A new configuration of composite nonlinear feedback control for nonlinear systems with input saturation

Abstract

This study proposes a U-control–based Composite Nonlinear Feedback (U-CNF) design procedure. This U-CNF control establishes a double feedback loop framework for generalisation and simplification in designing the CNF control systems. Two controllers, in terms of double dynamic inversion, are designed separately, (1) to stabilise and cancel the nonlinearities and dynamics (convert the plant into an identity matrix) in the inner closed-loop, and then (2) to improve the system transient response by specifying a second-order linear system with a monotonic nonlinear function to smoothly tune the damping ratio. Accordingly, the conventional CNF characteristics in a concise pathway are achieved. The properties show, under proper conditions, the U-CNF control is plant model-free control, applicable to nonaffine nonlinear dynamic systems, and robust against model uncertainty and external disturbance. For the initial bench tests of the first-time proposed U-CNF configuration, the simulated case studies are provided with a transparent procedure to demonstrate the consistency with the analytical results in the numerical computations and to present guidance for applications.

Keywords

Uncertain nonlinear systems composite nonlinear feedback sliding mode control U-model-based control nonlinear dynamic inversion dynamic stability nonlinear damping ratio

1. Introduction

It has been a tense research topic to accommodate transient response performance in design of nonlinear control systems (George et al., 2016; Isidori et al., 1995; Slotine and Li, 1991). One of the popular approaches is the Composite Nonlinear Feedback (CNF) (Chen et al., 2003; Lin et al., 1998; Mobayen and Ma, 2018). The CNF control is a scheme consisting of a linear feedback law and a nonlinear feedback law without any switching element. The linear feedback part is designed to yield a closed-loop system with a small damping ratio for a quick response, while at the same time not exceeding the actuator limits for desired command input levels. The nonlinear feedback law is used to increase the damping ratio of the closed-loop system as the system output approaches the target reference to reduce the overshoot caused by the linear part (Bayat et al., 2018; Rasoolinasab et al., 2020). From the structure of the CNF control law, it is clear that the CNF controller reduces to a linear controller when the gains in the nonlinear law vanish. Therefore, the additional nonlinear feedback makes it possible to change the feedback gains to improve the transient performance (Mobayen, 2018). The merits of the CNF control lie in its simple structure and using the linear controller as a basic element which is of especial interest to many researchers and practical engineers as it can be easily implemented.

Here presents the related review on CNF approaches to justify the motivation of the study. Linear plant based: This is the origin/foundation of the CNF control. Lin et al. (1998) established the CNF design procedure with fundamental theoretical proof, algorithm formulation and simulation demonstrations. Chen et al. (2003) contributed a seminal study for CNF theory and application, and presented a real CNF control of hard disk drive servo system. Surely, these studies provided foundation for expansion to CNF control of various nonlinear dynamic systems. Nonlinear plant based: A CNF control approach is proposed in (Lu and Lan, 2019) for strict-feedback nonlinear systems with input saturation, which the standard backstepping technique was used to accommodate the nonlinear plant model. It has been noted that this approach assumed that the plant model exactly known without uncertainties/disturbances. An approach with genetic algorithm and fuzzy logic control is studied in (Mollabashi et al., 2019) to determine constant controller coefficients and optimize the control effort. It is workable even extra design complexity induced. Augmented: These include CNF for time-delay systems (Ghaffari, 2020; Singh et al., 2018), singular linear systems (Jafari and Binazadeh, 2018; Lin et al., 2017), chaotic systems (Mobayen and Tchier, 2017), observer-based control (Jafari and Binazadeh, 2019), adaptive control (Kulkarni and Purwar, 2018) and event-triggered design (Zuo et al., 2021). These approaches integrated CNF with the other techniques to achieve the CNF performance while keep the other properties remained in the systems. Applications: CNF has found many applications in the motion control systems. In (Chen et al., 2019), a CNF control method is proposed for autonomous vehicles with motion planning by velocity prediction for lane-change strategy. In (Hou and Fantoni, 2017), a CNF control technique is proposed for multiple quadrotors with interactive leader-follower consensus. A robust CNF control approach is studied in (Jiang et al., 2020) for uncertain robot manipulators. A gain-scheduling CNF is developed in (Chen et al., 2020) for impaired driver assistance control of vehicle trajectory tracking. In (Hu et al., 2018), a robust CNF control technique is presented for path-following of a fully actuated marine surface vessel. It has been noted that most applications have used the nominal model as a basis for those CNF designs.

Motivation for proposing the study:

By analysing the above-referred works, the study notes the following aspects for the potential research:

1) Currently almost all CNF approaches take the parallel distributed compensation in a single loop. The other control system configurations could provide supplements to enhance CNF configuration and design.

2) Relieve the dependence of model structure and make it generally available for linear/nonlinear models. Model-free/Data driven approaches could be considered.

3) CNF controller performance is challenged in the presence of model uncertainties and external disturbances. More robust approaches could be considered.

4) The nonlinear function of varying damping ratio could be further explored.

5) With the CNF concept, some of other system parameters, such as undamped natural frequency, could be considered in a varying style to change the system performance accordingly.

To deal with the above issues, this study presents a U-model–based control (U-control) system analysis and design procedure. Here is a brief introduction to the approach, the U-model set (Zhang et al., 2020; Zhu and Guo, 2002) is a type of control-oriented equations in crisp relationship of the controller output $u$ and the system output $y$ , where the other terms in classical model sets are absorbed into the associated time-varying parameters. All the other exiting model sets can be converted into U-model realisation (Li et al., 2020), which provides a general control system design platform for a wide range of linear/nonlinear, continuous time/discrete time, polynomial/state space model-based dynamic plants. U-model provides a straightforward approach for nonlinear and dynamic inversion simultaneously. U-control, taking the U-model-based nonlinear dynamic inversion as its base stone, designs the whole control system, favourably achieving linear system performances, separately from plant (model), so that it significantly simplifies the nonlinear control system design procedures. U-control can apply linear control system design methodologies to all U-model described dynamic plant and provide flexible control system configurations (Hussain et al., 2019; Li et al., 2021). The kernel issue in U-control is the robust nonlinear dynamic inversion against model uncertainty and external disturbance. U-control is a relatively new method and is a supplement to many existing approaches in control system design. It should be noted that this is the first study applying U-control for a new configuration and design of the CNF control systems. The U-control technique will be explained in the following sections, where it is needed.

Major contributions of the study in line with the aforementioned analysis are listed below:

1) A U-model-based control framework is introduced to configurate a new CNF control scheme in cascade distributed compensation in double feedback loops. The inner loop for stabilisation and dynamic inversion and the outer loop for system performance specification with variable damping ratio are designed. This makes feasibly separating CNF tracking control into a stabilisation part and a damping tuning part in a cascade (feedback + feedback) structure, which is different from classical CNF parallel (feedback + feedback) distributed controller structure.

2) The U-control–based CNF is a type of universally technique applicable to wide range of uncertain nonlinear systems and is model-free in design/operation. The system stability is characterised in terms of Lyapunov theory and Hurwitz criteria.

3) A new approach for varying damping ratio, directly tuning the damping ratio in a standard second-order linear transfer function. This formulation enables to generate desired/reference state vector as well. This approach also provides a foundation for tuning the other system performance parameters, such as undamped natural frequency.

4) Case studies are presented with computational experiments to demonstrate/explain the U-CNF design procedure.

5) This study presents a fundamental configuration of U-CNF for feasibility study. Surely, it leaves a large space for rigorous theoretical proof and technique development to promote the following studies, which provides enhancement to CNF.

For the remainder of the study, Section 2 prepares the foundation with classical CNF structure formulation and U-control configuration. Section 3 develops the U-CNF design procedure including the formulation stabilisation, nonlinearity and dynamics cancellation and nonlinear damping ratio tuning. Section 4 presents the U-CNF properties and comparison with the other related approaches. Section 5 conducts the bench tests to validate the analytical results with MATLAB/Simulink based simulation demonstrations. Section 6 summarises the study with findings and observations, and gives a view of potential research issues.

2. Preparation

2.1. CNF

Figure 1 shows CNF control schematic configuration. For easy description, but not losing kernel generality, consider a Single-Input and Single-Output (SISO) second-order linear plant model P (Lin et al., 1998) in terms of

\begin{array}{l} P : \dot{x} = A x + B s a t (u) x \in ℝ^{2} \\ y = C x \end{array}

(2.1)

where the operator

s a t (u) \in ℝ \to ℝ

denotes the actuator saturation, and mapping of

s a t (u) = s i g n (u) \min {u_{m a x}, | u |}

, where

u_{m a x}

is the upper/lower threshold. For designing such control systems, it requires the following conditions satisfied with the matrices

A, B, C

1) The controllable pair $(A, B)$ hold;

2) The observable pair $(A, C)$ hold

Figure 1.

Conventional CNF control system.

The formulated linear controller is expressed as

u_{L} = F x + H r

(2.2)

where

(F, H)

are the designed state-feedback controller and feedforward controller, respectively.

F

is deliberately designed, by assigning the system eigenvalues, with low damping ratio.

The formulated nonlinear controller is expressed as

u_{N} = - ρ (x, r) B^{T} P (x - x_{d})

(2.3)

where is locally Lipschitz in

x

P

is a solution from a Lyapunov equation to guarantee the control system asymptotically stable and control input with the saturation limits,

x_{d}

is the desired/reference state vector.

Accordingly, the combination of the composite parallel feedback and feedback controller is determined as

u = u_{L} + u_{N}

(2.4)

Remark 2.1

This is a model-based design approach and nonlinear control which implicitly represents varying damping ratio.

2.2. U-control

Figure 2 shows U-model-based control (U-control) schematic configuration. To explain the U-control, consider the model matched U-control as shown in Figure 2(a) (Zhang et al., 2020; Zhu and Guo, 2002). Let $P$ be a general representation in any model of linear/nonlinear and polynomial/state space equations for the dynamic plants. Assumingly, the set of the plants has the most of properties as those claimed in the other representative works (Isidori et al., 1995),

1) The model inverse $P^{- 1}$ exists;

2) The Lipschitz continuity is satisfied, $P$ and its inverse $P^{- 1}$ are diffeomorphic and globally uniformly Lipschitz in ; that is,

‖ P (x_{1}) - P (x_{2}) ‖ \leq γ_{1} P ‖ x_{1} - x_{2} ‖, \forall x_{1}, x_{2} \in ℝ^{n}

‖ P^{- 1} (x_{1}) - P^{- 2} (x_{2}) ‖ \leq γ_{2} P^{- 1} ‖ x_{1} - x_{2} ‖, \forall x_{1}, x_{2} \in ℝ^{n}

where

x_{1}, x_{2}

are the states while

P

is in form of the state space equation and

γ_{1}

and

γ_{2}

are the Lipschitz coefficients. Here takes SISO (input and output ) prototype for explanation.

Figure 2.

(a) Model matched U-control, (b) Model mis-matched U-control.

The U-control system is functionally expressed as

\sum^{​} = (F, C (C_{1}, P^{- 1}), P) \Leftrightarrow \sum^{​} = (ℱ, C_{1}, I_{i p})

(2.5)

where

ℱ

is for U-control system structure,

C (*)

is the set of designed controllers and

I_{i p} = P^{- 1} P

is a unit constant or identity matrix. Figure 1 shows the model matched U-control system framework (Zhang et al., 2020), in which

C_{1}

is a linear invariant controller to be designed, and

P^{- 1}

is the inverter of the controlled plant

P

to be designed as well. It is noted that the U-control framework is applicable to both linear and nonlinear plants while the dynamic inverse

P^{- 1}

exists.

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

1. Design the dynamic inverter $P^{- 1}$ to achieve $I_{i p} = P^{- 1} P$ . This gives $\sum^{} = (ℱ, C_{1})$ .

2. Design the invariant controller $C_{1}$ under $\sum^{} = (ℱ, C_{1}, I_{i p})$ with a specified closed-loop linear transfer function $G$ .

This can be achieved in a typical formulation of linear control systems. Let the desired closed-loop transfer function $G = ω_{n}^{2} / s^{2} + 2 ζ ω_{n} + ω_{n}^{2}$ , where damping ratio $ζ$ and undamped natural frequency $ω_{n}$ can well specify the system dynamic/steady-state response. As $G = C_{1} / 1 + C_{1}$ , taking the dynamic inversion gives $C_{1} = G / 1 - G$ . Figure 2(b) shows the model mis-matched U-control. The inner loop plays role in robust dynamic inversion.

Remark 2.2

1) Similarly, the U-control objective is to make the system output to track the reference input with assigned performance from the invariant controller $C_{1}$ . (2) Compared with feedback linearization approach, U-control cancels the nonlinearity and dynamic together to give freedom of determining invariant controller independently. (3) U-control is supplement to many classical control system design approaches to enhance the design efficiency. This study expands it with CNF approach.

3. U-CNF control

Consider a general SISO nonlinear system model of

\begin{matrix} {\dot{x}}_{1} = f_{1} (x) \\ {\dot{x}}_{2} = f_{2} (x) \\ \begin{matrix} ⋮ \\ {\dot{x}}_{n} = f_{n} (x, s a t (u)) \\ y = x_{1} \end{matrix} \end{matrix} {\begin{cases} x \in ℝ^{n} \\ y \in ℝ^{1} \\ u \in ℝ^{1} \\ F = {[\begin{matrix} f_{1} & f_{2} & \begin{matrix} \dots & f_{n} \end{matrix} \end{matrix}]}^{T} \in ℝ^{n} \end{cases}

(3.1)

where the operator denotes the actuator saturation, and mapping of

s a t (u) = s i g n (u) \min {u_{m a x}, | u |}

and

u_{m a x}

is the upper/lower threshold.

Control objectives

Figure 3 shows the U-CNF control schematic configuration. The control system design objectives are listed below:

1) The output transient and steady-state performance are specified by a second-order linear dynamic transfer function with a nonlinear time-varying damping ratio and a fixed undamped natural frequency. Accordingly, it achieves smoothly fast/slow responses in different transient phase and with/without overshoot.

2) The system is stabilisable in proper conditions.

3) The system is robust against model bounded uncertainty and disturbance.

4) Because of control input constraint, the global solution of the tracking control problem is not feasible. Thus, a local solution to the tracking control is accommodated.

5) The output tracks a step reference in this study and the framework could be expanded to time-varying references in the follow-up studies.

Figure 3.

U-CNF control system framework.

Solution route

Inspection of Figure 3, there are two separate controller designs within the framework as follows:

1) Nonlinear dynamic inverter ${\hat{P}}^{- 1}$ , this is designed with a Double Sliding Mode Controller (DSMC) in order to achieve the robust Nonlinear Dynamic Inversion (NDI) to make the inner loop into an identity matrix, which is a condition to achieve all aforementioned objectives.

2) Invariant controller $C_{1} (ζ)$ stands for a time-varying second-order linear controller with a nonlinear damping ratio tuning referring to the error and reference amplitude.

3.1. DSMC design procedure for dynamic inversion

This section presents the Sliding Mode Control (SMC) scheme to derive a robust dynamic inverter ${\hat{P}}^{- 1} \in {\hat{I}}_{i p} = K_{i} ({\hat{P}}^{- 1}, P) \Leftrightarrow \dot{x} = {\dot{x}}_{d}, t_{1} \leq t$ . A widely-used SMC approach (Yan et al., 2017) is referred to formulate the dynamic inverter with $u = u_{e q} + u_{s w}$ , which is a prototype of combining equivalent input $u_{e q}$ and switching input $u_{s w}$ . Once the sliding surface is determined, then the Sliding Mode Inverter (SMI) can be obtained in a revised SMC formulation. Accordingly, the study proposes a DSMC method, a global SMC bound with interval $δ$ to drive errors into and remain in the stable interval, a local SMC line to attract the error within the stable interval towards zero exponentially monotonically. Figure 4 (part of the figure from (Yang et al., 2014)) shows the double sliding surface against the conventional one.

Figure 4.

System states (Red for classical SMC and Red+Blue for DSMC).

The DSMC design procedure is explained below:

1) Design a global sliding surface $S_{g}$ to specify the considered system having the desired performance once the system remains on the sliding mode. Realistically a small interval (boundary) $δ$ is assigned for the distance to the classical sliding surface $S$ , so that $S_{g}$ presents a sliding bound surface with thickness $δ$ .

For a nth-order dynamic plant, define a $(n - 1) t h$ order of state tracking error vector

E = x - x_{d} = {[\begin{matrix} e_{1} = x_{1} - x_{d} & \dot{e_{1}} = \dot{x_{1}} - {\dot{x}}_{d} & \begin{matrix} \dots & e_{n - 1}^{(n - 1)} = x_{1}^{(n - 1)} - x_{d}^{(n - 1)} \end{matrix} \end{matrix}]}^{T}

(3.2)

where

x_{1}^{(i)}

and

x_{d}^{(i)}

are the

i t h

-order derivatives of the plant model state

x_{1}

and desired state

x_{d}

, respectively. Then set up a classical sliding surface function

S

(Yan et al., 2017) in the form of

S = c_{1} e_{1} + c_{2} e_{2} + \dots + e_{n - 1}

(3.3)

where the coefficient vector

C = [\begin{matrix} c_{1} & c_{2} & \begin{matrix} \dots & c_{n - 2} \end{matrix} \end{matrix}] \in ℝ_{\geq 0}

is chosen in terms of Hurwitz stable. Then, the global sliding surface with the boundary

δ

in the sliding mode interval is designed as

S_{g} = S + δ_{1}, 0 \leq | δ_{1} | \leq | δ |

(3.4)

2) Design a switching controller $u_{s w}$ to drive the system states to the sliding surface (interval/bound) in the finite time and keep the system state motion on the surface thereafter.

Assign a Lyapunov function $V_{g} = \frac{1}{2} {(S_{g})}^{2} = \frac{1}{2} {(S + δ_{1})}^{2}$ and the corresponding derivative is given by ${\dot{V}}_{g} = \dot{S_{g}} S_{g} = \dot{S} (S + δ_{1})$ . Let $\dot{S} = f_{g} + u_{s w}$ , where $f_{g}$ represents all the neglected bounded terms in classical SMC design.

The derivative of the Lyapunov function gives

\dot{V_{g}} = \dot{S} (S + δ_{1}) = (f_{g} + u_{s w}) (S + δ_{1})

(3.5)

To satisfy ${\dot{V}}_{g} \leq 0$ for stability, choose

u_{s w} = - k_{g} s g n (S + δ_{1})

(3.6)

where

k_{g} \in ℝ_{> 0} > | f_{g} |

is a positive gain of the design choice and

s g n (*)

is the sign function.

3) Design a local sliding surface $S_{l}$ . Then, design an equivalent controller $u_{e q}$ with the condition of driving the system state motion towards $\dot{S_{l}} = 0,$ $S_{l} = 0$ , asymptotically.

For the local sliding surface, assign it as the classical Hurwitz stable manifold of (3.2), that is

S_{l} = S

(3.7)

Assign a Lyapunov function

V_{l} = 1 / 2 {(S)}^{2}

and the corresponding derivative is given by

{\dot{V}}_{l} = \dot{S} S

. Let

\dot{S} = f_{l} + u_{e q} = k_{2} S

(3.8)

where

f_{l}

represents all modelled terms and neglected bounded terms in classical SMC design. Let the matching condition of

f_{l} = k_{2} S

satisfied, where

k_{2}

is a bounded unknown tangent factor of

S

. The geometry of (3.8) is shown in Figure 5. To derive the equivalent controller

u_{e q}

satisfying the Lyapunov stability conditions of

\begin{matrix} V_{l} \geq 0 & {\dot{V}}_{l} \leq 0 \end{matrix}

, expand the derivative of the Lyapunov function as

{\dot{V}}_{l} = \dot{S} S = (f_{l} + u_{e q}) S = (k_{2} S + u_{e q}) S

(3.9)

To satisfy

{\dot{V}}_{l} \leq 0

, choose

u_{e q} = - k_{l} S, k_{l} \in ℝ_{> 0} > | k_{2} |

(3.10)

4) Finally, the DSMC is formulated as $u = u_{e q} + u_{s w}$

Figure 5.

Geometry of local sliding mode.

3.2. Dealing with saturation

In CNF approaches, as plant models are known, the saturation problem can be avoided by solving a Lyapunov equation with a given positive-definite matrix so that the control input staying within the saturation bounds (Lin et al., 1998). As the DSMC is developed in terms of model free (consider all as an uncertain/disturbance term), it is not possible to use the model-based approaches for dealing with saturation. In order to develop a model free/data driven based approach to avoid such control input saturation, it requires some conditions for the formulation.

Consider an assumption (Fulwani and Bandyopadhyay, 2013)

| u_{m a x} | \geq | D |

(3.11)

where

| u_{m a x} |

is the threshold/limit of the control input and

| D |

is the amplitude bound of the disturbance/uncertainty. This assumption means the available control amplitude

\forall x \in \sum^{​}

larger than the maximum amplitude of disturbance/uncertainty. This is necessary to enforce the sliding mode. Accordingly, it has an option to assign

k_{g} = k_{l} = | u_{m a x} |

3.3. Tuning damping ratio

Let the whole control system performance specified by standard second-order transfer function with the varying damping ratio

G = \frac{ω_{n}^{2}}{s^{2} + 2 ζ (r, e) ω_{n} s + ω_{n}^{2}}

(3.12)

To achieve the target by the closed-loop control, take the inversion of $G$ to give the corresponding invariant controller

C_{1} = \frac{G}{1 - G} = \frac{ω_{n}^{2}}{s^{2} + 2 ζ (r, e) ω_{n} s}

(3.13)

where the damping ratio is a monotonic nonlinear bounded function, locally Lipschitz in

y

. This study proposes the damping function as

ζ (r, e) = ζ_{1} e^{(- \frac{a | r - y |}{R})}, R = {\begin{cases} | r | r_{0} \neq y_{0} \\ 1 r_{0} = y_{0} \end{cases}

(3.14)

where

ζ_{1} e^{- a} \geq 0

is the starting damping ratio and

ζ_{1} e^{- 0} = ζ_{1} \geq 0

is the ending damping ratio.

a > 0

is the exponential changing rate for the damping ratio and

r_{0}

and

y_{0}

are the initial reference and output, respectively. Clearly, it is noted

\begin{matrix} ζ_{1} e^{- a} < ζ (r, e) < ζ_{1}, & 0 < \frac{| r - y |}{R} \leq ζ_{1} \end{matrix}

(3.15)

Remark 3.1

For a fixed exponential rate, $ζ_{1} = 1$ generates the fastest response with no overshoot to step reference input.

Remark 3.2

The style of tuning the damping ratio can be applied to tuning any other interested parameters in the system (applicable for higher-order dynamics) characteristic equation, such as the undamped natural frequency, which could be explored in the future studies.

The other role of the invariant controller is to generate the reference/desired state vector for SMC. To make it, convert the controller (3.13) into state-space controllable realisation

\begin{matrix} {\dot{x}}_{d 1} \\ {\dot{x}}_{d 2} \end{matrix} = [\begin{matrix} 0 & 1 \\ 0 & 2 ζ (r, e) ω_{n} \end{matrix}] [\begin{matrix} x_{d 1} \\ x_{d 2} \end{matrix}] + [\begin{matrix} 0 \\ ω_{n}^{2} \end{matrix}] e

(3.16)

Accordingly, the desired states are

\begin{matrix} x_{d 1} = \frac{ω_{n}^{2}}{s + 2 ζ ω_{n}} \frac{1}{s}, & {\dot{x}}_{d 1} = x_{d 2} = \frac{ω_{n}^{2}}{s + 2 ζ ω_{n}} \end{matrix}

4. U-CNF control properties

P1 Theorem 1: Assume the plant is Bounded-Input-Bounded-Output (BIBO) and its inverse exists, the DSMC is globally stable and makes ${\hat{I}}_{i p} = K_{i} ({\hat{P}}^{- 1}, P) \Leftrightarrow \dot{x} = {\dot{x}}_{d}, t_{1} \leq t$ asymptotically.

The DSMC design procedure actually is a process of the proof. The first Lyapunov stability $(\begin{matrix} V_{g} \geq 0, & {\dot{V}}_{g} \leq 0 \end{matrix})$ is used to force the state vector $x$ converge to the sliding mode bound $S_{g} = S + δ_{1}, 0 \leq | δ_{1} | \leq | δ |$ by the switching control. The second Lyapunov stability $(\begin{matrix} V_{l} \geq 0 & {\dot{V}}_{l} \leq 0 \end{matrix})$ is used to force the state vector $x$ in the sliding mode bound converge asymptotically to the final sliding mode line $S_{l} = S + δ_{1} = S, \forall δ_{1} = 0$ by the continuous equivalent control. Therefore, one obtains $\dot{x} - {\dot{x}}_{d} = 0 \Leftrightarrow {\hat{I}}_{i p} = K_{i} ({\hat{P}}^{- 1}, P)$ .

Corollary

If ${\hat{I}}_{i p} = K_{i} ({\hat{P}}^{- 1}, P)$ is asymptotically stable and the linear invariant controller $C_{1}$ is Hurwitz stable, the U-CNF control system is Hurwitz stable. This is because the closed-loop system is $G = C_{1} / 1 + C_{1}$ .

P2 Robustness is related to the two conditions, and , which $| f_{g} |$ and $| k_{2} |$ give the bounds of the robustness in the switching control and equivalent control. respectively, for the whole U-CNF control systems.

P3 Essentially, the DSMC is a prototype of model-free/data-driven SMC to make the NDI in the form of model free. There is no need for plant model structure and parameters under those commonly used assumptions with BIBO, inverse exit and plant dynamic order known and accessible. This type of data driven control dramatically reduces the complexity in dealing with models in SMC system design, for example, those derivations of equivalent control which are effective for general variable structure systems. If the bounds of $| f_{g} |$ and $| k_{2} |$ are unknown in advance, the gains can be conservatively selected large to satisfy the conditions and . To avoid the actuator saturation, assign $k_{g} = k_{l} = | u_{m a x} |$ .

P4 Selecting the sliding surface with the best-conditioned linear dynamics (Slotine and Li, 1991) can be achieved with the U-CNF control framework, by assigning varying damping ratio and undamped natural frequency within an external feedback loop.

P5 Comparison in the configuration, the CNF approaches take structures in feedback +feedback in parallel, plant model-based design, the U-CNF takes up feedback + feedback in cascade and plant model-free design.

P6 Comparison with classical dynamic inversion, U-dynamic inversion cancels both nonlinearities and dynamics. In contrast, feedback linearization approach removes the nonlinearity first and then designs the coordinate converted linear dynamic system (Slotine and Li, 1991). The U-control designs the control system independently from the U-dynamic inversion.

P7 In comparison with the classical SMC, the DSMC is (1) no need to use the nominal model to design $u_{e q}$ ; (2) To remove chattering, the local sliding mode $S_{l}$ shares the properties with low-pass filters (Slotine and Li, 1991); (3) Regarding the classical SMC, in general, small sliding layer $δ$ is required for better control accuracy, but stimulates chattering easily. Vice versa large $δ$ reduces the chattering, but increases steady-state errors possibly. The other factor is the gain associated with the switching control. Larger for more control accuracy and faster to sliding surface, but its amplitude is limited by saturation of system input, particularly in applications. Both are normally selected by trial and error. For the DSMC, both factors are also vital for the control system performance, but more relaxed in selection than those with SMC.

P8 Comparison with the approach of SMC Nonlinear Dynamic Inversion (SMCNDI) (Yang et al., 2014), the DSMC includes the SMCNDI as its special case while using nominal model to design equivalent control.

P9 In comparison with High-Gain Control (HGC) (Lin, 2009), the DSMC includes HGC as it is a special case while assigning the sliding surface boundary $| δ | > \max (| u |, | y |)$ and $k_{l} ≫ 0$ . The DSMC can achieve the similar results, but using lower gain of $k_{l}$ in conjunction with $k_{g}$ .

P10 The U-CNF control can be used to facilitate the requirement on the Time Scale Separation (TSS) (Chakrabortty and Arcak, 2009) in the control system design because the framework separates the designs of the inner control loop (dynamic inverter) and the outer control loop (invariant controller).

P11 The U-CNF control could be expanded to fractional-order systems with reference to the stability analysis results (Makhlouf, 2018; Jmal et al., 2020; Naifar et al., 2020; Naifar and Ben Makhlouf, 2021).

P12 A comparison table (i.e., Table 1)

Table 1.

Comparison of the main characteristics of the parallel and serial CNF approaches.

CNF	Characteristics
CNF	Parallel (collection of the exiting approaches)	Serial (U-control approach at the moment)
Dealing with NL models	Challenging	General and concise as the pre-processing of the cancellation of nonlinearity and dynamics. Similar degree of work as a basic linear control system design
Tuning damping ratio	Indirect, by solving Lyapunov equation or similar equalities	Directly tuning damping ration $ζ$ from a linear invariant controller
Model dependency	Most need nominal models at least	Model free except dynamic order
Robustness	Various approaches	SMC at the moment

lists the main characteristics of the parallel and serial CNF approaches.

5. Case studies

Two case studies are selected for demonstrating the validity and comparisons of the derived procedure and block function connections of the control system configuration. The second purpose of the demonstrations is to show the design procedures and knowhow for reference of potential applications and academic research.

Plants: The two exemplary SISO plants, expressed in the model of $\begin{matrix} \dot{x} = F (x, s a t (u)) \\ y = h (x) + d (t) \end{matrix}$ , are (1) both second-order $(x = {[\begin{matrix} x_{1} & x_{2} \end{matrix}]}^{T})$ dynamics with an external disturbance, $d (t) = 0.5$ for example 1 and $d (t) = 0$ for example 2, added at the outputs, (2) the dynamic order = 2 known and all the state variables accessible for control system design, (3) no pure time delay, (4) the actuator saturation thresholds $| u_{m a x} | = 2$ for example 1 and $| u_{m a x} | = 1$ for example 2.

Comparison system with fixed damping ratio: This transfer function is assigned as a second-order decayed oscillatory response with damping ratio of 0.6 and undamped natural frequency of 2, and the system output has no steady-state error to the step reference.

U-CNF control: Figure 6 shows the control system structure to facilitate the explanation of the design procedure. For the two case studies, the design takes two separate tasks in common, 1) designing the SMI with the inner loop to achieve ${\hat{I}}_{i p} = K_{i} ({\hat{P}}^{- 1}, P)$ , and 2) designing the nonlinear damping controller with the external loop to achieve the whole system performance.

Figure 6.

U-CNF control system.

Design of SMI: Define a tracking error vector as

E = x - x_{d} = {[\begin{matrix} e_{1} = x_{1} - x_{d} & {\dot{e}}_{1} = \dot{x_{1}} - {\dot{x}}_{d} \end{matrix}]}^{T} = {[\begin{matrix} e_{1} = x_{1} - x_{d 1} & {\dot{e}}_{1} = x_{2} - x_{d 2} \end{matrix}]}^{T}

(5.1)

Then, set up a classical sliding surface function

S

S = c e_{1} + {\dot{e}}_{1}

(5.2)

where the coefficients

c = 25, 50

for the cases 1 and 2 are chosen in terms of Hurwitz stability, and a fast inner-loop convergent speed, respectively.

With reference to Section 3, of the MSI having $u_{s w} = - k_{g} s g n (S + δ_{1})$ and $u_{e q} = - k_{l} S$ , for the simulation tests, assign $k_{g} = k_{l} = | u_{m a x} |$ to avoid actuator saturation while keeping the SMC feasible. Consequently, the integrated controller of DSMC system is formulated as $u = u_{e q} + u_{s w} = - | u_{m a x} | s g n (S + δ_{1}) - | u_{m a x} | S$ . Assign the sliding band boundary thickness $| δ | = 0.8$ .

Design of the nonlinear damping control: Specify the whole desired control system performance with a linear second-order transfer function $G = ω_{n}^{2} / s^{2} + 2 ζ (r, e) ω_{n} s + ω_{n}^{2}$ and accordingly the closed-loop implementation gives rise to the damping controller $C_{1} (ζ) = G / 1 - G = ω_{n}^{2} / s^{2} + 2 ζ (r, e) ω_{n} s$ . The desired states are also generated with $\begin{matrix} x_{d 1} = ω_{n}^{2} / s + 2 ζ ω_{n} 1 / s, & {\dot{x}}_{d 1} = x_{d 2} = ω_{n}^{2} / s + 2 ζ ω_{n} \end{matrix}$ . The specific selection of $ζ_{1}$ and $a$ for the damping function of $ζ (r, e) = ζ_{1} e^{(- \frac{a | r - y |}{R})}$ will be explained in the following case study sections:

Case 1

Nonaffine nonlinear plant

This is a simulated model to represent a class of nonaffine nonlinear systems.

Plant model

\begin{matrix} \dot{x_{1}} = x_{2} \\ \dot{x_{2}} = - 0.6 x_{2} - x_{1} x_{2} - u x_{2} + \sin (u) + 2 u + u^{3} \\ y = x_{1} + d (t) \end{matrix}

(5.3)

where the external disturbances,

d (t) = 0

and

d (t) = 0.5

added at the outputs, are used for the tests.

U-CNF control

1) Design of SMI: Completed at the beginning of the section, which is applicable for both case studies.

2) Design of the damping ratio controller: Generally completed at the beginning of the section. For setting up the nonlinear damping function, assign the ratio stating from $ζ = 0.55$ , that is, assign the exponential damping rate $- a = \ln (0.55) = - 0.5978$ , and ending at $ζ_{1} = 1$ . Take the same undamped natural frequency $ω_{n} = 2$ as that from the comparison system.

Simulations

Figure 7 is a plot package to show the system performance to a sequence of step references without external disturbance. Figure 8 shows a set of plots for the system performance with a constant external disturbance of amplitude of 0.5 throughout the simulation.

Figure 7.

System 1 performance to step references without disturbance, (a) Output response, (b) Control input, (c) Tracking error, (d) Damping ratio, (e) Sliding mode.

Figure 8.

System 1 performance to step references with disturbance, (a) Output response, (b) Control input, (c) System tracking error, (d) Damping ratio, (e) Sliding mode.

Discussions on the simulated plots

1) Dynamic and static performance are well obtained in accordance of the designed framework.

2) Damping ratio is tuned from the assigned starting damping, change rate to the end damping, which achieves the CNF merits. Further the nonlinear function explicitly indicates the damping ratio variation instead of implicitly tuning the damping in CNF approaches.

3) The control inputs are well maintained within threshold of the actuator saturation.

4) DSMC cancels both nonlinearities and dynamics to provide a fast control of the inner loop into $\dot{x} = {\dot{x}}_{d}$ .

5) In case of the unknown external constant disturbance $(d (t) = 0.5)$ , the U-CNF control had initial regulation before 2.5 times from damping, and correspondingly the control input had large corrections.

6) The U-CNF control design did not use nominal model except the assumption of the state vector measurable for control.

Case 2

Pitch control of F16 aircraft

To further demonstrate the characteristics and performance of the proposed U-CNF approach, the first example, in CNF foundation study (Lin et al., 1998), was selected to re-test. This case is backgrounded with an F-16 aircraft in flight at altitude of 20000 feet and a Mach number of 0.9.

Plant model

A longitudinal short period dynamic plant of the aircraft is modelled (Lin et al., 1998) below

[\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}] = [\begin{matrix} 0 & 1.0 \\ 0.373 & - 2.184 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] + [\begin{matrix} 0 \\ 1 \end{matrix}] u

y = [\begin{matrix} 32.8781 & 24.3282 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}]

(5.4)

The output is the pitch rate. The control input is the elevator deflection with a unit actuator saturation limit, which is normalized from a physical manoeuvre actuator of

\pm 12^{o}

in deflection. It should be noted that this is an open-loop unstable dynamic, as one of the poles

(\begin{matrix} s_{1} = 2.3432 & s_{2} = - 0.1592 \end{matrix})

is on the right half S (Laplace) plane.

U-CNF control

The design is the same as that presented in Case 1.

Simulations

Figure 9 is a plot package to show the system performance to a sequence of step references.

Figure 9.

System 2 performance to step response without disturbance, (a) Output response, (b) Control input, (c) tracking error, (d) Damping ratio, (e) Sliding mode.

Discussions on the simulated plots

1) These plots confirm that the generality of design is independent of plant models, which shows output responses are the same with once of design. Even plant models are changed, the same output response can still be achieved.

2) The same output response is generated as that presented from the original study (Lin et al., 1998). However, the U-CNF control design does not require nominal model and its damping is explicitly tuned with the damping ratio.

3) The DSMC contributes both stabilisation and dynamic cancellation, which the resultant identity matrix makes the nonlinear damping tuning significantly effective.

4) The assumption of state vector measurable can be relieved by using state observers, which will be explored in future research.

5) The case study provides a transparent illustration for follow-up applications.

5. Conclusions

To the author’s best knowledge, this is the first different configuration of CNF approach, which is a supplement to enhance the CNF research and applications with some distinct merits. In research techniques: Compared with the two popular control systems design frameworks, model-based and model-free. Most of the CNF approaches represent the model-based development, and the U-CNF approach represents model-free development. The kernel contribution of the U-CNF is bringing forward a new CNF configuration to integrate the robust nonlinear dynamic inversion and damping ratio tuning. Regarding the follow-up studies, even without requiring plant nominal model in design, this study has assumed the plant state vector measurable. To progress, the state observer-based U-CNF approaches should be studied. The other potential research is the U-CNF control of the time-delayed systems, which is a practically challenging topic. This first study has paid more attention to the configuration and feasibility establishment, surely rigour theoretical studies are required to prove the associated properties. In the hope, this study could promote wide interest in applications.

Footnotes

Acknowledgements

The authors would like to expresses their gratitude to the editors and the anonymous reviewers for their helpful comments and constructive suggestions with regard to the revision of the paper.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Data availability statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

ORCID iD

Saleh Mobayen

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