
Model-based control.For a linear plantGp, the controller Gc could be designed by means of(1)Gc=Gp−1G1−G.For a nonlinear plantGp, the controller could be designed as follows:(2)Gc=fGp,G,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)
The model of the plantGp 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 [1–3]. 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 ofG/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.


### 1.2. Model-Free/Data-Driven Control System Design

There are various approaches to model-free control system design. A few well-known designs are described below.Figure 2
ILC.(1)
PID control by the Ziegler–Nichols approach [4]: this heuristic method of tuning a PID controller Gc (see Figure 1) has the following features:No need for a model of the plantGp, 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 plantGp in design, even when mild conditions are required for the controlled plant.Requires iterative learning to improve the controllerGc with repeated reference stimulation; we finally achieve GcGp=Gp−1Gp=1.Advantages: this approach considers every possibility for integrating past control information into the next round of control design. There 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 conditionsDisadvantage 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 plantGp, even under mild constraints (e.g., an ultralocal model yv=F+αu where α is a coefficient and u is the controller output) on the controlled plants.This approach is an enhanced PID controller (u=−F−yv+kpe+ki∫e+kde˙/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.


### 1.3. Model-Independent Control System Design [7–11]

This framework (see Figure3) consists of the following: Gc1, linear invariant controller, and Gp−1, dynamic inversion of plant.(3)Gc1=G1−GGp=1,Gc=Gc1Gp−1.Figure 3
Model-independent control system design.
(a)(b)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.(iii)
Advantage 2: this approach complements most existing design approaches.(iv)
Disadvantages: this approach is sensitive to model uncertainty; robustness is the paramount issue in designing control systems.


## 1.1. Model-Based Control System Design

To show this framework, consider the general cascade feedback control system shown in Figure1, consisting of the following elements: Gp: plant, which could be modelled as a linear transfer function or a nonlinear dynamic equation in either the polynomial or state-space expression; Gc: classic controller; and G: closed-loop performance function, specified in advance by designers and/or users.Figure 1
Model-based control.For a linear plantGp, the controller Gc could be designed by means of(1)Gc=Gp−1G1−G.For a nonlinear plantGp, the controller could be designed as follows:(2)Gc=fGp,G,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)
The model of the plantGp 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 [1–3]. 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 ofG/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.


## 1.2. Model-Free/Data-Driven Control System Design

There are various approaches to model-free control system design. A few well-known designs are described below.Figure 2
ILC.(1)
PID control by the Ziegler–Nichols approach [4]: this heuristic method of tuning a PID controller Gc (see Figure 1) has the following features:No need for a model of the plantGp, 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 plantGp in design, even when mild conditions are required for the controlled plant.Requires iterative learning to improve the controllerGc with repeated reference stimulation; we finally achieve GcGp=Gp−1Gp=1.Advantages: this approach considers every possibility for integrating past control information into the next round of control design. There 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 conditionsDisadvantage 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 plantGp, even under mild constraints (e.g., an ultralocal model yv=F+αu where α is a coefficient and u is the controller output) on the controlled plants.This approach is an enhanced PID controller (u=−F−yv+kpe+ki∫e+kde˙/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.


## 1.3. Model-Independent Control System Design [7–11]

This framework (see Figure3) consists of the following: Gc1, linear invariant controller, and Gp−1, dynamic inversion of plant.(3)Gc1=G1−GGp=1,Gc=Gc1Gp−1.Figure 3
Model-independent control system design.
(a)(b)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.(iii)
Advantage 2: this approach complements most existing design approaches.(iv)
Disadvantages: this approach is sensitive to model uncertainty; robustness is the paramount issue in designing control systems.


## 2. Discrete TimeU-Model Set

TheU-model expresses an explicit input-output relationship Uyt,ut−1,Ψt at time t with time-varying parameters Ψt to absorb dynamics implicitly. This is a control-oriented model and is derived from existing principle models or data-fitting models. This 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 U-state-space models.


### 2.1.U-Models

Definition: for a single input (u∈ℝ single) and single output (y∈ℝ) dynamic system ∑, assign a triplet U,Ψ,fU, where U=Uαα∈ut−1 is a vector of appropriate dimension and Ψ=Ψαα∉ut−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 fU=fUαα∈Ψ,U is a mapping fU:ut−1⟶Ψ∪Uyt∈ℝ from the input space to the output space.


#### 2.1.1.U-Model Realisation from Classic Polynomials

Consider a general classical SISO polynomial model in the form of(4)yt=fPΦ∗,Θ,Φ∗=ΦYt−1,Ut−1,where yt∈ℝut−1∈ℝ are the output/input, respectively, at t∈ℤ+ and Φ∗=ϕ0∗⋯ϕL∗∈ℝL+1, where Yt−1,Ut−1 are expanded from the output and input, respectively, in the proper dimensions. Let ϕi∗=ϕiyt−1,…,yt−nut−1,…ut−n∀i=0⋯n, where n is the plant dynamic order and Θ=θ0⋯θL∈ℝL+1 is the associated parametric vector. Let the function fP:u⟶y be a polynomial mapping of the input space to the output space. The vector form of the expanded equation (4) is given as follows:(5)yt=ΦTΘ=∑l=0Lϕl∗θl,where the bases ϕl∗ are the smooth functions in the space expanded from the past inputs/outputs, for example, y3t−2ut−1, u3t−1, yt−1yt−5, and the associated coefficients θl 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 aU-model from this classical polynomial, set up an absorbing rule.Absorbing rule: letμ:RL+1⟶RM+1 be a map from a polynomial fP to its U-polynomial fU and suppose that its inverse μ−1 exists; therefore, it has(6)fPP∗,Θ⟶μfUΨ∗,Uut−1.The mapping has some proper algebra properties as [8].(7)a∀P∗,Θ,Ψ∗,Uut−1,fP∗=fU∗⟹P∗,Θ=Ψ∗,Uut−1b∀P∗,Θ∈fP,∃Ψ∗,Uut−1∈fU,μP∗,Θ=Ψ∗,Uut−1cμ−1⋅μ=I.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.The absorbing rule is a formation ofΨ∗ from the polynomial fP with reference to ut−1: first identify a control basis function Uut−1 and then absorb all the other associated functions as a coefficient that varies with time.Therefore, using the absorbing rule, realisingfU mapped from polynomial fP (5) gives the following:(8)yt=ΨTU=∑j=0MψjtUjut−1.This function is expanded from the above nonlinear functionfP as a polynomial in terms of ut−1. M is the number of items associated with input ut−1 and the time-varying parameter vector Ψ∗=ψ0t⋯ψMt∈ℝM+1 is a function derived from absorbing the other regression terms and the coefficients.Example 1.
Consider the polynomial model as shown below:(9)yt=0.2∗sinyt−1+ut−1exp−y2t−1−0.8yt−2ut−2u3t−1.
Absorbing the terms associated withut−1 into the vector Ψ∗ gives the corresponding U-model realisation as follows:(10)yt=ψ0t+ψ1tU1ut−1+ψ2tU2ut−1,where(11)ψ0t=0.2∗sinyt−1,ψ1t=exp−y2t−1,ψ2t=−0.8yt−2ut−2,U1ut−1=ut−1,U2ut−1=u3t−1.


#### 2.1.2.U-Model Realisation from Classic Rational Models

Rational model, also known as total nonlinear model [13], is a ratio of two polynomials as follows:(12)yt=frΦ∗,Θ=fpnΦn∗,ΘnfpdΦd∗,Θd.Herefr is a rational function, the ratio of the fpn/numerator polynomial and fpd/denominator polynomial, which are maps of the input space into the output space. The 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 theU-polynomial model conversion, formulate the U-rational model expression as follows:(13)yt=ΨnTUnΨdTUd=∑j=0MnψjntUjnut−1∑j=0MdψjdtUjdut−1.To obtain the model inversion for solving the roots, expand the model as follows:(14)yt∑j=0MdψjdtUjdut−1=∑j=0MnψjntUjnut−1.Example 2.
Consider the rational model as follows:(15)yt=0.1y3t−2+sinut−1+0.5u3t−11+cos2yt−2+u2t−1.
Absorbing the terms associated withut−1 into the vectors Ψn∗,Ψd∗ gives the corresponding U-model realisation as follows:(16)fpnut−1=ψ0nt+ψ1ntU1nut−1+ψ2ntU2nut−1,fpdut−1=ψ0dt+ψ1dtU1dut−1,where(17)ψ0nt=0.1y3t−2,ψ0dt=1+cos2yt−2,ψ1nt=1,ψ2n=0.5,ψ1dt=1,(18)U1nut−1=sinut−1,U2nut−1=u3t−1,U1dut−1=u2t−1.


#### 2.1.3.U Realisation from a Classical State-Space Mode-Multilayer U-Model

For a general SISO state-space system model, it has(19)Xt+1=FXt,ut,yt=hXt,where X∈ℝnu∈ℝy∈ℝ denotes the state, the control, and the output at time t∈ℤ+, respectively. F∈ℝn 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 U-model expression as follows:(20)x1t+1=∑j=0M1ψ1jtU1jx2t,⋮,xn−1t+1=∑j=0Mn−1ψn−1jtUn−1jxnt,xnt+1=∑j=0MnψnjtUnjut,yt=hXt.For each line,Mj is the number of terms associated with the next line state variable xj+1t and ψijt=ψj0t⋯ψjMjt∈ℝMj+1i∈1…n are time-varying parameter vector functions absorbing the other state variables. In the penultimate line, Mn consists of the terms associated with control ut and the time-varying vectors ψn0t⋯ψnMnt∈ℝMn+1 absorb all the states associated with the control vector Un0t⋯UnMnt∈ℝMn+1. Therefore, 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:(21)x1t+1=x2t+0.1x1tx2t,x2t+1=−0.1x1t−0.7x2t+ut,yt=x1t.Take realisation of the corresponding multilayerU-model by using the absorbing rule as below:(22)x1t+1=ψ11tU11x2t,x2t+1=ψ20t+ψ21U12ut,yt=x1t,where(23)ψ11t=1+0.1x1t,ψ20t=−0.1x1t−0.7x2t,ψ21t=1,U11x2t=x2t,U12ut=ut.


### 2.2. Inversion ofU-Polynomial Models

For simplicity, consider the SISO polynomialU-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 ut−1.Iteratively, the root searching computation gives rise to the following formulation:(24)uk+1t−1=ukt−1−yt−∑j=0Mλjtukjt−1d∑j=0Mλjtujt−1/dut−1ukt−1=ukjt−1.Here, indexk is the iteration handle: generate the (k+1)th results from the kth iteration, k≻0. There 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 asroots, can be used to find accurate roots of the U-model equations.


### 2.3. Inversion ofU-State Space Models

For simplicity, consider the SISOU-state space model (20). Inversion is a multilayer root solving procedure involving a back-stepping routine whenever x1t+1 is known; each line of the equation iteratively uses the Newton–Raphson algorithm to obtain x1t+1x2t+1⋯ in back-stepping order.


## 2.1.U-Models

Definition: for a single input (u∈ℝ single) and single output (y∈ℝ) dynamic system ∑, assign a triplet U,Ψ,fU, where U=Uαα∈ut−1 is a vector of appropriate dimension and Ψ=Ψαα∉ut−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 fU=fUαα∈Ψ,U is a mapping fU:ut−1⟶Ψ∪Uyt∈ℝ from the input space to the output space.


### 2.1.1.U-Model Realisation from Classic Polynomials

Consider a general classical SISO polynomial model in the form of(4)yt=fPΦ∗,Θ,Φ∗=ΦYt−1,Ut−1,where yt∈ℝut−1∈ℝ are the output/input, respectively, at t∈ℤ+ and Φ∗=ϕ0∗⋯ϕL∗∈ℝL+1, where Yt−1,Ut−1 are expanded from the output and input, respectively, in the proper dimensions. Let ϕi∗=ϕiyt−1,…,yt−nut−1,…ut−n∀i=0⋯n, where n is the plant dynamic order and Θ=θ0⋯θL∈ℝL+1 is the associated parametric vector. Let the function fP:u⟶y be a polynomial mapping of the input space to the output space. The vector form of the expanded equation (4) is given as follows:(5)yt=ΦTΘ=∑l=0Lϕl∗θl,where the bases ϕl∗ are the smooth functions in the space expanded from the past inputs/outputs, for example, y3t−2ut−1, u3t−1, yt−1yt−5, and the associated coefficients θl 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 aU-model from this classical polynomial, set up an absorbing rule.Absorbing rule: letμ:RL+1⟶RM+1 be a map from a polynomial fP to its U-polynomial fU and suppose that its inverse μ−1 exists; therefore, it has(6)fPP∗,Θ⟶μfUΨ∗,Uut−1.The mapping has some proper algebra properties as [8].(7)a∀P∗,Θ,Ψ∗,Uut−1,fP∗=fU∗⟹P∗,Θ=Ψ∗,Uut−1b∀P∗,Θ∈fP,∃Ψ∗,Uut−1∈fU,μP∗,Θ=Ψ∗,Uut−1cμ−1⋅μ=I.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.The absorbing rule is a formation ofΨ∗ from the polynomial fP with reference to ut−1: first identify a control basis function Uut−1 and then absorb all the other associated functions as a coefficient that varies with time.Therefore, using the absorbing rule, realisingfU mapped from polynomial fP (5) gives the following:(8)yt=ΨTU=∑j=0MψjtUjut−1.This function is expanded from the above nonlinear functionfP as a polynomial in terms of ut−1. M is the number of items associated with input ut−1 and the time-varying parameter vector Ψ∗=ψ0t⋯ψMt∈ℝM+1 is a function derived from absorbing the other regression terms and the coefficients.Example 1.
Consider the polynomial model as shown below:(9)yt=0.2∗sinyt−1+ut−1exp−y2t−1−0.8yt−2ut−2u3t−1.
Absorbing the terms associated withut−1 into the vector Ψ∗ gives the corresponding U-model realisation as follows:(10)yt=ψ0t+ψ1tU1ut−1+ψ2tU2ut−1,where(11)ψ0t=0.2∗sinyt−1,ψ1t=exp−y2t−1,ψ2t=−0.8yt−2ut−2,U1ut−1=ut−1,U2ut−1=u3t−1.


### 2.1.2.U-Model Realisation from Classic Rational Models

Rational model, also known as total nonlinear model [13], is a ratio of two polynomials as follows:(12)yt=frΦ∗,Θ=fpnΦn∗,ΘnfpdΦd∗,Θd.Herefr is a rational function, the ratio of the fpn/numerator polynomial and fpd/denominator polynomial, which are maps of the input space into the output space. The 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 theU-polynomial model conversion, formulate the U-rational model expression as follows:(13)yt=ΨnTUnΨdTUd=∑j=0MnψjntUjnut−1∑j=0MdψjdtUjdut−1.To obtain the model inversion for solving the roots, expand the model as follows:(14)yt∑j=0MdψjdtUjdut−1=∑j=0MnψjntUjnut−1.Example 2.
Consider the rational model as follows:(15)yt=0.1y3t−2+sinut−1+0.5u3t−11+cos2yt−2+u2t−1.
Absorbing the terms associated withut−1 into the vectors Ψn∗,Ψd∗ gives the corresponding U-model realisation as follows:(16)fpnut−1=ψ0nt+ψ1ntU1nut−1+ψ2ntU2nut−1,fpdut−1=ψ0dt+ψ1dtU1dut−1,where(17)ψ0nt=0.1y3t−2,ψ0dt=1+cos2yt−2,ψ1nt=1,ψ2n=0.5,ψ1dt=1,(18)U1nut−1=sinut−1,U2nut−1=u3t−1,U1dut−1=u2t−1.


### 2.1.3.U Realisation from a Classical State-Space Mode-Multilayer U-Model

For a general SISO state-space system model, it has(19)Xt+1=FXt,ut,yt=hXt,where X∈ℝnu∈ℝy∈ℝ denotes the state, the control, and the output at time t∈ℤ+, respectively. F∈ℝn 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 U-model expression as follows:(20)x1t+1=∑j=0M1ψ1jtU1jx2t,⋮,xn−1t+1=∑j=0Mn−1ψn−1jtUn−1jxnt,xnt+1=∑j=0MnψnjtUnjut,yt=hXt.For each line,Mj is the number of terms associated with the next line state variable xj+1t and ψijt=ψj0t⋯ψjMjt∈ℝMj+1i∈1…n are time-varying parameter vector functions absorbing the other state variables. In the penultimate line, Mn consists of the terms associated with control ut and the time-varying vectors ψn0t⋯ψnMnt∈ℝMn+1 absorb all the states associated with the control vector Un0t⋯UnMnt∈ℝMn+1. Therefore, 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:(21)x1t+1=x2t+0.1x1tx2t,x2t+1=−0.1x1t−0.7x2t+ut,yt=x1t.Take realisation of the corresponding multilayerU-model by using the absorbing rule as below:(22)x1t+1=ψ11tU11x2t,x2t+1=ψ20t+ψ21U12ut,yt=x1t,where(23)ψ11t=1+0.1x1t,ψ20t=−0.1x1t−0.7x2t,ψ21t=1,U11x2t=x2t,U12ut=ut.


## 2.1.1.U-Model Realisation from Classic Polynomials

Consider a general classical SISO polynomial model in the form of(4)yt=fPΦ∗,Θ,Φ∗=ΦYt−1,Ut−1,where yt∈ℝut−1∈ℝ are the output/input, respectively, at t∈ℤ+ and Φ∗=ϕ0∗⋯ϕL∗∈ℝL+1, where Yt−1,Ut−1 are expanded from the output and input, respectively, in the proper dimensions. Let ϕi∗=ϕiyt−1,…,yt−nut−1,…ut−n∀i=0⋯n, where n is the plant dynamic order and Θ=θ0⋯θL∈ℝL+1 is the associated parametric vector. Let the function fP:u⟶y be a polynomial mapping of the input space to the output space. The vector form of the expanded equation (4) is given as follows:(5)yt=ΦTΘ=∑l=0Lϕl∗θl,where the bases ϕl∗ are the smooth functions in the space expanded from the past inputs/outputs, for example, y3t−2ut−1, u3t−1, yt−1yt−5, and the associated coefficients θl 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 aU-model from this classical polynomial, set up an absorbing rule.Absorbing rule: letμ:RL+1⟶RM+1 be a map from a polynomial fP to its U-polynomial fU and suppose that its inverse μ−1 exists; therefore, it has(6)fPP∗,Θ⟶μfUΨ∗,Uut−1.The mapping has some proper algebra properties as [8].(7)a∀P∗,Θ,Ψ∗,Uut−1,fP∗=fU∗⟹P∗,Θ=Ψ∗,Uut−1b∀P∗,Θ∈fP,∃Ψ∗,Uut−1∈fU,μP∗,Θ=Ψ∗,Uut−1cμ−1⋅μ=I.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.The absorbing rule is a formation ofΨ∗ from the polynomial fP with reference to ut−1: first identify a control basis function Uut−1 and then absorb all the other associated functions as a coefficient that varies with time.Therefore, using the absorbing rule, realisingfU mapped from polynomial fP (5) gives the following:(8)yt=ΨTU=∑j=0MψjtUjut−1.This function is expanded from the above nonlinear functionfP as a polynomial in terms of ut−1. M is the number of items associated with input ut−1 and the time-varying parameter vector Ψ∗=ψ0t⋯ψMt∈ℝM+1 is a function derived from absorbing the other regression terms and the coefficients.Example 1.
Consider the polynomial model as shown below:(9)yt=0.2∗sinyt−1+ut−1exp−y2t−1−0.8yt−2ut−2u3t−1.
Absorbing the terms associated withut−1 into the vector Ψ∗ gives the corresponding U-model realisation as follows:(10)yt=ψ0t+ψ1tU1ut−1+ψ2tU2ut−1,where(11)ψ0t=0.2∗sinyt−1,ψ1t=exp−y2t−1,ψ2t=−0.8yt−2ut−2,U1ut−1=ut−1,U2ut−1=u3t−1.


## 2.1.2.U-Model Realisation from Classic Rational Models

Rational model, also known as total nonlinear model [13], is a ratio of two polynomials as follows:(12)yt=frΦ∗,Θ=fpnΦn∗,ΘnfpdΦd∗,Θd.Herefr is a rational function, the ratio of the fpn/numerator polynomial and fpd/denominator polynomial, which are maps of the input space into the output space. The 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 theU-polynomial model conversion, formulate the U-rational model expression as follows:(13)yt=ΨnTUnΨdTUd=∑j=0MnψjntUjnut−1∑j=0MdψjdtUjdut−1.To obtain the model inversion for solving the roots, expand the model as follows:(14)yt∑j=0MdψjdtUjdut−1=∑j=0MnψjntUjnut−1.Example 2.
Consider the rational model as follows:(15)yt=0.1y3t−2+sinut−1+0.5u3t−11+cos2yt−2+u2t−1.
Absorbing the terms associated withut−1 into the vectors Ψn∗,Ψd∗ gives the corresponding U-model realisation as follows:(16)fpnut−1=ψ0nt+ψ1ntU1nut−1+ψ2ntU2nut−1,fpdut−1=ψ0dt+ψ1dtU1dut−1,where(17)ψ0nt=0.1y3t−2,ψ0dt=1+cos2yt−2,ψ1nt=1,ψ2n=0.5,ψ1dt=1,(18)U1nut−1=sinut−1,U2nut−1=u3t−1,U1dut−1=u2t−1.


## 2.1.3.U Realisation from a Classical State-Space Mode-Multilayer U-Model

For a general SISO state-space system model, it has(19)Xt+1=FXt,ut,yt=hXt,where X∈ℝnu∈ℝy∈ℝ denotes the state, the control, and the output at time t∈ℤ+, respectively. F∈ℝn 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 U-model expression as follows:(20)x1t+1=∑j=0M1ψ1jtU1jx2t,⋮,xn−1t+1=∑j=0Mn−1ψn−1jtUn−1jxnt,xnt+1=∑j=0MnψnjtUnjut,yt=hXt.For each line,Mj is the number of terms associated with the next line state variable xj+1t and ψijt=ψj0t⋯ψjMjt∈ℝMj+1i∈1…n are time-varying parameter vector functions absorbing the other state variables. In the penultimate line, Mn consists of the terms associated with control ut and the time-varying vectors ψn0t⋯ψnMnt∈ℝMn+1 absorb all the states associated with the control vector Un0t⋯UnMnt∈ℝMn+1. Therefore, 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:(21)x1t+1=x2t+0.1x1tx2t,x2t+1=−0.1x1t−0.7x2t+ut,yt=x1t.Take realisation of the corresponding multilayerU-model by using the absorbing rule as below:(22)x1t+1=ψ11tU11x2t,x2t+1=ψ20t+ψ21U12ut,yt=x1t,where(23)ψ11t=1+0.1x1t,ψ20t=−0.1x1t−0.7x2t,ψ21t=1,U11x2t=x2t,U12ut=ut.


## 2.2. Inversion ofU-Polynomial Models

For simplicity, consider the SISO polynomialU-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 ut−1.Iteratively, the root searching computation gives rise to the following formulation:(24)uk+1t−1=ukt−1−yt−∑j=0Mλjtukjt−1d∑j=0Mλjtujt−1/dut−1ukt−1=ukjt−1.Here, indexk is the iteration handle: generate the (k+1)th results from the kth iteration, k≻0. There 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 asroots, can be used to find accurate roots of the U-model equations.


## 2.3. Inversion ofU-State Space Models

For simplicity, consider the SISOU-state space model (20). Inversion is a multilayer root solving procedure involving a back-stepping routine whenever x1t+1 is known; each line of the equation iteratively uses the Newton–Raphson algorithm to obtain x1t+1x2t+1⋯ in back-stepping order.


## 3.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]. This 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-by-step procedure for U-control implementation.


### 3.1.U-Control Framework

LetGp 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)
The model inverseGp−1 exists(2)
Lipschitz continuity is satisfied, andGp and its inverse Gp−1 are diffeomorphic and globally uniformly Lipschitz in ℝn; that is,Gx1−Gx2≤γ1Gx1−x2,∀x1,x2∈ℝnG−1x1−G−1x2≤γ2G−1x1−x2,∀x1,x2∈ℝn, where x1,x2 are the states with Gp in expression of state space equation and γ1,γ2 are the Lipschitz coefficients.For simplicity, but not losing generality, take consideration of a SISO (inputu∈ℝ1 and output y∈ℝ1) 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(25)∑=FlfbcGc1Gip,where Flfbc is a linear feedback loop with functions, linear virtual controller Gc1:y⟶u, and virtual unit plant Gip=1:u⟶y.ThisU-control system structure proposes a model-independent control procedure, because the designs of Gc1:y⟶u and Gip=1:u⟶y are independent. These two independent designs are explained below.For design of the virtual linear controllerGc1:y⟶u, referring to Figure 3(a), it gives(26)Gc1=G1−G=1−G−1G,where G is a specified closed-loop transfer function with proper dynamic/static responses.For design of the virtual unit plantGip=1:u⟶y, designing/formulating the plant inverse GP−1 gives(27)Gip=GP−1Gp=1:u⟶y.Remark 1.
Regarding the merit of the design prototype, the establishedU-control system framework (25) has two independent inversion designs: (1) linear controller Gc1:y⟶u without involving any plant model structures; therefore, it is also named as linear invariant controller [9]; (2) virtual plant unitisation Gip=GP−1Gp=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). Therefore, the two designs are separately independent and connected within a linear feedback control loop.Remark 2.
Regarding the efficiency of theU-control system design, linear controller Gc1:y⟶u is once-off design irrespective of plant model types and parameters. Plant inverter GP−1 is formable for polynomial and state-space equations in U-model and numerically solvable for the roots to achieve Gip=GP−1Gp=1:u⟶y. Consequently, the control system design is reduced to the determination of the plant inverse GP−1 once the linear controller designed. Consequently, the design procedure is that once-off Gc1:y⟶u design and GP−1⟹Gip=GP−1Gp=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). This aspect can be explained through an inverse function Ψ−1; for U-control systems, it is split into two separate functions of Gc1=Ψ1−1G (linear dynamic inversion) and Gp−1=Ψ2−1Gp(U-model-based root solving). For the other popular control system design approaches, it is at least a function of Ψ−1Gc1,Gp, 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 theU-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, well-designed control systems could takeU-control as a plug-in box to expand to control different types of plants.Remark 5.
AsU-control is fundamentally based on the assumption GP−1Gp=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.


### 3.2. Design Procedure

With reference to the aforementioned description and the block diagram in Figure4, here we list a step-by-step design procedure.(1)
Establish a stable linear feedback control system structured in Figure4. 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 plantGip=1:u⟶y that has been achieved. determine a linear invariant controller Gc1 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 Gc1.(3)
Convert plant model intoU-model realisation ∑U−model with reference to the formulations presented in Section 2.(4)
To achieveGip=1:u⟶y to guarantee the desired output ydesiredt=vt, determine the controller output by solving an equation vt−∑U−model=0; that is, ut∈vt−∑U−model=0.(5)
Locate/connect the blocks in Figure5.Figure 4
U-control framework.Figure 5
U-control implementation.


### 3.3.U-Model-Based Adaptive Control

This 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.Figure 6
U-model-based adaptive control.


### 3.4. Robustness Analysis ofU-Control

This section presents the robustness analysis ofU-control based on discrete-time H∞ using linear matrix inequalities (LMI) technique. Consider the state-space equation in terms of multilayer U-realisation (20) with an external disturbance vector Wt=w1t⋯wntT as(28)x1t+1=∑j=0M1Ψ1jtU1jx2t+w1t,⋮,xn−1t+1=∑j=0Mn−1Ψn−1jtUn−1jxnt+wn−1t,x1t+1=∑j=0MnΨnjtUnjut+wnt,yt=hXt.Remark 1.
Assume that the elements of the external disturbance vector are bounded; that is,wit<di,∀i=1⋯n, where di is a positive constant.
To provide the robustness analysis, take one single line state ofxn=t+1 from state-space equation (28) at first, and then extend the analysis to the other state variables xit+1. Accordingly, take out(29)Twx:xnt+1=∑j=0MnΨnjtUnjXnt+wnjt,yt=hxt.
The control objective is to minimize the effect of the external disturbancewn on the state vector xn. This study takes the discrete-time H∞ robust control technique into consideration, where the robust control condition is(30)Twx<β⇒supxnL2wnL2<β,where β is a known constant defining the upper boundary of H∞ performance index. Equation (30) can be rewritten as(31)xnL22<β2wnL22,or equivalently(32)β−1xnL22−βwnL22<0,with(33)xnL22=∑t=0∞xnTtxnt,wnL22=∑t=0∞wnTtwnt.
From the above formulations, we have(34)⟹∑t=0∞β−1xnTtxnt−βwnTtwnt<0.
Construct the positive-definite Lyapunov function with(35)Vxnt=xnTtQxnt>0,where Q>0. Suppose that the gradient of the Lyapunov function ∇Vxnt is satisfied in the following inequality:(36)∇Vxnt+β−1xnTtxnt−βwnTtwnt<0.
In order to prove condition of (36), take summation (Σ) of all terms as(37)∑t=0∞∇Vxnt+∑t=0∞β−1xnTtxnt−βwnTtwnt<0.
Since the first term of (37) is positive, the second term is always negative; that is,(38)∑t=0∞β−1xnTtxnt−βwnTtwnt<0.which is the same as condition (34). Then, inequality (36) is a correct assumption.
Determine the gradient of the Lyapunov function by(39)∇Vxnt=xnTt+1Qxnt+1−xnTt+1QxnTt.
By substituting (39) into (36), we have(40)xnTt+1Qxnt+1−xnTtQxnt+β−1xnTtQxnt−βwnTtwnt<0.
Now, substitutingxnt+1 from (28) into (40), we have(41)∑j=0MnΨnjtUnjxnt+wnjtTQΨnjtUnjxnt+wnjt−∑j=0MnΨnjt−1Unjxnt−1+wnjt−1TQΨnjt−1Unjxnt−1+wnjt−1+∑j=0Mnβ−1Ψnjt−1Unjxnt−1+wnjt−1TΨnjt−1Unjxnt−1+wnjt−1−∑j=0MnβwnjTtwnjt<0.
In what follows, for simplicity, shorten the following notations as(42)Ψnjt≜Ψt,Ψnjt−1≜Ψt−1,Unjxnt≜Ut,Unjxnt−1≜Ut−1,wnjt≜wt,wnjt−1≜wt−1.
Then (41) is expressed as(43)∑UtTψtTQψtUt+UtTψtTQwt+wtTQψtUt+wtTQwt−∑Ut−1Tψt−1TQψt−1Ut−1+Ut−1Tψt−1TQwt−1+wt−1TQψt−1Ut−1+wt−1TQwt−1+∑β−1Ut−1Tψt−1Tψt−1Ut−1+Ut−1Tψt−1Twt−1+wt−1Tψt−1Ut−1+wt−1Twt−1−∑βwtTwt<0.
Furthermore, it can be expressed in terms of quadratic form(44)∑UtTψtTQψtUt+∑UtTψtTQwt+∑wtTQψtUt+∑wtTQ−βIwt+∑Ut−1Tψt−1Tβ−1I−Qψt−1Ut−1+∑Ut−1Tψt−1Tβ−1I−Qwt−1+∑wt−1Tβ−1I−Qψt−1Ut−1−∑wt−1Tβ−1I−Qwt−1<0,and then matrix form of(45)∑Utψt∑wt∑Ut−1ψt−1∑wt−1HH∑Utψt∑wt∑Ut−1ψt−1∑wt−1<0,with(46)H=QQ00QQ−βI0000β−1I−Qβ−1I−Q00β−1I−Qβ−1I−Q<0.
Now, applying the Schur complement [20] on (46), we get(47)F22<0,F11−F12F22−1F21<0,where(48)F11=QQQQ−βI,F12=F21=0000,F22β−1I−Qβ−1I−Qβ−1I−Qβ−1I−Q<0.
Then condition of (47) can be simplified as(49)β−1I−Qβ−1I−Qβ−1I−Qβ−1I−Q<0,QQQQ−βI<0.
Defining a new variableγ:=β−1 in the first inequality of (49), it changes to(50)LMIQ,γ:γI−QγI−QγI−QγI−Q<0⟹γI−Q<0,where the optimal values Q∗ and γ∗ can be calculated via Matlab LMI toolbox. The optimal value of β is correspondingly given by β∗=γ∗−1. Applying Schur complement on the second inequality of (49), we have(51)Q<0,Q−βI−QQ−1Q<0⟶−βI<0,which yields −γI<0. Then, from (51), it gives(52)LMIQ,γ:Q00−γI<0⟹Q∗,γ∗,where the existence of optimal solutions for Q∗ and γ∗ means the robustness of the U-model system versus external disturbances.
The robustness analysis for the remainder of equations of state-space model (28) can be proved similar to the above-presented procedure. In addition, Virtual Equivalent System (VES) methods [21–23] can also be used for robustness analysis of U-model control systems.


## 3.1.U-Control Framework

LetGp 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)
The model inverseGp−1 exists(2)
Lipschitz continuity is satisfied, andGp and its inverse Gp−1 are diffeomorphic and globally uniformly Lipschitz in ℝn; that is,Gx1−Gx2≤γ1Gx1−x2,∀x1,x2∈ℝnG−1x1−G−1x2≤γ2G−1x1−x2,∀x1,x2∈ℝn, where x1,x2 are the states with Gp in expression of state space equation and γ1,γ2 are the Lipschitz coefficients.For simplicity, but not losing generality, take consideration of a SISO (inputu∈ℝ1 and output y∈ℝ1) 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(25)∑=FlfbcGc1Gip,where Flfbc is a linear feedback loop with functions, linear virtual controller Gc1:y⟶u, and virtual unit plant Gip=1:u⟶y.ThisU-control system structure proposes a model-independent control procedure, because the designs of Gc1:y⟶u and Gip=1:u⟶y are independent. These two independent designs are explained below.For design of the virtual linear controllerGc1:y⟶u, referring to Figure 3(a), it gives(26)Gc1=G1−G=1−G−1G,where G is a specified closed-loop transfer function with proper dynamic/static responses.For design of the virtual unit plantGip=1:u⟶y, designing/formulating the plant inverse GP−1 gives(27)Gip=GP−1Gp=1:u⟶y.Remark 1.
Regarding the merit of the design prototype, the establishedU-control system framework (25) has two independent inversion designs: (1) linear controller Gc1:y⟶u without involving any plant model structures; therefore, it is also named as linear invariant controller [9]; (2) virtual plant unitisation Gip=GP−1Gp=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). Therefore, the two designs are separately independent and connected within a linear feedback control loop.Remark 2.
Regarding the efficiency of theU-control system design, linear controller Gc1:y⟶u is once-off design irrespective of plant model types and parameters. Plant inverter GP−1 is formable for polynomial and state-space equations in U-model and numerically solvable for the roots to achieve Gip=GP−1Gp=1:u⟶y. Consequently, the control system design is reduced to the determination of the plant inverse GP−1 once the linear controller designed. Consequently, the design procedure is that once-off Gc1:y⟶u design and GP−1⟹Gip=GP−1Gp=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). This aspect can be explained through an inverse function Ψ−1; for U-control systems, it is split into two separate functions of Gc1=Ψ1−1G (linear dynamic inversion) and Gp−1=Ψ2−1Gp(U-model-based root solving). For the other popular control system design approaches, it is at least a function of Ψ−1Gc1,Gp, 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 theU-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, well-designed control systems could takeU-control as a plug-in box to expand to control different types of plants.Remark 5.
AsU-control is fundamentally based on the assumption GP−1Gp=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.


## 3.2. Design Procedure

With reference to the aforementioned description and the block diagram in Figure4, here we list a step-by-step design procedure.(1)
Establish a stable linear feedback control system structured in Figure4. 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 plantGip=1:u⟶y that has been achieved. determine a linear invariant controller Gc1 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 Gc1.(3)
Convert plant model intoU-model realisation ∑U−model with reference to the formulations presented in Section 2.(4)
To achieveGip=1:u⟶y to guarantee the desired output ydesiredt=vt, determine the controller output by solving an equation vt−∑U−model=0; that is, ut∈vt−∑U−model=0.(5)
Locate/connect the blocks in Figure5.Figure 4
U-control framework.Figure 5
U-control implementation.


## 3.3.U-Model-Based Adaptive Control

This 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.Figure 6
U-model-based adaptive control.


## 3.4. Robustness Analysis ofU-Control

This section presents the robustness analysis ofU-control based on discrete-time H∞ using linear matrix inequalities (LMI) technique. Consider the state-space equation in terms of multilayer U-realisation (20) with an external disturbance vector Wt=w1t⋯wntT as(28)x1t+1=∑j=0M1Ψ1jtU1jx2t+w1t,⋮,xn−1t+1=∑j=0Mn−1Ψn−1jtUn−1jxnt+wn−1t,x1t+1=∑j=0MnΨnjtUnjut+wnt,yt=hXt.Remark 1.
Assume that the elements of the external disturbance vector are bounded; that is,wit<di,∀i=1⋯n, where di is a positive constant.
To provide the robustness analysis, take one single line state ofxn=t+1 from state-space equation (28) at first, and then extend the analysis to the other state variables xit+1. Accordingly, take out(29)Twx:xnt+1=∑j=0MnΨnjtUnjXnt+wnjt,yt=hxt.
The control objective is to minimize the effect of the external disturbancewn on the state vector xn. This study takes the discrete-time H∞ robust control technique into consideration, where the robust control condition is(30)Twx<β⇒supxnL2wnL2<β,where β is a known constant defining the upper boundary of H∞ performance index. Equation (30) can be rewritten as(31)xnL22<β2wnL22,or equivalently(32)β−1xnL22−βwnL22<0,with(33)xnL22=∑t=0∞xnTtxnt,wnL22=∑t=0∞wnTtwnt.
From the above formulations, we have(34)⟹∑t=0∞β−1xnTtxnt−βwnTtwnt<0.
Construct the positive-definite Lyapunov function with(35)Vxnt=xnTtQxnt>0,where Q>0. Suppose that the gradient of the Lyapunov function ∇Vxnt is satisfied in the following inequality:(36)∇Vxnt+β−1xnTtxnt−βwnTtwnt<0.
In order to prove condition of (36), take summation (Σ) of all terms as(37)∑t=0∞∇Vxnt+∑t=0∞β−1xnTtxnt−βwnTtwnt<0.
Since the first term of (37) is positive, the second term is always negative; that is,(38)∑t=0∞β−1xnTtxnt−βwnTtwnt<0.which is the same as condition (34). Then, inequality (36) is a correct assumption.
Determine the gradient of the Lyapunov function by(39)∇Vxnt=xnTt+1Qxnt+1−xnTt+1QxnTt.
By substituting (39) into (36), we have(40)xnTt+1Qxnt+1−xnTtQxnt+β−1xnTtQxnt−βwnTtwnt<0.
Now, substitutingxnt+1 from (28) into (40), we have(41)∑j=0MnΨnjtUnjxnt+wnjtTQΨnjtUnjxnt+wnjt−∑j=0MnΨnjt−1Unjxnt−1+wnjt−1TQΨnjt−1Unjxnt−1+wnjt−1+∑j=0Mnβ−1Ψnjt−1Unjxnt−1+wnjt−1TΨnjt−1Unjxnt−1+wnjt−1−∑j=0MnβwnjTtwnjt<0.
In what follows, for simplicity, shorten the following notations as(42)Ψnjt≜Ψt,Ψnjt−1≜Ψt−1,Unjxnt≜Ut,Unjxnt−1≜Ut−1,wnjt≜wt,wnjt−1≜wt−1.
Then (41) is expressed as(43)∑UtTψtTQψtUt+UtTψtTQwt+wtTQψtUt+wtTQwt−∑Ut−1Tψt−1TQψt−1Ut−1+Ut−1Tψt−1TQwt−1+wt−1TQψt−1Ut−1+wt−1TQwt−1+∑β−1Ut−1Tψt−1Tψt−1Ut−1+Ut−1Tψt−1Twt−1+wt−1Tψt−1Ut−1+wt−1Twt−1−∑βwtTwt<0.
Furthermore, it can be expressed in terms of quadratic form(44)∑UtTψtTQψtUt+∑UtTψtTQwt+∑wtTQψtUt+∑wtTQ−βIwt+∑Ut−1Tψt−1Tβ−1I−Qψt−1Ut−1+∑Ut−1Tψt−1Tβ−1I−Qwt−1+∑wt−1Tβ−1I−Qψt−1Ut−1−∑wt−1Tβ−1I−Qwt−1<0,and then matrix form of(45)∑Utψt∑wt∑Ut−1ψt−1∑wt−1HH∑Utψt∑wt∑Ut−1ψt−1∑wt−1<0,with(46)H=QQ00QQ−βI0000β−1I−Qβ−1I−Q00β−1I−Qβ−1I−Q<0.
Now, applying the Schur complement [20] on (46), we get(47)F22<0,F11−F12F22−1F21<0,where(48)F11=QQQQ−βI,F12=F21=0000,F22β−1I−Qβ−1I−Qβ−1I−Qβ−1I−Q<0.
Then condition of (47) can be simplified as(49)β−1I−Qβ−1I−Qβ−1I−Qβ−1I−Q<0,QQQQ−βI<0.
Defining a new variableγ:=β−1 in the first inequality of (49), it changes to(50)LMIQ,γ:γI−QγI−QγI−QγI−Q<0⟹γI−Q<0,where the optimal values Q∗ and γ∗ can be calculated via Matlab LMI toolbox. The optimal value of β is correspondingly given by β∗=γ∗−1. Applying Schur complement on the second inequality of (49), we have(51)Q<0,Q−βI−QQ−1Q<0⟶−βI<0,which yields −γI<0. Then, from (51), it gives(52)LMIQ,γ:Q00−γI<0⟹Q∗,γ∗,where the existence of optimal solutions for Q∗ and γ∗ means the robustness of the U-model system versus external disturbances.
The robustness analysis for the remainder of equations of state-space model (28) can be proved similar to the above-presented procedure. In addition, Virtual Equivalent System (VES) methods [21–23] can also be used for robustness analysis of U-model control systems.


## 4. Simulation Examples

This 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. The main purposes for designing the simulation tests of theU-control are as follows:(1)
To demonstrate the principle of model-independent design inU-control.(2)
To demonstrate the once-off design of the invariant controller which specifies the whole closed loop 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 ofU-control, particularly in the design of nonlinear control systems.(4)
To demonstrate thatU-control can supplement/enhance classic pole placement control.From previous sections, the design is divided into two parallel blocks: (1) designing the linear invariant controlGc1 (thus vt) by reversing the specified closed-loop transfer function and (2) determining the control input ut−1 by reversing the plant U-model equation.For familiarisation of different notations used inU-control, this simulation section takes in yt−j=yt−j,j∈ℤ+, ut−j=ut−j,j∈ℤ+, vt−j=vt−j,j∈ℤ+, xjt+1=xjt+1,j∈ℤ+, and ψjt=ψj,j∈ℤ+.


### 4.1. Design Invariant ControlGc1

In a popular approach, the conventional pole placement control [24] assigns the closed-loop characteristic equation in terms of Z transform:(53)Az=z2+a1z+a2=z2−1.3205z+0.4966.Equivalently the poles are located at0.6603±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 closed-loop transfer function as(54)Bz=bz,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(55)b=Azz=1=1−1.3205+0.4966=0.1761.Thereby, the resultant transfer function is specified as(56)YzRz=Gz=0.1761z−11−1.3205z−1+0.4966z−2.It should be noted that when the conditionGip=1:u⟶y is satisfied, it gives(57)VzRz=Gz=0.1761z−11−1.3205z−1+0.4966z−2,where Vz is the Z transform of the controller Gc1 output as shown in Figure 3.To determine the linear invariant controllerGc1, temporarily, let the plant Gp=1 or Gip=Gp−1Gp=1:u⟶y. Then take inverse of the transfer function G to yield(58)Gc1=Gz1−Gz=bz−11+a1−cz−1+a2z−2=0.1761z−11−1.4966z−1+0.4966z−2.The rest of the control system design will formulate the specific plant inverseGp−1 in form of U-model for each selected example, which will be implemented in each related subsection.


### 4.2. Hammerstein Model: A SISO Nonlinear Polynomial [7]

The 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]. The simulation example selected [7] is as follows:(59)yt=0.5yt−1+xt−1+0.1xt−2,xt=1+ut−1−ut−12+0.2ut−13,where yt,ut,xt are the plant output, input, and intermediate variable for the static nonlinear component output, respectively.As explained above, the first step inU-control system design is generic to determine the linear invariant controller Gc1, that is, independent of the plant model and universally designed (as was done in the beginning of this section). The second step of the design is specifically working out the controller output ut−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:(60)yt=ψ0+ψ1ut−1+ψ2ut−12+ψ3ut−13,where(61)ψ0=0.5yt−1+1+0.3xt−2,ψ1=1,ψ2=−1,ψ3=0.2.Then replace the outputyt with the virtual controller output vt (i.e., the desired output). Subsequently, it determines one of the roots by solving (60) as the controller output. This gives the following formula:(62)ut−1∈rootsvt−ψ0+ψ1u+t−1ψ2ut−12+ψ3ut−13=0.Figure7 illustrates the simulation results.Figure 7
(a) Plant output. (b) Control input.
(a)(b)


### 4.3. Nonlinear State-Space Model

The control of strict-feedback 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]. The simulation example for the state-space model is as follows:(63)x1t=x2t−1+0.1x1t−1x2t−1,x2t=−0.1x1t−1−0.7x2t−1+ut−1,yt=x1t,where yt,ut,xt denote the plant output and input and xt is a state vector, respectively. This represents a second-order nonlinear dynamic plant.Again, in the second step ofU-control system design, it requires to work out the specific controller output ut−1 by inverting the plant U-model. The realised multilayer U-model is expressed as follows:(64)x1t=ψ11x2t−1,x2t=ψ20+ψ21ut−1,yt=x1t,where ψ11=1+0.1x1t, ψ20=−0.1x1t−1−0.7x2t−1, and ψ21=1.This is a two-layerU-model structure. Accordingly, using back-stepping routine with initial x1t=vt works out the controller output ut−1 by inverting each line of the equations, as specified in step 4 in U-control design procedure. The simulation results are shown in Figure 8.Figure 8
(a) Plant output. (b) Control input.
(a)(b)


### 4.4. Extended Total Nonlinear Model [9]

The control of nonlinear rational systems, which are modelled as ratios of two nonlinear polynomials, is even more challenging. Until a recent analyticalU-model-based approach [9], these models were previously taken as examples of complex systems in neurocontrol system design. The difficulty is that rational model sets are subject to total nonlinearity (both in the parameters/identification and in input/control) [13]. The selected simulation example [9] with dynamics (time delay) and transcendental nonlinearities was as follows:(65)yt=0.5yt−1+sinut−1+ut−11+exp−y2t−1,where yt,ut are the plant output and input, respectively. Once again, by applying the absorbing rule, it yields the following U-rational model:(66)yt1+exp−yt−12=0.5yt−1+sinut−1+ut−1⟹,ψ0dyt=ψ0n+ψ1nsinut−1+ψ2nut−1.With the same linear invariant controller Gc1 used as before, replacing the output yt of (66) with the desired output vt of (58) gives the following:(67)ψ0dvt=ψ0n+ψ1nsinut−1+ψ2nut−1.Subsequently, the control inputut−1 is obtained by the following:(68)ψ0dvt−ψ0n−ψ1nsinut−1−ψ2nut−1=0.Figure9 illustrates the simulation results. Again, the bench test confirms the performance of the U-control.Figure 9
(a) Plant output. (b) Control input.
(a)(b)


## 4.1. Design Invariant ControlGc1

In a popular approach, the conventional pole placement control [24] assigns the closed-loop characteristic equation in terms of Z transform:(53)Az=z2+a1z+a2=z2−1.3205z+0.4966.Equivalently the poles are located at0.6603±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 closed-loop transfer function as(54)Bz=bz,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(55)b=Azz=1=1−1.3205+0.4966=0.1761.Thereby, the resultant transfer function is specified as(56)YzRz=Gz=0.1761z−11−1.3205z−1+0.4966z−2.It should be noted that when the conditionGip=1:u⟶y is satisfied, it gives(57)VzRz=Gz=0.1761z−11−1.3205z−1+0.4966z−2,where Vz is the Z transform of the controller Gc1 output as shown in Figure 3.To determine the linear invariant controllerGc1, temporarily, let the plant Gp=1 or Gip=Gp−1Gp=1:u⟶y. Then take inverse of the transfer function G to yield(58)Gc1=Gz1−Gz=bz−11+a1−cz−1+a2z−2=0.1761z−11−1.4966z−1+0.4966z−2.The rest of the control system design will formulate the specific plant inverseGp−1 in form of U-model for each selected example, which will be implemented in each related subsection.


## 4.2. Hammerstein Model: A SISO Nonlinear Polynomial [7]

The 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]. The simulation example selected [7] is as follows:(59)yt=0.5yt−1+xt−1+0.1xt−2,xt=1+ut−1−ut−12+0.2ut−13,where yt,ut,xt are the plant output, input, and intermediate variable for the static nonlinear component output, respectively.As explained above, the first step inU-control system design is generic to determine the linear invariant controller Gc1, that is, independent of the plant model and universally designed (as was done in the beginning of this section). The second step of the design is specifically working out the controller output ut−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:(60)yt=ψ0+ψ1ut−1+ψ2ut−12+ψ3ut−13,where(61)ψ0=0.5yt−1+1+0.3xt−2,ψ1=1,ψ2=−1,ψ3=0.2.Then replace the outputyt with the virtual controller output vt (i.e., the desired output). Subsequently, it determines one of the roots by solving (60) as the controller output. This gives the following formula:(62)ut−1∈rootsvt−ψ0+ψ1u+t−1ψ2ut−12+ψ3ut−13=0.Figure7 illustrates the simulation results.Figure 7
(a) Plant output. (b) Control input.
(a)(b)


## 4.3. Nonlinear State-Space Model

The control of strict-feedback 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]. The simulation example for the state-space model is as follows:(63)x1t=x2t−1+0.1x1t−1x2t−1,x2t=−0.1x1t−1−0.7x2t−1+ut−1,yt=x1t,where yt,ut,xt denote the plant output and input and xt is a state vector, respectively. This represents a second-order nonlinear dynamic plant.Again, in the second step ofU-control system design, it requires to work out the specific controller output ut−1 by inverting the plant U-model. The realised multilayer U-model is expressed as follows:(64)x1t=ψ11x2t−1,x2t=ψ20+ψ21ut−1,yt=x1t,where ψ11=1+0.1x1t, ψ20=−0.1x1t−1−0.7x2t−1, and ψ21=1.This is a two-layerU-model structure. Accordingly, using back-stepping routine with initial x1t=vt works out the controller output ut−1 by inverting each line of the equations, as specified in step 4 in U-control design procedure. The simulation results are shown in Figure 8.Figure 8
(a) Plant output. (b) Control input.
(a)(b)


## 4.4. Extended Total Nonlinear Model [9]

The control of nonlinear rational systems, which are modelled as ratios of two nonlinear polynomials, is even more challenging. Until a recent analyticalU-model-based approach [9], these models were previously taken as examples of complex systems in neurocontrol system design. The difficulty is that rational model sets are subject to total nonlinearity (both in the parameters/identification and in input/control) [13]. The selected simulation example [9] with dynamics (time delay) and transcendental nonlinearities was as follows:(65)yt=0.5yt−1+sinut−1+ut−11+exp−y2t−1,where yt,ut are the plant output and input, respectively. Once again, by applying the absorbing rule, it yields the following U-rational model:(66)yt1+exp−yt−12=0.5yt−1+sinut−1+ut−1⟹,ψ0dyt=ψ0n+ψ1nsinut−1+ψ2nut−1.With the same linear invariant controller Gc1 used as before, replacing the output yt of (66) with the desired output vt of (58) gives the following:(67)ψ0dvt=ψ0n+ψ1nsinut−1+ψ2nut−1.Subsequently, the control inputut−1 is obtained by the following:(68)ψ0dvt−ψ0n−ψ1nsinut−1−ψ2nut−1=0.Figure9 illustrates the simulation results. Again, the bench test confirms the performance of the U-control.Figure 9
(a) Plant output. (b) Control input.
(a)(b)


## 5. Conclusion

U-control has been featured in several publications. This 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 inU-control(2)
Explain/demonstrate a universal design for multiple plant model structures(3)
Explain/demonstrateU-control workability and effectiveness/efficiency, particularly dealing with nonlinear plant control(4)
Explain/demonstrateU-control as a supplement to classic control system design frameworksIn terms of research techniques, compared with the two most popular control system design frameworks, model-based and model-free, this model-independent design effectively relieves the complexity involved in inverting the controller and plant together. The 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. The most critical issue with this design framework is its robustness because it relies on havingGp−1Gp=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, U-control 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 [28–32].



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