Conditional funnel control: A simple approach to prescribed performance asymptotic tracking and its industrial applications

Abstract

Funnel control is remarkably a low-complexity control technique which ensures a model-free tracking of reference signal for nonlinear systems. It also ensures a prescribed transient behaviour of error signal; however, it only guarantees the error signal to remain within or close to the funnel boundary rather than to converge to zero. Therefore, for ensuring the asymptotic convergence of the error to zero, in this paper, we introduce the concept of conditional error in funnel control without any complex condition or requirement(s). Thus, we first introduce the idea of conditional funnel control for relative degree one systems and then, we extend this idea for arbitrary relative degree nonlinear systems. Mathematical analyses are provided to ensure the boundedness of the tracking error within the funnel boundary as well as the occurrence of asymptotic convergence. For validating the efficacy of the proposed scheme, we provide numerical examples to compare the numerical results with the recently proposed schemes. Finally, we demonstrate the industrial applications (through simulation case-studies) of the proposed method for a Continuous Stirred Tank Reactor (CSTR), Bioreactor and Flexible Joint Manipulator.

Keywords

Asymptotic tracking model-free control funnel control conditional error CSTR bioreactor flexible joint manipulator

Introduction

Tracking of reference signals is one of the main objectives when designing a control system, where characteristics such as prescribed transient behaviour along with asymptotic tracking and steady-state accuracy are desired. For instance, spacecraft rendezvous and docking, interception of missiles, assembling of machine parts and multi-agent system formation are among the few applications where such control scheme is crucial (Song et al., 2017; Zhao et al., 2022). For a relative degree one and minimum phase system, simple high-gain controllers such as $u = - ke$ can attain the asymptotic tracking, where k is the high-gain of the controllers and e is the tracking error. Instead of using a constant value, adaptive rules can be used for obtaining the value of the gain k. One such simple rule is $\overset{\cdot}{k} = e^{2} (t)$ ; however, this adaptive rule results in an increase of the magnitude of the gain unboundedly (Chowdhury and Khalil, 2019). This drawback is addressed in λ-tracking control algorithm (Ilchmann and Ryan, 1994). But, the λ-tracking controller does not give asymptotic tracking. In addition, λ-tracking has two main drawbacks: (1) the transient behaviour cannot be directly controlled, and thus, no guarantee of the transient behaviour is ensured and (2) the value of the gain increases monotonically; thus, there is a large value of gain even for error values. Both of these drawbacks of the λ-tracking controller are solved by the Funnel Control (Lee and Trenn, 2019) which has been proposed by Ilchmann et al. (2002).

Funnel control provides a simple approach for a model free prescribed performance tracking of relative degree one, positive high-frequency gain and minimum phase nonlinear systems (Ilchmann et al., 2002; Lee and Trenn, 2019). It relies on the principle of a high-gain feedback control and can be viewed as an advancement to adaptive high-gain control, in which the monotonic adaptive gain is replaced by a time varying function known as the funnel function which increases the value of the gain only when error approaches the funnel boundary. Thus, it ensures the attainment of two main objectives: (1) the error evolves with in a performance funnel through which the transient behaviour can be controlled and (2) the error converge to a strip of $[- ψ, ψ]$ (rather than to zero) in finite time where $ψ > 0$ and remains within that strip thereafter. Due to these benefits, funnel control has been utilized in many applications such as linear electrical circuits (Berger and Reis, 2014), industrial servo systems (Hackl, 2017), cruise control (Berger and Rauert, 2020) and under-actuated multi-body systems (Berger et al., 2021).

The basic funnel control algorithm has been expanded in many different directions; for instance, bang-bang funnel control (Liberzon and Trenn, 2013), funnel control of differential-algebraic systems (Berger et al., 2012), model predictive-based funnel control (Berger et al., 2024a), zeroing barrier function based funnel control (Verginis, 2023), reinforcement learning based funnel control (Elhaki, 2023), etc. In these variations, only the ultimate-uniform boundedness of the tracking error can be achieved (Zhao et al., 2022). However, asymptotic tracking of funnel control has been solved in Ilchmann and Ryan (2006) but only for linear systems. Solutions to this problem have also been proposed in Lee and Trenn (2019) and Trenn (2019). However, all these methods are based on the idea of the shrinkage of the funnel boundary which amplifies the error signal as time approaches to infinity. This in turn unnecessarily amplifies the noises in the measurement signal (Berger et al., 2024b). Another approach for asymptotic tracking in Zhao et al. (2022) uses an adaptive algorithm, but it requires the parameterized form of the system as well as certain prior knowledge of the system dynamics and is complicated from the implementation point of view. Recently, an internal model principle-based asymptotic tracking of the funnel control is proposed in Berger et al. (2024b); however, this technique depends on the internal model approach and is only applicable to linear systems.

Motivated by the discussions above, this paper focuses on attaining an asymptotic convergence of the tracking error through funnel control while avoiding the drawbacks such as shrinkage of the funnel boundary to zero (which is required in Lee and Trenn, 2019 and Trenn, 2019), prior knowledge of the system dynamics (which is required in Zhao et al., 2022) and complexity of the algorithm. To achieve these objectives, we present a simple algorithm that introduces a concept of ‘conditional error’ and thereby ‘conditional funnel gain’ in the classical funnel control technique to obtain a Conditional Funnel Control (CFC) algorithm which can be used to control the relative degree one nonlinear systems. The idea is to introduce an integral action only when the trajectory reaches the boundary $[- μ, μ]$ , where $μ > 0$ . This conditional integral action causes the trajectory to approach zero asymptotically within a funnel boundary; hence, this technique neither shrinks the funnel boundary nor degrades the prescribed transient performance. The analysis of the proposed scheme is also presented. Since, the proposed CFC is limited for relative degree one nonlinear systems, we extend the theoretical results by applying the proposed CFC technique to modify the virtual output–based funnel control algorithm of Chowdhury and Khalil (2019); thus, introducing a virtual output–based CFC algorithm for arbitrary relative degree nonlinear systems. For comparative numerical analysis, we simulate numerical examples and compare the results of the proposed algorithm with that of existing techniques. For demonstrating the effectiveness of the proposed and modified algorithms, we present their industrial applications to control four benchmark processes: (1) Temperature Control of a Continuous Stirred Tank Reactor (CSTR); (2) Product Concentration Control in a CSTR; (3) Biomass Control in a Bioreactor and (4) Position Control of a Flexible Joint Manipulator, which validate the efficacy of the proposed schemes. Thus, the main contributions of this paper are:

A novel conditional error-based funnel control algorithm is proposed for industrial process of relative degree one systems.

The proposed method achieves the asymptotic convergence of error to zero and is simple from implementation point of view.

The extension of the proposed algorithm is also proposed for industrial processes of arbitrary relative degree systems.

A detailed closed-loop stability analysis has been provided.

The application of the proposed algorithms has been demonstrated for industrial processes such as: (a) CSTRs, (b) Bioreactors and (c) Flexible Joint Manipulator.

The rest of this paper is structured as follows: in section ‘Notations and preliminaries’, notations and preliminaries for the paper are defined; in section ‘Funnel control’, funnel control is briefly explained; in section ‘Main results’, the main results of the paper are presented; in section ‘Extension of the main results,’ the main results are extended for arbitrary relative degree nonlinear systems; in section ‘Examples and case studies’, numerical examples and case studies are discussed; and finally, in section ‘Conclusion’, conclusions of the paper are presented.

Notations and preliminaries

In this paper, the following standard notations are used: $R$ denotes a set of real numbers; $R_{+}$ denotes a set of positive real numbers; $R_{\geq 0}$ denotes a set of real numbers greater than or equal to 0; $W^{1, \infty} (R_{\geq 0}, R_{+})$ represents the class of bounded functions with bounded derivatives; sup (.) and inf (.) respectively represents the supremum and inferimum of their arguments; t denotes the time in seconds; ‖ . ‖ and | . | respectively defines the norm and absolute value; $sat (.)$ represents a saturation function.

Definition 1: A function $α (w)$ , defined for $w \in [0, b)$ , belongs to a class $K$ if: (1) it is continuous, (2) it is strictly increasing and (3) $α (0)$ = 0 (Khalil, 2002).

Consider the following class of nonlinear systems:

\begin{matrix} \overset{\cdot}{η} (t) = \bar{h} (η, y) \\ {\overset{\cdot}{x}}_{i} (t) = x_{i + 1} (t), 1 \leq i \leq ρ - 1 \\ {\overset{\cdot}{x}}_{ρ} (t) = \bar{f} (η, x) + \bar{g} (η, x) u (t) \\ y (t) = x_{1} (t) \end{matrix}

(1)

where $η \in R^{n - ρ}$ is the internal dynamics, $x = {[\begin{matrix} x_{1} & x_{2} & \dots & x_{ρ} \end{matrix}]}^{T} \in R^{ρ}$ is the state of the system, $\bar{f} (η, x) \in R$ , $\bar{g} (η, x) \in R$ and $\bar{h} (η, y) \in R$ are the dynamic functions and $u \in R$ is the input and y is the output of the system. By transforming the system dynamics into error coordinates, we get,

\begin{matrix} \overset{\cdot}{\tilde{η}} (t) = h (\tilde{η}, e_{1} + r) \\ {\overset{\cdot}{e}}_{i} (t) = e_{i + 1} (t), 1 \leq i \leq ρ - 1 \\ {\overset{\cdot}{e}}_{ρ} (t) = f (\tilde{η}, x) + g (\tilde{η}, x) u (t) - r^{(ρ)} (t) \end{matrix}

(2)

where $\tilde{η} = η - η_{0}$ , $η_{0}$ is the equilibrium point of η, $e_{1} = y - r$ is the error signal and r is a reference signal, $x = e + \vec{r}$ where $e = {[\begin{matrix} e_{1} & e_{2} \dots & e_{ρ} \end{matrix}]}^{T}$ , $\vec{r} = {[\begin{matrix} r & r^{(1)} & \dots & r^{(ρ)} \end{matrix}]}^{T}$ , $f (\tilde{η}, x) \in R$ , $g (\tilde{η}, x) \in R$ and $h (\tilde{η}, y) \in R$ are respectively the resulting transformed functions from $\bar{f} (η, x)$ , $\bar{g} (η, x)$ and $\bar{h} (η, y)$ . The following assumptions have been made on the system.

Assumption 1: The internal dynamics of the system is stable, that is, the system is minimum phase. Thus, there exist a continuously differentiable function $V_{η} (η)$ and class $K$ functions $α_{1}$ , $α_{2}$ , $α_{3}$ and γ such that,

α_{1} (\tilde{η}) \leq V_{η} (\tilde{η}) \leq α_{2} (\tilde{η})

\frac{\partial V_{η}}{\partial t} + \frac{\partial V_{η}}{\partial η} h (\tilde{η}, e_{1} + r) \leq - α_{3} (| | \tilde{η} | |) \forall | | \tilde{η} | | \geq γ (| e_{1} + r |)

It may be noted in Assumption 1 that $e_{1} + r$ is the driving input, and thus, with $e_{1} + r = 0$ , $\tilde{η} = 0$ is an asymptotically stable equilibrium point.

Assumption 2: The functions $\bar{f} (.)$ , $f (.)$ , $\bar{g} (.)$ , $g (.)$ , $\bar{h} (.)$ and $h (.)$ are Lipschitz in their arguments. It is also assumed that there exists a unique equilibrium point $x = \bar{x}$ , $y = r_{ss}$ and $η = \bar{η}$ such that $\bar{f} (\bar{η}, \bar{x}) + \bar{g} (\bar{η}, \bar{x}) \bar{u} = 0$ and $\bar{h} (\bar{η}, r_{ss}) = 0$ , where $r_{ss}$ is defined in Assumption 3.

Assumption 3: The reference signal and its derivative are bounded, that is, $r^{(i)} \leq c_{ri} > 0$ for $i = 1 to ρ - 1$ . Furthermore, it is assumed that $lim_{t \to \infty} \vec{r} (t) = {\vec{r}}_{ss} = {[\begin{matrix} r_{ss} & 0 & 0 & \dots & 0 \end{matrix}]}^{T}$ .

Assumption 4: For all values of $\tilde{η}$ and x, $b \geq g (\tilde{η}, x) \geq a$ , where $a > 0$ and $b > 0$ are the constants.

Assumption 4 is a standard assumption in the theory of funnel control which facilitates the proofs of stability and convergence of the closed-loop system.

Funnel control

The idea behind funnel control is to increase the feedback gain as the error reaches the prescribed boundary which is known as the funnel boundary (Ilchmann et al., 2002). Let ϕ be a function such that (Chowdhury and Khalil, 2019):

ϕ : = {φ ϵ W^{1, \infty} (R_{\geq 0}, R_{+}) | \forall τ \geq 0 : φ (τ) > 0 and lim_{τ \to \infty} inf φ (τ) > 0}

For a relative degree one system, the objective of funnel control is to make the error signal $e_{1}$ attains a prescribed performance defined by:

F_{φ} : = {(t, e) \in R_{\geq 0} \times R | φ (t) | e_{1} | < 1}

The funnel boundary is defined as $ψ (t) = 1 / φ (t)$ and the funnel control is defined as:

u (t) = - k_{1} (t) e_{1} (t)

(3)

where $k_{1}$ is selected as:

k_{1} (t) = \frac{1}{ψ (t) - | e_{1} (t) |}

(4)

Remark 1: From the definition of ϕ, it is clear that φ and $\overset{\cdot}{φ}$ are bounded. Furthermore, the funnel remains positive for all time, that is, $ψ (t) > 0 \forall t \geq 0$ .

Remark 2: From Remark 1 and Assumption 4, we deduce that $\frac{d}{dt} {ψ (t) |}_{0}^{\infty}$ is bounded which implies that the funnel boundary $ψ (t)$ is globally Lipschitz with $L > 0$ being a Lipschitz constant, that is, for a time interval $[t_{0}, t_{1}]$ ,

| ψ (t_{1}) - ψ (t_{0}) | \leq L (t_{1} - t_{0})

Main results

In this section, we first present the Conditional Funnel Control for relative degree $ρ = 1$ nonlinear systems and then we present the stability analysis of the closed-loop system.

Conditional Funnel Control (CFC)

To achieve the asymptotic convergence of the error signal to zero without compromising the transient behaviour, we first define a conditional error as:

e_{c} (t) = e_{1} (t) + l σ (t)

(5)

where $σ (t)$ is defined as follows (Seshagiri and Khalil, 2005):

\overset{\cdot}{σ} (t) = - l σ (t) + μ sat (\frac{e_{c}}{μ})

(6)

where l and μ are the positive constants, and $σ (0) = 0$ . We then modify the funnel control defined in equation (3) to be:

u (t) = - k_{c} (t) e_{c} (t)

(7)

and the feedback gain $k_{1} (t)$ in (4) to be Conditional Funnel Gain, $k_{c} (t)$ :

k_{c} (t) = \frac{1}{ψ (t) - | e_{c} (t) |}

(8)

Since the initial value of the integrator is selected by the designer and the integral action begins when the error approaches the boundary $[- μ, μ], σ (0) = 0$ is plausible as it ensures that the tracking error will start within the funnel region. Thus, this facilitates the stability analysis of the closed-loop system. Notice in the conditional funnel gain given in equation (8) that the objective of the CFC is to force the conditional error ( $e_{c}$ ) to stay within the funnel region which is in contrast to funnel gain given in equation (4) where the objective of the funnel control is to force the error ( $e_{1}$ ) to stay within funnel region. It may also be noticed in equation (6) that the integral action begins when $| e_{c} |$ approaches μ; thus, it does not degrade the prescribed transient performance of the error. The overall scheme is given in Figure 1.

Figure 1.

Block diagram of the proposed control scheme.

Remark 3 (Asymptotic Funnel Control based on the shrinkage of the funnel boundary): For relative degree one systems, the authors in Lee and Trenn (2019) and Trenn (2019) introduce the following asymptotic-funnel control method based on the shrinkage of the funnel boundary to zero:

\begin{matrix} u = \frac{η (t)}{1 - | η (t) |} \\ η (t) = ω (t) [y - r] \\ ω (t) = \frac{1}{ψ (t)} \end{matrix}}

(9)

The algorithm presented in equation (9) works by shrinking the funnel boundary to zero as time approaches to infinity, that is, $ψ (t) \to 0$ as $t \to \infty$ . Thus, as $ψ (t) \to 0$ , $ω (t) \to \infty$ which in turn amplifies the noises present in the output measurement signal y. In comparison with the CFC given in equations (5)–(7), there is no such amplification of noise signals.

Remark 4 (Adaptive Asymptotic Funnel Control): Consider a relative degree one system that can be represented in parameterized form: ${\overset{\cdot}{x}}_{1} = {\hat{θ}}_{1}^{T} {\hat{ϕ}}_{1} (x_{1}, t) + {\hat{g}}_{1} u$ , where ${\hat{θ}}_{1}$ and ${\hat{ϕ}}_{1}$ are respectively the vectors of unknown parameters and known smooth functions and ${\hat{g}}_{1}$ is a control coefficient. For the system being considered, the authors in Zhao et al. (2022) introduce the following adaptive asymptotic-funnel control:

\begin{matrix} u = \frac{1}{{\bar{μ}}_{1}} [- \hat{c} s (t) - {\hat{μ}}_{2}] - {\hat{θ}}_{1} (t) {\hat{ϕ}}_{1} (x_{1}, t) + \overset{\cdot}{r} \\ \overset{\cdot}{\hat{θ}} (t) = s (t) {\hat{μ}}_{1} \hat{Γ} {\hat{ϕ}}_{1} (t) \\ s (t) = \frac{\hat{ζ} (t)}{1 - {\hat{ζ}}^{2} (t)} \\ \hat{ζ} (t) = β (t) \hat{η} (e_{1}) \\ \hat{η} (e_{1}) = \frac{e_{1}}{\sqrt{e_{1}^{2} + \hat{l}}} \\ {\hat{μ}}_{1} = \hat{μ} β (t) \hat{ρ} (t) \\ {\hat{μ}}_{2} = \hat{μ} \overset{\cdot}{β} (t) \hat{η} (e_{1}) \\ \hat{μ} = \frac{1 + {\hat{ζ}}^{2} (t)}{{[1 - {\hat{ζ}}^{2} (t)]}^{2}} \\ \hat{ρ} (t) = \frac{l}{{({e_{1}}^{2} + \hat{l})}^{\frac{3}{2}}} \\ e_{1} = x_{1} - r \end{matrix}}

(10)

where $ψ (t) = \frac{\sqrt{\hat{l}} β (t)}{\sqrt{1 - β^{2} (t)}}$ , $β (t)$ is a time varying function and $\hat{c}$ , $\hat{Γ}$ and $\hat{l}$ are the design constants. Notice that the algorithm requires prior knowledge of the system dynamical function ${\hat{ϕ}}_{1} (x_{1}, t)$ and it can only be applied to the systems that can be represented in parameterized form. Furthermore, the complexity of the algorithm makes it difficult to implement in comparison with the proposed algorithm given in equations (5)–(8).

Stability analysis

For the closed-loop analysis of the proposed scheme, we first present Lemma 1 to show that the dynamics of the conditional integrator, and hence, the internal dynamics are bounded. We then provide Proposition 1 which extends the discussions of Lemma 1. Similar to (Chowdhury and Khalil, 2019), with the help of Lemma 1 and Proposition 1, we present Theorem 1 to show that: (1) the conditional error evolves within the funnel boundary and remains within it for all the future time and (2) the conditional funnel gain $k_{c} (t)$ is bounded. We then extend this discussion through Corollary 1 to show that the error signal also evolves within the funnel region and remains within it for all the future time. We finally present Theorem 2 to discuss that the conditional error approaches the boundary $[- μ, μ]$ where the integrator becomes active which brings the error to zero as time tends to infinity. Now, by using equations (6) and (7), the closed-loop system (2) for $ρ = 1$ can now be represented as:

\begin{matrix} \overset{\cdot}{\tilde{η}} = h (\tilde{η}, e_{1} + r) \\ {\overset{\cdot}{e}}_{c} (t) = f (\tilde{η}, e_{1} + r) - g (\tilde{η}, e_{1} + r) k_{c} e_{c} (t) - \overset{\cdot}{r} - l^{2} σ (t) \\ + l μ sat (\frac{e_{c}}{μ}) \\ \overset{\cdot}{σ} (t) = - l σ + μ sat (\frac{e_{c}}{μ}) \end{matrix}

(11)

Since the funnel function $ψ (t)$ is finite, we can define $λ_{f} = sup_{t \in [0, \infty)} ψ (t) > 0$ . Furthermore, define $ε = inf_{t \in [0, \infty)} ψ (t)$ .

Lemma 1: For $μ \leq | e_{c} | \leq λ_{f}$ , there exists $c_{0} > 0$ such that the sets ${V_{σ} \leq c_{0} {(\frac{μ}{l})}^{2}}$ and ${V_{η} \leq α_{4} (λ_{f} + μ + c_{r 1})}$ remain positively invariant, where $α_{4} (.)$ is a class $K$ function such that $α_{4} = α_{2} ° γ$ , $V_{σ}$ is defined as $V_{σ} = \frac{1}{2} σ^{2}$ and $V_{η}$ is defined in Assumption 1.

Proof: By taking the time-derivative of $V_{σ}$ , we get,

{\overset{\cdot}{V}}_{σ} = σ [- l σ + μ sat (\frac{e_{c}}{μ})]

(12)

Equation (12) can be represented as:

{\overset{\cdot}{V}}_{σ} \leq - l (1 - θ) {| σ |}^{2}, \forall | σ | \geq \frac{μ}{l θ} sat (\frac{e_{c}}{μ})

(13)

where $0 < θ < 1$ . Thus, we conclude from inequality (13) that the set ${V_{σ} \leq c_{0} {(\frac{μ}{l})}^{2}}$ is positively invariant. Also, from Assumption 1, ${\overset{\cdot}{V}}_{η} \leq 0, \forall | | \tilde{η} | | \geq γ (λ_{f} + μ + c_{r 1})$ . Therefore, ${V_{η} \leq α_{4} (λ_{f} + μ + c_{r 1})}$ is also a positively invariant set. This completes the proof. □

Define the following compact sets:

\begin{matrix} Ω_{f} = {V_{η} \leq α_{4} (λ_{f} + μ + c_{r 1})} \times {V_{σ} \leq c_{0} {(\frac{μ}{l})}^{2}} \\ \times {| e_{c} | \leq λ_{f}} \\ Ω_{0} = {V_{η} \leq α_{4} (λ_{0} + c_{r 1})} \times {V_{σ} = 0} \times {| e_{c} | \leq λ_{0}} \end{matrix}

where $0 < λ_{0} < ψ (0)$ , and thus, $Ω_{0} \subset Ω_{f}$ .

Proposition 1: Since $λ_{0} < ψ (0)$ , for $(\tilde{η} (0), σ (0), e_{c} (0)) \in Ω_{0}$ , there is a strict positive distance between $e_{c} (0)$ and the funnel boundary at $t = 0$ . Thus, we are able to find a constant $d_{c} > 0$ such that $ψ (0) - | e_{c} (0) | > d_{c}$ . Since, $σ (0) = 0$ , we have $ψ (0) - | e_{1} (0) | > d_{c}$ which shows that at $t = 0$ , there is also a strict positive distance between the error $e_{1} (t)$ and the funnel boundary $ψ (t)$ . □

Remark 5: From Assumption 3, the reference signal and its derivative are bounded. Therefore, for all $(\tilde{η} (t), σ (t), e_{c} (t)) \in Ω_{f}$ , there exists $χ > 0$ such that $| f (\tilde{η}, e_{1} + r) - \overset{\cdot}{r} - l^{2} σ (t) + l μ sat (\frac{e_{c}}{μ}) | \leq χ$ .

Theorem 1: Consider a closed-loop system (9). Suppose Assumptions 1–4 are satisfied, $φ (t) \in ϕ$ and $(\tilde{η} (0), σ (0), e_{c} (0)) \in Ω_{0}$ , then by Remark 1, there exists $κ_{f} > 0$ such that $| e_{c} | \leq ψ (t) - κ_{f}$ which ensures that the conditional error $e_{c}$ evolves within the funnel region and remains in it for all $t \in [0, \infty)$ . Furthermore, the conditional funnel gain $k_{c} (t)$ remains bounded for all $t \in [0, \infty)$ .

Proof: According to the theorem of existence and uniqueness for ordinary differential equation (Khalil, 2002: Theorem 3.1), the solution exists for the closed-loop system over $t \in [0, T^{*})$ . From Proposition 1, we have $ψ (0) - | e_{c} (0) | > d_{c}$ . Thus, there exists a time $t_{1}^{*} \leq T^{*}$ such that $ψ (t) - | e_{c} (t) | > d_{c 0} \forall t \in (0, t_{1}^{*}]$ , where $d_{c 0} < d_{c}$ . Since $Ω_{0} \subset Ω_{f}$ , $(\tilde{η} (t), σ (t), e_{c} (t)) \in Ω_{f}$ for $| e_{c} | \leq λ_{f}$ , thus, $\tilde{η}$ and σ are bounded. Define $κ_{f}$ as:

κ_{f} : = min {ε_{0}, \frac{a \tilde{ε}}{[L + χ]}, d_{c 0}}

where $ε_{0} < ε$ , $\tilde{ε} = ε - ε_{0}$ and L and χ are respectively defined in Remarks 2 and 5. Now, we show that the following inequality holds:

ψ (t) - | e_{c} (t) | \geq κ_{f} \forall t \in [t_{1}^{*}, T^{*})

(14)

Similar to Chowdhury and Khalil (2019), we use an argument of contradiction (a standard procedure in funnel control Ilchmann, 2013) to show that the inequality (14) holds. Thus, seeking a contradiction in inequality (14), we assume that there exists a time interval $[t_{2}^{*}, t_{3}^{*}]$ during which the inequality (14) does not hold, where $t_{2}^{*} > t_{1}^{*}$ . By the continuity of the function $ψ (t) - | e_{c} (t) |$ , we divide the time interval $[t_{1}^{*}, t_{3}^{*}]$ such that,

ψ (t) - | e_{c} (t) | \geq κ_{f} \forall t \in [t_{1}^{*}, t_{2}^{*}]

(15)

and

κ_{f} \geq ψ (t) - | e_{c} (t) | > κ_{f 0} \forall t \in [t_{2}^{*}, t_{3}^{*}]

(16)

where $κ_{f 0} < κ_{f}$ . From inequalities (15) and (16), $t_{2}^{*}$ exists such that $ψ (t_{2}^{*}) - | e_{c} (t_{2}^{*}) | = κ_{f}$ . Since, $d_{c} > d_{c 0} > κ_{f} > κ_{f 0}$ , we have, for time $t \in [0, t_{3}^{*}]$ , $ψ (t) - | e_{c} (t) | > κ_{f 0}$ which implies that $| e_{c} (t) | < ψ (t) - κ_{f 0} < λ_{f} - κ_{f 0}$ . Thus, $| e_{c} (t) | < λ_{f}$ which implies that $(\tilde{η} (t), σ (t), e_{c} (t)) \in Ω_{f}$ . Furthermore, from inequality (16) and by using the definition of $κ_{f}$ , we get $| e_{c} (t) | \geq ψ (t) - κ_{f} \geq \tilde{ε}$ . By multiplying equation (8) with $e_{c} (t)$ and using the result $| e_{c} (t) | \geq \tilde{ε}$ , we get:

k_{c} (t) | e_{c} (t) | = \frac{| e_{c} (t) |}{ψ (t) - | e_{c} (t) |} \geq \frac{\tilde{ε}}{κ_{f}}

(17)

Multiplying ${\overset{\cdot}{e}}_{c}$ by $e_{c}$ and by using equation (8), we get:

e_{c} {\overset{\cdot}{e}}_{c} = e_{c} [f (.) - g (.) \frac{e_{c}}{ψ (t) - | e_{c} (t) |} - \overset{\cdot}{r} - l^{2} σ + l μ sat (\frac{e_{c}}{μ})]

(18)

By Assumptions 2 and 4 and Remark 5 and using inequality (17), we can write equation (18) as:

e_{c} {\overset{\cdot}{e}}_{c} \leq | e_{c} | [χ - g (.) \frac{| e_{c} |}{ψ (t) - | e_{c} (t) |}] \leq | e_{c} | [χ - a \frac{\tilde{ε}}{κ_{f}}]

(19)

By using the definition of $κ_{f}$ , we can write inequality (19) as:

e_{c} {\overset{\cdot}{e}}_{c} \leq - L | e_{c} |

(20)

By integrating inequality (20) over the interval $t \in [t_{2}^{*}, t_{3}^{*}]$ , we get,

| e_{c} (t_{3}^{*}) | - | e_{c} (t_{2}^{*}) | \leq - L (t_{3}^{*} - t_{2}^{*})

(21)

Using Remark 2, we have,

| ψ (t_{3}^{*}) - ψ (t_{2}^{*}) | \leq L (t_{3}^{*} - t_{2}^{*})

(22)

Thus, by using inequalities (21) and (22), we get,

| e_{c} (t_{3}^{*}) | - | e_{c} (t_{2}^{*}) | \leq - L (t_{3}^{*} - t_{2}^{*}) \leq - | ψ (t_{3}^{*}) - ψ (t_{2}^{*}) | \leq ψ (t_{3}^{*}) - ψ (t_{2}^{*})

(23)

which implies that,

ψ (t_{3}^{*}) - | e_{c} (t_{3}^{*}) | \geq ψ (t_{2}^{*}) - | e_{c} (t_{2}^{*}) |

(24)

By using $ψ (t_{2}^{*}) - | e_{c} (t_{2}^{*}) | = κ_{f}$ , we can write the inequality (24) as $ψ (t_{3}^{*}) - | e_{c} (t_{3}^{*}) | \geq κ_{f}$ . Thus, we arrive at the contradiction to $ψ (t_{3}^{*}) - | e_{c} (t_{3}^{*}) | \leq κ_{f}$ which proves that the inequality (14) holds true. By using the facts $ψ (0) - | e_{c} (0) | > d_{c}$ , $ψ (t) - | e_{c} (t) | > d_{c 0} \forall t \in [0, t_{1}^{*}]$ , $d_{c} > d_{c 0}$ and $ψ (t) - | e_{c} (t) | \geq κ_{f} \forall t \in [t_{1}^{*}, T_{1}^{*}]$ and by using $ψ (t) - | e_{c} (t) | > d_{c 0} \geq κ_{f}$ , we get,

ψ (t) - | e_{c} (t) | \geq κ_{f} \forall t \in [0, T_{1}^{*})

(25)

Inequality (25) shows that the conditional error $e_{c} (t)$ evolves within the funnel region; hence, it is bounded on $[0, T_{1}^{*})$ . Since $ψ (t) < λ_{f}$ , $| e_{c} (t) | \leq ψ (t) - κ_{f} < λ_{f} - κ_{f}$ which implies that $| e_{c} (t) | \leq λ_{f} - κ_{f} \forall t \in [0, T_{1}^{*})$ and hence $(η (t), σ (t), e_{c} (t)) \in Ω_{f} \forall t \in [0, T_{1}^{*})$ . Since the trajectories of the system and the reference signal along its derivatives are bounded, we conclude by using the property of maximal solution of differential equation that the solution remains valid for all $t \in [0, \infty)$ , that is, $T_{1}^{*} = \infty$ . Thus,

ψ (t) - | e_{c} (t) | \geq κ_{f} \forall t \in [0, \infty)

(26)

By using inequality (26), the funnel gain (8) satisfies:

k_{c} (t) | e_{c} (t) | = \frac{| e_{c} (t) |}{ψ (t) - | e_{c} (t) |} \leq \frac{λ_{f} - κ_{f}}{κ_{f}} \forall t \in [0, \infty)

(27)

Thus, the funnel gain remains bounded for all $t \in [0, \infty)$ . This completes the proof. □

Corollary 1: Consider the closed-loop system (11), for $μ \leq | e_{c} (t) | \leq λ_{f}$ and $ψ (t) - | e_{c} (t) | \geq κ_{f} \forall t \in [0, \infty)$ there exists $μ = \bar{μ}$ and $κ_{f}^{*} \leq κ_{f} - c_{0}^{*} \bar{μ}$ such that $| e (t) | \leq ψ (t) - κ_{f}^{*}$ which ensures that the error $e_{1}$ also evolves within the funnel region and remains in it for all $t \in [0, \infty)$ , where $c_{0}^{*} = {(2 c_{0})}^{1 / 2}$ and $c_{0}$ is defined in Lemma 1.

Proof: Since $σ (0) = 0$ and from Lemma 1, the set ${V_{σ} \leq c_{0} {(\frac{μ}{l})}^{2}}$ is positively invariant; thus, we have $| e_{c} (t) - e_{1} (t) | = l | σ (t) | \leq c_{0}^{*} \bar{μ}$ . From Theorem 1, we can write inequality (26) as:

| e_{c} (t) | + c_{0}^{*} \bar{μ} \leq ψ (t) - κ_{f} + c_{0}^{*} \bar{μ} \forall t \geq 0

(28)

Since $| e_{1} (t) | = | e_{c} (t) - l σ (t) | \leq | e_{c} (t) | + l | σ (t) | \leq | e_{c} (t) | + c_{0}^{*} \bar{μ}$ , we can write inequality (28) as,

| e_{1} (t) | \leq ψ (t) - κ_{f} + c_{0}^{*} \bar{μ} \forall t \geq 0

(29)

Since $κ_{f}^{*} \leq κ_{f} - c_{0}^{*} \bar{μ}$ , we have,

| e_{1} (t) | \leq ψ (t) - κ_{f}^{*} \forall t \geq 0

(30)

Inequality (30) shows that the error signal evolves within the funnel region and remains within it for all the future time. This completes the proof. □

Remark 6: The funnel function starts from $ψ (t) = ψ (0) > 0$ at $t = 0$ and there exists $t \geq t_{4}^{*}$ such that the funnel function reaches to $ε^{*} > 0$ , that is, $ψ (t_{4}^{*}) \leq ε^{*} \forall t \geq t_{4}^{*}$ . From Theorem 1, it is established that the conditional error $e_{c} (t)$ remains within the funnel boundary $ψ (t)$ , that is, $| e_{c} (t) | \leq ε^{*} - κ_{f} \forall t \in [t_{4}^{*}, \infty)$ ; thus, there exists $μ^{*} \geq ε^{*}$ such that at $t \geq t_{4}^{*}$ , $| e_{c} |$ enters the boundary $[- μ^{*}, μ^{*}]$ , that is, $| e_{c} (t_{4}^{*}) | \leq μ^{*} - κ_{f}$ .

Proposition 2: From Remark 6, $e_{c} (t)$ enters the boundary $[- μ^{*}, μ^{*}]$ ; thus, ${\tilde{e}}_{c} (t)$ evolves within the funnel region and the equilibrium point $e_{c} = e_{c 0}$ of the closed-loop system is enclosed inside that boundary, where ${\tilde{e}}_{c} = e_{c} - e_{c 0}$ .

Proof: For $| e_{c} (t) | < μ^{*}$ , equation (6) becomes:

\overset{\cdot}{σ} (t) = - l σ + e_{c} \forall t \geq t_{4}^{*}

(31)

Thus, by using the proof of Lemma 1, we can show that the trajectories of the system reach the set $Ω_{μ}$ in finite time, where $Ω_{μ}$ is defined by:

Ω_{μ} = {V_{η} \leq α_{4} (c_{2} μ^{*} + c_{r 1})} \times {V_{σ} \leq c_{1} {(\frac{μ^{*}}{l})}^{2}} \times {| e_{c} | < μ^{*}}

where $c_{2} = 2$ . Since, from Assumptions 2 and 3, there exists a unique equilibrium point $e_{c} = e_{c 0}$ of the closed-loop system. Since $| {\tilde{e}}_{c} |$ satisfies the inequality $- | {\tilde{e}}_{c} | + | e_{c 0} | \geq - | e_{c} |$ , we can transform the inequality (26) to:

ψ (t) - | {\tilde{e}}_{c} (t) | \geq {\tilde{κ}}_{f} \forall t \in [0, \infty)

(32)

where ${\tilde{κ}}_{f} = κ_{f} - | e_{c 0} |$ . This ensures that ${\tilde{e}}_{c} (t)$ also evolves within the funnel region. The controller brings the system trajectory $| e_{c} |$ from $| e_{c} (0) |$ into the boundary $[- μ^{*}, μ^{*}]$ ; thus, we conclude from Assumption 2 that the equilibrium point $e_{c 0}$ is also enclosed inside that boundary. This completes the proof. □

It may be noted that $k_{c} (t)$ is a dynamic term, and therefore, the proof for asymptotic stability is challenging. Thus, in order to prove the asymptotic stability of CFC, we divide $k_{c} (t)$ into a constant gain and a dynamic gain that is:

k_{c} (t) = k_{c 0} + k_{c 1} (t)

(33)

where $k_{c 0} > 0$ is a constant gain and $k_{c 1} (t)$ is a dynamic gain.

Theorem 2: Suppose Assumptions 1 through 4 are satisfied, then there exists $l = l^{*}$ such that the equilibrium point of the closed-loop system is asymptotically stable.

Proof: Inside the boundary $[- μ^{*}, μ^{*}]$ , ${\overset{\cdot}{e}}_{c}$ in equation (11) reduces to:

{\overset{\cdot}{e}}_{c} = F (.) - g (.) k_{c 0} e_{c} \forall t \geq t_{4}^{*}

(34)

where $F (.) = f (.) - \overset{\cdot}{r} - l^{2} σ + l e_{c} - g (.) k_{c 1} e_{c}$ . From Assumption 3, reference signal and its derivative are bounded. From Remark 5 and Assumption 4, we have $| f (.) - \overset{\cdot}{r} - l^{2} σ + l e_{c} | \leq χ$ and $g (.) \leq b$ and from Theorem 1, $k_{c 1} e_{c}$ remains bounded; thus, we conclude that $| F (.) | \leq χ^{*}$ , where $χ^{*} > 0$ . At equilibrium point ( $e_{c 0}, σ_{0}$ ), ${\overset{\cdot}{e}}_{c} = 0$ ; thus, from Assumption 2, we get, $e_{c 0} = \frac{F_{0}}{g_{0} k_{c 0}}$ , where $σ_{0} = \frac{e_{c 0}}{l}$ is the equilibrium point of equation (34) and $F_{0}$ and $g_{0}$ are respectively the constant values of $F (.)$ and $g (.)$ at the equilibrium point ( $e_{c 0}, σ_{0}$ ). By taking time-derivative of ${\tilde{e}}_{c} = e_{c} - e_{c 0}$ , we get:

{\overset{\cdot}{\tilde{e}}}_{c} = \tilde{F} (.) - g (.) k_{c 0} {\tilde{e}}_{c} \forall t \geq t_{4}^{*}

(35)

where $\tilde{F} (.) = F (.) - g (.) k_{c 0} e_{c 0}$ . Since $| F (.) | \leq χ^{*}$ , $g (.) \leq b$ and at equilibrium point ( $e_{c 0}, σ_{0}$ ), $\tilde{F} (.) = 0$ , there exists $λ_{1} < k_{c 0}$ , $λ_{2}$ , $λ_{3}$ and $λ_{4}$ such that $| \frac{\tilde{F} (.)}{g (.)} | \leq λ_{1} | {\tilde{e}}_{c} | + λ_{2} | \tilde{σ} | + λ_{3} | \tilde{r} | + λ_{4} | \overset{\cdot}{r} |$ , where $\tilde{σ} = σ - σ_{0}$ and $\tilde{r} = r - r_{ss}$ . Furthermore, the time-derivative of $\tilde{σ}$ results:

\overset{\cdot}{\tilde{σ}} (t) = - l \tilde{σ} + {\tilde{e}}_{c} \forall t \geq t_{4}^{*}

(36)

Select the Lyapunov function $V = \frac{1}{2} {\tilde{e}}_{c}^{2} + \frac{1}{2} {\tilde{σ}}^{2}$ . By taking its time-derivative and using standard operation and Assumption 4, we get:

\overset{\cdot}{V} \leq - ζ^{T} Π ζ + b (λ_{3} | \tilde{r} | + λ_{4} | \overset{\cdot}{r} |) | {\tilde{e}}_{c} |

(37)

where $ζ = {[\begin{matrix} {\tilde{e}}_{c} & \tilde{σ} \end{matrix}]}^{T}$ and $Π = [\begin{matrix} a (k_{c 0} - λ_{1}) & - \frac{(b λ_{2} + 1)}{2} \\ - \frac{(b λ_{2} + 1)}{2} & l \end{matrix}]$ . Thus, by selecting $l = l^{*}$ , Π can be made positive definite. Since $| \tilde{r} | \to 0$ and $| \overset{\cdot}{r} | \to 0$ as $t \to \infty$ , we conclude that the trajectories $e_{c}$ and σ respectively reach their equilibrium points $e_{c 0}$ and $σ_{0}$ . Since $e_{1} (t) = {\tilde{e}}_{c} (t) - l \tilde{σ} (t)$ , we conclude that $e_{1} (t) \to 0$ as $t \to \infty$ . From this together with Assumption 1, we deduce that $| | \tilde{η} | |$ also approaches zero as $e_{1} (t) \to 0$ . This completes the proof. □

Extension of the main results

In this section, we extend the theory of section ‘Main results’ to present the CFC algorithm for arbitrary relative degree nonlinear systems.

Virtual output–based Conditional Funnel Control

For arbitrary relative degree nonlinear systems given in equation (2), following algorithm has been proposed in Chowdhury and Khalil (2019):

\begin{matrix} u (t) = \frac{1}{ϵ^{ρ} k_{ρ} g (.)} [- z_{2} (t) - \sum_{i = 3}^{ρ} k_{i - 1} z_{i} (t) + ϵ υ (t)] \\ υ (t) = - \frac{s (t)}{ψ (t) - | s (t) |} \\ s (t) = z_{1} (t) + \sum_{i = 2}^{ρ} k_{i} z_{i} (t) \\ z_{j} (t) = ϵ^{j - 1} e_{j} (t), j = 1 to ρ \end{matrix}}

(38)

where $k_{i} > 0$ and $ϵ > 0$ are the design constants and $s (t)$ is known as the Virtual Output. For the asymptotic tracking problem of arbitrary relative degree nonlinear systems, we apply the results of section ‘Main results’ to modify the algorithm given in equation (38) as:

\begin{matrix} u (t) = \frac{1}{ϵ^{ρ} k_{ρ} g (.)} [- z_{2} (t) - \sum_{i = 3}^{ρ} k_{i - 1} z_{i} (t) + ϵ υ (t)] \\ υ (t) = - {\bar{k}}_{c} s_{c} (t) \\ {\bar{k}}_{c} (t) = \frac{1}{ψ (t) - | s_{c} (t) |} \\ s_{c} (t) = z_{1} (t) + \sum_{i = 2}^{ρ} k_{i} z_{i} (t) + l \bar{σ} (t) \\ z_{j} (t) = ϵ^{j - 1} e_{j} (t), j = 1 to ρ \\ \overset{\cdot}{\bar{σ}} (t) = - l \bar{σ} (t) + μ sat (\frac{s_{c} (t)}{μ}) \end{matrix}}

(39)

where $\bar{σ} (0) = 0$ and $s_{c} (t)$ and ${\bar{k}}_{c} (t)$ are respectively known as the Conditional Virtual Output and the Conditional Funnel Gain. By applying equation (39) to equation (2), the closed-loop system becomes:

\begin{matrix} \overset{\cdot}{\tilde{η}} = h (\tilde{η}, z_{1} + r) \\ ϵ \overset{\cdot}{z} = Az + B s_{c} \\ {\overset{\cdot}{s}}_{c} = ϵ^{ρ - 1} k_{ρ} [f (.) - r^{(ρ)} - l^{2} \bar{σ} + l μ sat (\frac{s_{c}}{μ})] + v \end{matrix}}

(40)

where A and B are defined as:

A = [\begin{matrix} 0 & 1 & \dots & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ \\ 0 & \dots & \dots & 0 & 1 \\ - \frac{1}{k_{ρ}} & - \frac{k_{2}}{k_{ρ}} & \dots & \dots & \frac{k_{ρ - 1}}{k_{ρ}} \end{matrix}], B = [\begin{matrix} 0 \\ 0 \\ ⋮ \\ 0 \\ \frac{1}{k_{ρ}} \end{matrix}]

Since we apply the result of section ‘Main results’ to modify the control algorithm given in equation (39), we need to modify the Remark 5 as Remark 7:

Remark 7: From Assumption 3, the reference signal and its derivatives are bounded. Therefore, for all $(\tilde{η} (t), z (t), \bar{σ} (t), s_{c} (t)) \in {\bar{Ω}}_{f}$ , there exists $\bar{χ} > 0$ such that $| ϵ^{ρ - 1} k_{ρ} [f (\tilde{η}, x) - r^{(ρ)} - l^{2} \bar{σ} (t) + l μ sat (\frac{s_{c}}{μ})] | \leq \bar{χ}$ .

The compact sets $Ω_{f}$ and $Ω_{0}$ are respectively modified as:

\begin{matrix} {\bar{Ω}}_{f} = {V_{η} \leq α_{4} (c_{1} λ_{f} + μ + c_{r 1})} \times {V_{z} \leq c λ_{f}^{2}} \\ \times {V_{\bar{σ}} \leq {\bar{c}}_{0} {(\frac{μ}{l})}^{2}} \times {| s_{c} | \leq λ_{f}} \\ {\bar{Ω}}_{0} = {V_{η} \leq α_{4} ({\bar{c}}_{1} λ_{0} + c_{r 1})} \times {V_{z} = \bar{c} λ_{0}^{2}} \times {V_{\bar{σ}} \leq 0} \\ \times {| s_{c} | \leq λ_{0}} \end{matrix}

where ${\bar{c}}_{0}$ , $c_{1}$ , ${\bar{c}}_{1}$ , c and $\bar{c}$ are the positive constants, $V_{\bar{σ}} = \frac{1}{2} {\bar{σ}}^{2}$ , $V_{z} = z^{T} Pz$ where P is a symmetric positive definite matrix resulting from $PA + A^{T} P = - Q$ and Q is also a symmetric positive definite matrix.

Theorem 3: Consider a closed-loop system (40). Suppose Assumptions 1–4 are satisfied, $φ (t) \in ϕ$ and $(\tilde{η} (0), z (0), \bar{σ} (0), s_{c} (0)) \in {\bar{Ω}}_{0}$ , then by Remark 1, there exists ${\bar{κ}}_{f} > 0$ such that $| s_{c} | \leq ψ (t) - {\bar{κ}}_{f}$ and the conditional funnel gain ${\bar{k}}_{c} (t)$ remains bounded for all $t \in [0, \infty)$ . Furthermore, there exists ${\bar{κ}}_{f}^{*} > 0$ such that $| e_{1} | \leq ψ (t) - {\bar{κ}}_{f}^{*}$ .

Proof: Following Lemma 1, it can be shown that the sets ${V_{z} \leq c λ_{f}^{2}}$ and ${V_{\bar{σ}} \leq {\bar{c}}_{0} {(\frac{μ}{l})}^{2}}$ remain positively invariant. Then, by following (Chowdhury and Khalil, 2019: section 4.1) and Theorem 1 (discussed in section ‘Stability Analysis’) and by using Remark 7, it can be shown that $| s_{c} | \leq ψ (t) - {\bar{κ}}_{f}$ . Furthermore, by following Corollary 1 (discussed in section ‘Stability Analysis’) and (Chowdhury and Khalil, 2019: Theorem 1), it can respectively be shown that $| s_{c} - s | = l | \bar{σ} | \leq μ$ and $| s - e_{1} | \leq \bar{k} ϵ$ hold where $\bar{k} > 0$ . Thus, by using the results, $| s_{c} | \leq ψ (t) - {\bar{κ}}_{f}$ , $| s_{c} - s | = l | \bar{σ} | \leq μ$ and $| s - e_{1} | \leq \bar{k} ϵ$ , it can be shown that $| e_{1} | \leq ψ (t) - {\bar{κ}}_{f}^{*}$ . This completes the proof. ■

Examples and case studies

In this section, we first present examples to compare the results of the algorithm proposed in section ‘Main results’ with the results of algorithms given in Lee and Trenn (2019) and Trenn (2019), Zhao et al. (2022) and Ilchmann et al. (2002). To show the practicability of the theoretical results, we then present four case studies in which the industrial applications: (1) Temperature Control of a CSTR; (2) Product Concentration Control in a CSTR; (3) Biomass Control in a Bioreactor and (4) Position Control of a Flexible Joint Manipulator, are demonstrated. For these case studies, we compare the results of the algorithm proposed in section ‘Extension of the main results’ with those of algorithm given in Chowdhury and Khalil (2019). In order to preserve the transient performance of the proposed funnel algorithms presented in sections ‘Main results’ and ‘Extension of the main results’, the parameters μ and l should respectively be selected as small as possible and as large as possible.

Examples

Example 1: Consider the following system:

\overset{\cdot}{y} (t) = 2 + \sin (y) + u (t)

(41)

For CFC given in equations (5)–(8), select $l = 40$ , $μ = 0.05$ , a constant reference signal $r = - 1$ , $y (0) = 1$ and $ψ (t)$ as:

ψ (t) = (ψ_{i} - ψ_{f}) \exp (- 3 t) + ψ_{f}

(42)

where $ψ_{i} = 3.5$ and $ψ_{f} = 0.05$ . For this example, we also implement the algorithm presented in Lee and Trenn (2019) and Trenn (2019) given in equation (9) and the same $ψ (t)$ given in equation (42) is selected with the same $ψ_{i}$ , but with $ψ_{f} = 0$ which is the main condition for implementing the algorithm. The simulation results are shown in Figure 2. Both the algorithms force the error to converge towards zero, but notice in Figure 2(d) that the $ω (t)$ in equation (9) grows exponentially which can amplify the noises in the measurement signal $y (t)$ . We further implement the classical funnel control (Ilchmann et al., 2002) given in equations (3) and (4) and the same $ψ (t)$ , given in equation (42), is selected. The simulation results are shown in Figure 2(e) and (f). Notice that the classical funnel algorithm does not converge the error to zero.

Figure 2.

(a) Example 1: Plot of conditional error $e_{c}$ , error e and funnel boundary ψ for the Proposed Algorithm given in equations (5)–(8). (b) Example 1: Plot of error e and funnel boundary ψ for the algorithm given in equation (9). (c) Example 1: Plot of outputs of the Proposed Algorithm given in equation (7) and the Controller given in equation (9). (d) Example 1: Plot of gains k and $ω = 1 / ψ$ respectively given in equations (8) and (9). (e) Example 1: Plot of error e and funnel boundary ψ for the Classical Funnel Control given in equations (4) and (5). (f) Example 1: Plot of output of the Classical Funnel Control given in equations (4) and (5).

Example 2: Consider the following system (Zhao et al., 2022):

\overset{\cdot}{y} (t) = \hat{ϕ} (t) + u (t)

(43)

where $\hat{ϕ} (t)$ is a dynamical function and is given as $\hat{ϕ} (t) = y (t)$ . For CFC given in equations (5)–(8), select $l = 20$ , $μ = 0.7$ , a constant reference signal $r = 2$ , $y (0) = 1$ and $ψ (t)$ as:

ψ (t) = \frac{\sqrt{\hat{l}} β (t)}{\sqrt{1 - β^{2}}}

(44a)

β (t) = \frac{1}{(1 - b_{f}) \exp (- 1.5 t) + b_{f}}

(44b)

where $b_{f} = 0.05$ , $l = 1 - b_{f}^{2}$ . For this example, we also implement the algorithm presented in Zhao et al. (2022) given in equation (10). The design constants are $\hat{c} = 5$ and $\hat{Γ} = 1$ (Zhao et al., 2022), whereas $β (t)$ is given in equation (44b). The simulation results are shown in Figure 3. Both the algorithms make the error signal converge towards zero; however, as mentioned in Remark 4 that the algorithm given in equation (10) is complex and requires prior knowledge of the system dynamical function $\hat{ϕ}$ in comparison with the proposed algorithm given in equations (5)–(8). We further implement the classical funnel control (Ilchmann et al., 2002) given in equations (3) and (4). The simulation results are shown in Figure 3(d) and (e). Notice that the classical funnel algorithm again fails to converge the error to zero.

Figure 3.

(a) Example 2: Plot of conditional error $e_{c}$ , error e and funnel boundary ψ for the Proposed Algorithm given in equations (5)–(8). (b) Example 2: Plot of error e and funnel boundary ψ for algorithm given in equation (10). (c) Example 2: Plot of outputs of the Proposed Algorithm given in equation (7) and the Controller given in equation (10). (d) Example 2: Plot of error e and funnel boundary ψ for the Classical Funnel Control given in equations (3) and (4). (e) Example 2: Plot of output of the Classical Funnel Control given in equations (3) and (4).

Case studies

It may be noticed that in all the figures displayed in this section, the virtual output–based funnel control algorithm given in equation (38) is represented by ‘Virtual FC’ whereas virtual output–based Conditional Funnel Control algorithm given in equation (39) is represented by ‘Virtual Conditional FC’.

Case Study 1: Temperature Control of a CSTR: CSTRs have been widely used in chemical as well as in other process industries. In this case study, we present the application of the proposed scheme, CFC, along the comparison with classical funnel control (Ilchmann et al., 2002) for regulating the temperature of a given below CSTR model (Colantonio et al., 1995):

\begin{matrix} {\overset{\cdot}{x}}_{1} = - x_{1} + D_{a} (1 - x_{1}) \exp (\frac{x_{2}}{γ^{- 1} x_{2} + 1}) \\ {\overset{\cdot}{x}}_{2} = - x_{2} + B D_{a} (1 - x_{1}) \exp (\frac{x_{2}}{γ^{- 1} x_{2} + 1}) - β x_{2} + β u \\ y = x_{2} \end{matrix}}

(45)

where $x_{1}$ and $x_{2}$ are respectively the reactant concentration and reactor temperature, $γ = 20$ is an activation energy, $D_{a} = 0.078$ is a Damkohler number, $B = 8$ is an adiabatic temperature rise and $β = 0.3$ is a heat transfer coefficient. All the parameters of reactor model are dimensionless and the relative degree of a given CSTR model is $ρ = 1$ . The objective is to set and maintain the reactor temperature $x_{2}$ precisely to reference r. Consider it is required to track the reference signal $r = 2.0 + 0.8 \tanh (\frac{t}{0.5})$ and the initial value of the system is $[\begin{matrix} x_{1} (0) & x_{2} (0) \end{matrix}] = [\begin{matrix} 0.3246 & 2.0 \end{matrix}]$ . For this benchmark system, we first implement the classical funnel control (Ilchmann et al., 2002) given in equations (3) and (4) with the funnel function $ϕ (t) = \frac{1}{ψ (t)}$ as:

ϕ (t) = {[0.8 e^{- \frac{t}{0.5}} + 0.2]}^{- 1}

(46)

We then implement the proposed algorithm given in equations (5)–(8) with the design parameters $l = 10$ and $μ = 0.1$ and with the same funnel function $ϕ (t)$ as given in equation (46). The simulation results are shown in Figure 4 where classical control given in equations (3) and (4) is represented by ‘Classical FC’ while Conditional Funnel Control given in equations (5)–(8) is represented by ‘Conditional FC’. It can be observed in Figure 4(a) that the proposed algorithm forces the error to converge to zero as time progresses but the classical funnel control sets the error to a non-zero value rather than converging towards zero.

Figure 4.

(a) Case Study 1: Plot of error e and funnel boundary ψ for the Classical FC (equations (4) and (5)) and Conditional FC (equations (6)–(8)). (b) Case Study 1: Plot of controller output u for the Classical FC (equations (4) and (5)) and Conditional FC (equations (6)–(8)).

Case Study 2: Product Concentration Control in a CSTR: In Case Study 1, we consider the problem of controlling the temperature of a CSTR. Now, we present a case study in order to control the concentration of the product being produced in a CSTR. Consider the following chemical reaction:

A ⇄ B \to C

which is carried out in an isothermal CSTR. The dimensionless dynamic model of the system is (Chen and Dai, 2001):

\begin{matrix} {\overset{\cdot}{\bar{x}}}_{1} = 1 - {\bar{x}}_{1} - D_{a 1} {\bar{x}}_{1} + D_{a 2} {\bar{x}}_{2}^{2} \\ {\overset{\cdot}{\bar{x}}}_{2} = D_{a 1} {\bar{x}}_{1} - {\bar{x}}_{2} - D_{a 2} {\bar{x}}_{2}^{2} - D_{a 3} {\bar{x}}_{2}^{2} + u \\ {\overset{\cdot}{\bar{x}}}_{3} = D_{a 3} {\bar{x}}_{2}^{2} - {\bar{x}}_{3} \\ y = {\bar{x}}_{3} \end{matrix}}

(47a)

where $D_{a 1} = 3.0$ , $D_{a 2} = 0.5$ and $D_{a 3} = 1.0$ are the system parameters, ${\bar{x}}_{1}$ , ${\bar{x}}_{2}$ and ${\bar{x}}_{3}$ are respectively the dimensionless concentration of species A, B and C. By applying the coordinate transformation $[\begin{matrix} η & x_{1} & x_{2} \end{matrix}] = [\begin{matrix} {\bar{x}}_{1} & {\bar{x}}_{3} & D_{a 3} {\bar{x}}_{2}^{2} - {\bar{x}}_{3} \end{matrix}]$ , we get the following dynamic model:

\begin{matrix} \overset{\cdot}{η} = 1 - η - D_{a 1} η + D_{a 2} x_{1}^{2} \\ {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f (\bar{x}) + g (\bar{x}) u \\ y = x_{1} \end{matrix}}

(47b)

where $f (\bar{x}) = 2 D_{a 3} {\bar{x}}_{2} (D_{a 1} {\bar{x}}_{1} - {\bar{x}}_{2} - D_{a 2} {\bar{x}}_{2}^{2} - D_{a 3} {\bar{x}}_{2}^{2}) - D_{a 3} {\bar{x}}_{2}^{2} + {\bar{x}}_{3}$ and $g (\bar{x}) = 2 D_{a 3} {\bar{x}}_{2}$ . The transformed model given in equation (47b) has a relative degree $ρ = 2$ . The objective is to track and maintain the concentration of the product C, that is, $x_{1}$ , precisely to a reference signal r. Consider it is required to track the reference signal $r = 1.5 + 0.5 \tanh (\frac{t}{0.5})$ and the initial value of the system is $[η (0) x_{1} (0) x_{2} (0)] = [0.0313 1.5 0]$ . For this benchmark system, we first implement the virtual output–based funnel control given in equation (38) with the design parameters $k_{2} = 1$ , $ϵ = 0.5$ and the funnel function $ϕ (t) = \frac{1}{ψ (t)}$ given in equation (46). We then implement the modified algorithm, virtual output–based CFC given in equation (39), with the design parameters $k_{2} = 1$ , $ϵ = 0.5$ , $l = 10$ , $μ = 0.2$ and with the same funnel function $ϕ (t)$ as given in (46). The simulation results are shown in Figure 5. It can be observed in Figure 5(a) that the modified algorithm given in equation (39) makes the error to approach zero as time progresses but the algorithm given in equation (38) adjusts the error to a non-zero value rather than converging towards zero.

Figure 5.

(a) Case Study 2: Plot of error e and funnel boundary ψ for the Virtual FC (equation (38)) and Virtual Conditional FC (equation (39)). (b) Case Study 2: Plot of controller output u for the Virtual FC (equation (38)) and Virtual Conditional FC (equation (39)).

Case Study 3: Biomass Control in a Bioreactor: In this case study, we consider the problem of controlling the concentration of the biomass in a bioreactor. Consider the following dimensionless dynamic model of a bioreactor (Seilisteanu et al., 2007):

\begin{matrix} {\overset{\cdot}{\bar{x}}}_{1} = \bar{μ} ({\bar{x}}_{2}) {\bar{x}}_{1} - D_{a} {\bar{x}}_{1} \\ {\overset{\cdot}{\bar{x}}}_{2} = - k_{c} \bar{μ} ({\bar{x}}_{2}) {\bar{x}}_{1} - D_{a} {\bar{x}}_{2} + u \\ \bar{μ} ({\bar{x}}_{2}) = \frac{μ^{*} {\bar{x}}_{2}}{K_{m} + {\bar{x}}_{2}} \\ y = {\bar{x}}_{1} \end{matrix}}

(48a)

where $D_{a} = 0.2$ is the dilution rate, ${\bar{x}}_{1}$ and ${\bar{x}}_{2}$ are respectively the concentrations of biomass and substrate, $k_{c} = 12$ is the yield coefficient, $μ^{*} = 2.1$ is the maximum specific rate and $K_{M} = 10$ is the Michaelis–Menten constant. By applying the coordinate transformation $[\begin{matrix} x_{1} & x_{2} \end{matrix}] = [\begin{matrix} {\bar{x}}_{1} & \bar{μ} ({\bar{x}}_{2}) {\bar{x}}_{1} - D_{a} {\bar{x}}_{1} \end{matrix}]$ , we get the following dynamic model:

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f (\bar{x}) + g (\bar{x}) u \\ y = x_{1} \end{matrix}}

(48b)

ϕ (t) = {[4.8 e^{- \frac{t}{0.5}} + 0.2]}^{- 1}

(49)

We then implement the modified algorithm, virtual output–based Conditional Funnel Control given in equation (39), with the design parameters $k_{2} = 1$ , $ϵ = 0.1$ , $l = 10$ , $μ = 0.2$ and with the same funnel function $ϕ (t)$ as given in equation (49). The simulation results are shown in Figure 6. It can be observed in Figure 6(a) that the modified algorithm given in equation (39) again converges the error to zero as time progresses but the algorithm given in equation (38) regulates the error to a non-zero value.

Figure 6.

(a) Case Study 3: Plot of error e and funnel boundary ψ for the Virtual FC (equation (38)) and Virtual Conditional FC(equation (39)). (b) Case-Study 3: Plot of controller output u for the Virtual FC (equation (38)) and Virtual Conditional FC (equation (39)).

Case Study 4: Position Control of a Flexible Joint Manipulator: Consider a single link manipulator coupled with the actuator through spring. The dynamics of the manipulator is:

I {\overset{\cdot}{q}}_{1} + Mgl \sin (q_{1}) + k (q_{1} - q_{2}) = 0

(50a)

and the dynamics of the actuator is:

J {\overset{\cdot\cdot}{q}}_{2} - k (q_{1} - q_{2}) = u

(50b)

where $q_{1}$ and $q_{2}$ are respectively the positions of the link and motor-shaft, I and J are respectively the inertias of the link and motor-shaft, k is the spring constant, M and l are respectively the mass and length of the link and u is the input to the system. The values of the model parameters are: $I = J = 1$ , $Mgl = 10$ and $k = 1600$ (Spong, 1989). By considering the change of variables: $x_{1} = q_{1}$ , $x_{2} = {\overset{\cdot}{q}}_{1}$ , $x_{3} = - I^{- 1} Mgl \sin (q_{1}) - I^{- 1} k (q_{1} - q_{2})$ and $x_{4} = - I^{- 1} Mgl \cos (q_{1}) {\overset{\cdot}{q}}_{1} - I^{- 1} k ({\overset{\cdot}{q}}_{1} - {\overset{\cdot}{q}}_{2})$ , the dynamics of the flexible link manipulator given in equations (50a) and (50b) are converted into the following normal form:

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} \\ {\overset{\cdot}{x}}_{3} = x_{4} \\ {\overset{\cdot}{x}}_{4} = f (x) + bu \\ y = x_{1} \end{matrix}}

(50c)

where $f (x) = - [a_{1} \cos (x_{1}) + a_{2}] x_{3} + a_{3} \sin (x_{1}) [x_{2}^{2} - a_{4}]$ , $a_{1} = Mgl$ , $a_{2} = (I^{- 1} + J^{- 1}) k$ , $a_{3} = I^{- 1} Mgl$ , $a_{4} = J^{- 1} k$ and $b = {(IJ)}^{- 1} k$ . The resulting transformed system given in equation (50c) has a relative degree $ρ = 4$ . The objective is to track and maintain the position of a link, that is, $x_{1}$ , precisely to a reference signal r. Consider, it is required to track the reference signal $r = 5 + \sin (2 π t) e^{- t}$ and the initial value of the system is $[\begin{matrix} x_{1} (0) & x_{2} (0) & x_{3} (0) & x_{4} (0) \end{matrix}] = [\begin{matrix} 3 & - 1 & 1 & - 1 \end{matrix}]$ . For this benchmark system, we first implement the virtual output–based funnel control given in equation (38) with the design parameters $k_{2} = k_{3} = 3$ , $k_{4} = 1$ , $ϵ = 0.04$ and the funnel function $ϕ (t) = \frac{1}{ψ (t)}$ as (Chowdhury and Khalil, 2019):

ϕ = {[(2 π - 0.2618) e^{- \frac{t}{0.8189}} + 0.2618]}^{- 1}

(51)

We then implement the modified algorithm, virtual output–based Conditional Funnel Control given in equation (39), with the design parameters $k_{2} = k_{3} = 3$ , $k_{4} = 1$ , $ϵ = 0.04$ , $l = 20$ , $μ = 0.15$ and with the same funnel function $ϕ (t)$ as given in equation (51). The simulation results are shown in Figure 7. For this case study, it can be observed in Figure 7(a) that the modified algorithm given in equation (39) again makes the error to approach zero as time progresses whereas the algorithm given in equation (38) does not have the ability to force the error to approach zero asymptotically.

Figure 7.

(a) Case Study 4: Plot of error e and funnel boundary ψ for the Virtual FC (equation (38)) and Virtual Conditional FC (equation (39)). (b) Case Study 4: Plot of controller output u for the Virtual FC (equation (38)) and Virtual Conditional FC (equation (39)).

Conclusion

This paper has presented a funnel control algorithm for nonlinear system which ensures the asymptotic tracking of error signal to zero. This is achieved by introducing the concept of conditional error in the classical funnel control. The proposed algorithm is very simple to implement for relative degree one systems and it does not generate high gain; thus, it does not amplify noises in the measurement signal. For asymptotic convergence of arbitrary relative degree nonlinear systems, in this work, we extend the theory to modify the virtual output–based funnel control algorithm (Chowdhury and Khalil, 2019), and thus presented the virtual output–based CFC algorithm. The simulation results show the effectiveness of the proposed method in comparison with other recently proposed algorithms. However, we consider that the reference signal reaches a constant value at time tends to infinity; thus, for future work, this limitation may be removed by considering a more general reference signal. Furthermore, a fault tolerant asymptotic funnel control may be considered as future research endeavour.

Footnotes

Declaration of conflicting interests

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

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Fahad Wallam

Data availability statement

Data sharing is not applicable to this article, as no data sets were generated or analysed during the current study.

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