General Formula for Event-Based Stabilization of Nonlinear Systems with Delays in the State

Abstract

In this chapter, a universal formula is proposed for event-based stabilization of nonlinear systems affine in the control and with delays in the state. The feedback is derived from the seminal law proposed by E. Sontag (1989) and then extended to event-based control of affine nonlinear undelayed systems. Under the assumption of the existence of a control Lyapunov–Krasovskii functional (CLKF), the proposal enables smooth (except at the origin) asymptotic stabilization while ensuring that the sampling intervals do not contract to zero. Global asymptotic stability is obtained under the small control property assumption. Moreover, the control can be proved to be smooth anywhere under certain conditions. Simulation results highlight the ability of the proposed formula. The particular linear case is also discussed.

Appendix

Proofs of the present contribution were previously presented in [8]. They are recalled here.

1.1 Proof of Theorem 3

The proof follows the one developed in [24] for event-based control of systems without delays (4.1). First, let define hereafter

$$\begin{aligned} \psi (x) := \sqrt{\alpha _d(x)^2 + \varOmega (x)\beta _d(x)\varDelta (x)\beta _d(x)^T}. \end{aligned}$$

(4.29)

Let begin establishing $\gamma$ is smooth on ${\mathscr {X}}^*$. For this, consider the algebraic equation

$$\begin{aligned} P(x_d,\zeta ) := \beta _d(x_d)\varDelta (x_d)\beta _d(x_d)^T \zeta ^2 - 2 \alpha _d(x_d)\zeta - \varOmega (x_d) = 0. \end{aligned}$$

(4.30)

Note first that $\zeta = \gamma (x)$ is a solution of (4.30) for all $x_d \in {\mathscr {X}}$. It is easy to prove that the partial derivative of P with respect to $\zeta$ is always strictly positive on ${\mathscr {X}}^*$

$$\begin{aligned} \frac{\partial P}{\partial \zeta } := 2 \beta _d(x_d)\varDelta (x_d)\beta _d(x_d)^T \zeta - 2 \alpha _d(x_d). \end{aligned}$$

(4.31)

Indeed, when $\Vert \beta _d(x_d)\Vert =0$, (4.6) gives $\frac{\partial P}{\partial \zeta } = - 2 \alpha _d(x_d) \ge 2 \lambda (|\chi _0|) > 0$ and when $\Vert \beta _d(x_d)\Vert \ne 0$, (4.18) gives $\frac{\partial P}{\partial \zeta } = 2 \sqrt{\alpha _d(x_d)^2 + \varOmega (x_d)\beta _d(x_d)\varDelta (x_d)\beta _d(x_d)^T} > 0$ replacing $\zeta$ in (4.31) by the expression of $\gamma$ (since $\zeta = \gamma (x)$ is a solution of (4.30)). Therefore $\frac{\partial P}{\partial \zeta }$ never vanishes at each point of the form $\{(x_d,\gamma (x_d)) | x_d \in {\mathscr {X}}^*\}$. Furthermore, P is smooth w.r.t. $x_d$ and $\zeta$ since so are $\alpha _d$, $\beta _d$, $\varOmega$ and $\varDelta$. Hence, using the implicit function theorem, $\gamma$ is smooth on ${\mathscr {X}}^*$.

The decrease of the CLKF of the form (4.5) when applying the event-based feedback (4.16)–(4.17) is easy to prove. For this, let consider the time interval $[t_i,t_{i+1}]$, that is the interval separating two successive events. Recall that $x_{di}$ denotes the value of the state when the ${i^\mathrm{th}}$ event occurs and $t_i$ the corresponding time instant, as defined in (4.15). At time $t_i$, when the event occurs, the time derivative of the CLKF, i.e., (4.7), after the update of the control, is

$$\begin{aligned} \frac{dV}{dt}(x_{di}) = \alpha _d(x_{di}) + \beta _d(x_{di}) \upsilon (x_{di}) = - \psi (x_{di}) < 0 \end{aligned}$$

when substituting (4.18) in (4.16), where $\psi$ is defined in (4.29). More precisely, defining a compact set not containing the origin, that is $\varSigma = \{ x_d \in CP([-r,0],\mathscr {X}): d \le \Vert x_d\Vert \le D \}$, where $CP([-r,0],\mathscr {X})$ denotes the space of piecewise continuous functions from $[-r,0]$ into $\mathscr {X}$, d and D are some constant in $\mathbb {R}^+$. If V is a CLKF for the system of the form (4.4) then for all $0< \delta < D$ there exists $\varepsilon > 0$ such that $\alpha _d(\chi _d) \ge -\frac{1}{2} \lambda (|\chi _0|) \Rightarrow |\beta _d(\chi _d)| \ge \varepsilon$ for $\chi _d \in \varSigma$. This gives

$$\begin{aligned} \dot{V} \le - \lambda (|x|). \end{aligned}$$

One can refer to Lemma 1 in [17], and [16], for further details. With this updated control, the event function (4.17) hence becomes strictly positive

$$\begin{aligned}&\varepsilon (x_{di},x_{di}) = (1-\sigma ) \psi (x_{di}) > 0, \end{aligned}$$

since $\sigma \in [0,1[$, where $\psi$ is defined in (4.29). Furthermore, the event function necessarily remains positive before the next event by continuity, because an event will occur when $\varepsilon (x_d,x_{di}) = 0$ (see Definition 3). Therefore, on the interval $[t_i,t_{i+1}]$, one has

$$\begin{aligned} \varepsilon (x_d,x_{di})&= - \alpha _d(x_d) - \beta _d(x_d) \upsilon (x_{di}) - \sigma \psi (x_d),\\&= - \frac{dV}{dt}(x_d) - \sigma \psi (x_d) \ge 0, \end{aligned}$$

which ensures the decrease of the CLKF on the interval since $\sigma \psi (x_d) \ge 0$, where $\psi$ is defined in (4.29). Moreover, $t_{i+1}$ is necessarily bounded since, if not, V should converge to a constant value where $\frac{dV}{dt} = 0$, which is impossible thanks to the inequality above. The event function precisely prevents this phenomena detecting when $\frac{dV}{dt}$ is close to vanish and updates the control if it happens, where $\sigma$ is a tunable parameter fixing how “close to vanish” has to be the time derivative of V.

To prove that the event-based control is MSI , one has to prove that for any initial condition in an a priori given set, the sampling intervals are below bounded. First of all, notice that events only occur when $\varepsilon$ becomes negative (with $x_d \ne 0$). Therefore, using the fact that when $\beta _d(x_d) = 0$, $\alpha _d(x_d) < -\lambda (|\chi _0|)$ (because V is a CLKF as defined in Definition 2), it follows from (4.17), on $\{ x_d \in {\mathscr {X}}^*\,|\, \Vert \beta _d(x_d) \Vert = 0 \}$, that

$$\begin{aligned} \varepsilon (x_d,x_{di}) = - \alpha _d(x_d) - \sigma |\alpha _d(x_d)| = (1-\sigma ) \lambda (|\chi _0|) > 0, \end{aligned}$$

because $\sigma \in [0,1[$ and $\lambda (s) > 0$ for $s>0$. Therefore, there is no event on the set $\{x_d \in {\mathscr {X}} | \Vert \beta _d(x_d)\Vert = 0\} \cup \{0\}$. The study is then restricted to the set ${\mathscr {S}}_d^*= \{x_d \in {\mathscr {X}}^*| \Vert \beta _d(x_d)\Vert \ne 0\}$, where $\varOmega$ and $\varDelta$ are strictly positive by assumption. Rewriting the time derivative of the CLKF along the trajectories yields

$$\begin{aligned} \frac{dV}{dt}(x_d)&= \alpha _d(x_d) + \beta _d(x_d) \upsilon (x_{di}), \nonumber \\&= - \psi (x_d) + \beta _d(x_d) \bigl ( \upsilon (x_{di}) - \upsilon (x_d) \bigr ), \end{aligned}$$

(4.32)

when using the definition of $\upsilon (x_d)$ in (4.16) and (4.18), where $\psi$ is defined in (4.29). Let respectively define the level and the set

$$\begin{aligned}&\vartheta _i := V(x_{di}), \qquad \forall x_{di} \in {\mathscr {S}}_d, \\&{\mathscr {V}}_{\vartheta _i} := \{x_d \in {\mathscr {X}} | V(x_d) \le \vartheta _i\}. \end{aligned}$$

From the choice of the event function, it follows from (4.32) that $x_d$ belongs to ${\mathscr {V}}_{\vartheta } \subset {\mathscr {V}}_{\vartheta _i}$. Note that if $x_{di}$ belongs to ${\mathscr {S}}_d$, this is not necessarily the case for $x_d$ that can escape from this set. First see that, since (i) $\varOmega (x_d)$ is such that $\alpha _d(x_d)^2 + \varOmega (x_d) \beta _d(x_d) \varDelta (x_d) \beta _d(x_d)^T > 0$ for all $x_d \in {\mathscr {S}}_d^*$, and (ii) $\alpha _d(x_d)$ is necessarily nonzero on the frontier of ${\mathscr {S}}_d$ (except possibly at the origin)

$$\begin{aligned}&\frac{dV}{dt}(x_{di}) = - \psi (x_{di}) \le - \inf _{\underset{\small {\text{ s.t. }} V(x_{di})=\vartheta _i}{x_{di} \in {\mathscr {S}}_d}} \psi (x_{di}) =: -\varphi (\vartheta _i) < 0. \end{aligned}$$

(4.33)

Considering now the second time derivative of the CLKF

$$\begin{aligned}&\ddot{V}(x_d) = \left( \frac{\partial \alpha _d}{\partial x_d}(x_d) + \upsilon (x_{di})^T \frac{\partial \beta _d^T}{\partial x_d}(x_d) \right) \varTheta (x_d,x_{di}), \\&\text {with} \quad \varTheta (x_d,x_{di}) := \varPhi (x_\tau ) + g(x_\tau ) \upsilon (x_{di}), \nonumber \end{aligned}$$

(4.34)

where $\varPhi$ is defined in (4.4). By continuity of all the involved functions (except for $\varGamma$ in $\varPhi$ which is piecewise continuous but bounded by assumption), both terms can be bounded for all $x_d \in {\mathscr {V}}_{\vartheta _i}$ by the following upper bounds $\rho _1(\vartheta _i)$ and $\rho _2(\vartheta _i)$ such that

$$\begin{aligned}&\rho _1(\vartheta _i) := \sup _{\begin{array}{c} x_{di} \in {\mathscr {S}}_d \text{ s.t. } V(x_{di})=\vartheta _i \\ x_d \in {\mathscr {V}}_{\vartheta _i} \end{array}} \left\| \frac{\partial \alpha _d}{\partial x_d}(x_d) + \upsilon (x_{di})^T \frac{\partial \beta _d^T}{\partial x_d}(x_d) \right\| , \nonumber \\&\rho _2(\vartheta _i) := \sup _{\begin{array}{c} x_{di} \in {\mathscr {S}}_d \text{ s.t. } V(x_{di})=\vartheta _i \\ x_d \in {\mathscr {V}}_{\vartheta _i} \end{array}} \left\| \varTheta (x_d,x_{di}) \right\| , \nonumber \end{aligned}$$

where $\varTheta$ is defined in (4.34). Therefore, $\dot{V}$ is strictly negative at any event instant $t_i$ and cannot vanish until a certain time $\underline{\tau }(\vartheta _i)$ is elapsed (because its slope is positive). This minimal sampling interval is only depending on the level $\vartheta _i$. A bound on $\underline{\tau }(\vartheta _i)$ is given by the inequality

$$\begin{aligned} \frac{dV}{dt}(x_d) \le \frac{dV}{dt}(x_{di}) + \rho _1\rho _2(t-t_i),\qquad \forall x_d \in {\mathscr {V}}_{\vartheta _i}, \end{aligned}$$

that yields

$$\begin{aligned} \underline{\tau }(\vartheta _i) \ge \frac{\varphi (\vartheta _i)}{\rho _1(\vartheta _i)\rho _2(\vartheta _i)} > 0, \end{aligned}$$

where $\varphi$ is defined in (4.33). As a consequence, the event-based feedback (4.16)–(4.17) is semi-uniformly MSI . This ends the proof of Theorem 3.

1.2 Proof of Property 4

To prove the continuity of $\upsilon$ at the origin, one only needs to consider the points in $\mathscr {S}$ since $\upsilon (x_d) = 0$ if $\Vert \beta _d(x_d)\Vert = 0$. Then (4.16) gives

(4.35)

With the small control property (see Property 1), for any $\varepsilon > 0$, there is $\mu > 0$ such that for any $x_d \in {\mathscr {B}}(\mu )\backslash \{0\}$, there exists some u with $\Vert u\Vert \le \varepsilon$ such that $L_f^{*}V(x_d) + [L_gV_1(x_d)]^T u = \alpha _d(x_d) + \beta _d(x_d) u < 0$ and therefore $|\alpha _d(x_d)|<\Vert \beta _d(x_d)\Vert \varepsilon$. It follows

$$\begin{aligned} \Vert \upsilon (x_d)\Vert \le \frac{2\varepsilon \Vert \beta _d(x_d)\Vert \Vert \varDelta (x_d)\beta _d(x_d)^T\Vert }{\beta _d(x_d)\varDelta (x_d)\beta _d(x_d)^T} + \sqrt{\varOmega (x_d)\Vert \varDelta (x_d)\Vert }. \end{aligned}$$

Since the function $(v_1,v_2) \rightarrow \frac{\Vert v_1\Vert \Vert v_2\Vert }{v_1^Tv_2}$ is continuous w.r.t. its two variables at the origin where it equals 1, since $\varOmega$ and $\varDelta$ are also continuous, since $\varOmega (x_d) \Vert \varDelta (x_d)\Vert$ vanishes at the origin, for any $\varepsilon '$, there is some $\mu '$ such that $\forall x_d \in {\mathscr {B}}(\mu ')\backslash \{0\}$, $\Vert \upsilon (x_d)\Vert \le \varepsilon '$ which ends the proof of continuity.

1.3 Proof of Property 5

With $\varOmega$ defined as in (4.20), the feedback in (4.16) becomes

$$\begin{aligned} \upsilon (x_d) = - \beta _d(x_d)\varDelta (x_d)\omega (x_d) \end{aligned}$$

if the condition (4.19) is satisfied, which is obviously smooth on $\mathscr {X}$. Note that the expression of $\varOmega$ in (4.20) comes from the solution of (4.30), where $\omega$ only has to be smooth.