Designing Controllers with Predefined Convergence-Time Bound Using Bounded Time-Varying Gains

Abstract

Recently, there has been a great deal of attention in a class of controllers based on time-varying gains, called prescribed-time controllers, that steer the system’s state to the origin in the desired time, a priori set by the user, regardless of the initial condition. Furthermore, such a class of controllers has been shown to maintain a prescribed-time convergence in the presence of disturbances even if the disturbance bound is unknown. However, such properties require a time-varying gain that becomes singular at the terminal time, which limits its application to scenarios under quantization or measurement noise. This chapter presents a methodology to design a broader class of controllers, called predefined-time controllers, with a prescribed convergence-time bound. Our approach allows designing robust predefined-time controllers based on time-varying gains while maintaining uniformly bounded time-varying gains. We analyze the condition for uniform Lyapunov stability under the proposed time-varying controllers.

Appendix

1.1 Auxiliary Lemmas

Let us introduce the following Lemmas, on some properties of matrix $\textbf{Q}_{\rho }$ and the time-varying matrix $\textbf{K}_{\rho }(t)$.

Lemma 1

Let $\textbf{D}_{\rho }\in \mathbb {R}^{n\times n}$ and $\textbf{Q}_{\rho }\in \mathbb {R}^{n\times n}$ be defined as in (32). Then, $\textbf{Q}_{\rho }\in \mathbb {R}^{n\times n}$ is a lower triangular matrix satisfying

$$\begin{aligned} \textbf{J}+\textbf{A} =\textbf{Q}_{\rho }(\textbf{J}-\alpha \textbf{D}_{\rho })\textbf{Q}_{\rho }^{-1} \end{aligned}$$

(39)

where

$$\begin{aligned} \textbf{A}=\textbf{b}_{n}\textbf{b}_{1}^{T}(\textbf{J}-\alpha \textbf{D}_{\rho })^{n}\textbf{Q}_{\rho }^{-1} \end{aligned}$$

(40)

with $\textbf{b}_{n}=[0,\cdots ,0,1]^T\in \mathbb {R}^{n\times 1}$.

Proof

Notice that by construction $\textbf{Q}_{\rho }$ is a lower triangular matrix with ones over the diagonal. Moreover, $\textbf{J}$ is an upper shift matrix, thus

$$\begin{aligned} \textbf{J} \textbf{Q}_{\rho }= \begin{bmatrix} \textbf{b}_{1}^{T}(\textbf{J}-\alpha \textbf{D}_{\rho })\\ \vdots \\ \textbf{b}_{1}^{T}(\textbf{J}-\alpha \textbf{D}_{\rho })^{n-1}\\ \textbf{0}_n^T \end{bmatrix} \text {\,\, and\,\, } \textbf{J} \textbf{Q}_{\rho } +\textbf{A} \textbf{Q}_{\rho }= \begin{bmatrix} \textbf{b}_{1}^{T}(\textbf{J}-\alpha \textbf{D}_{\rho })\\ \vdots \\ \textbf{b}_{1}^{T}(\textbf{J}-\alpha \textbf{D}_{\rho })^{n-1}\\ \textbf{b}_{1}^{T}(\textbf{J}-\alpha \textbf{D}_{\rho })^{n} \end{bmatrix} \end{aligned}$$

where $\textbf{0}_n\in \mathbb {R}^n$ is a zero vector. Therefore, $\textbf{J} \textbf{Q}_{\rho } -\textbf{A} \textbf{Q}_{\rho }=\textbf{Q}_{\rho }(\textbf{J}-\alpha \textbf{D}_{\rho })$ which completes the proof.    $\blacksquare$

Lemma 2

Let $\kappa (t)$ be given as in (7), with $\eta$ as in (18), and let

$$\begin{aligned} \textbf{K}_{\rho }(t):=\text {diag}(\kappa (t)^{-\rho },\kappa (t)^{1-\rho },\ldots ,\kappa (t)^{n-\rho -1}), \end{aligned}$$

where $\rho \in [0,n]$. Then, the following identities hold:

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\textbf{K}_{\rho }(t)^{-1} &= -\alpha \kappa (t)\textbf{D}_{\rho }\textbf{K}_{\rho }(t)^{-1} \end{aligned}$$

(41)

$$\begin{aligned} \textbf{K}_{\rho }^{-1}(t)\textbf{J}\textbf{K}_{\rho }(t)&=\kappa (t)\textbf{J}. \end{aligned}$$

(42)

Proof

A direct calculation yields

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\textbf{K}_{\rho }(t)^{-1} &= \frac{\textrm{d}}{\textrm{d}t}\text {diag}(\kappa (t)^{\rho },\kappa (t)^{\rho -1},\ldots ,\kappa (t)^{\rho -n+1})\\ &=\dot{\kappa }(t)\kappa (t)^{-1}\text {diag}(\rho \kappa (t)^{\rho },\rho -1\kappa (t)^{\rho -1},\ldots ,\rho -n+1\kappa (t)^{\rho -n+1}). \end{aligned}$$

Since $\dot{\kappa }(t)\kappa (t)^{-1}=\alpha \kappa (t)$, Eq. (41) follows trivially by definition of $\textbf{D}_{\rho }$.

Now, to show that (42) holds, notice that since $\textbf{J}$ is an upper shift matrix. Thus,

$$\begin{aligned} \textbf{K}_{\rho }^{-1}(t)\textbf{J}\textbf{K}_{\rho }(t)&=\kappa (t) \textbf{K}_{\rho }^{-1}(t) \begin{bmatrix} 0 &{} \kappa (t)^{-\rho } &{} 0 &{} \cdots &{}0 &{} 0\\ 0 &{} 0 &{} \kappa (t)^{1-\rho } &{} \cdots &{}0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} \kappa (t)^{n-3+\rho } &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} \kappa (t)^{n-2+\rho }\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ \end{bmatrix}\\ &=\kappa (t)\textbf{J}, \end{aligned}$$

which completes the proof.    $\blacksquare$

1.2 Some Admissible Auxiliary Controllers

Theorem 2

([1, Theorem 3]) Consider a controller

$$\begin{aligned} u = -\left[ (a|x|^{p} + b|x|^{q})^k+\zeta \right] \text{ sign }(x), \end{aligned}$$

(43)

where $\zeta \ge \varDelta$, $a,b,p,q,k>0$ are system parameters which satisfy the constraints $kp<1$, and $kq>1$. Then, the origin of (8) under the controller (43) is fixed-time stable and the settling-time function satisfies $\sup _{x_0 \in \mathbb {R}} T(x_0)=\gamma$, where

$$\begin{aligned} \gamma =\frac{\varGamma \left( m_p\right) \varGamma \left( m_q\right) }{a^{k}\varGamma (k) (q-p)}\left( \frac{a}{b}\right) ^{m_p}, \end{aligned}$$

(44)

with $m_p=\frac{1-k p}{q-p}$ and $m_q=\frac{k q-1}{q-p}$.

Theorem 3

([1, Theorem 4]) Consider a second-order perturbed chain of integrators, and let $a_1,a_2,b_1,b_2,p,q,k>0$, $kp<1$, $kq>1$, $T_{f_1},T_{f_2}>0$, $\zeta \ge \varDelta$, and

$$ \gamma _1=\frac{\varGamma \left( \frac{1}{4}\right) ^2 }{2a_1^{1/2}\varGamma \left( \frac{1}{2}\right) }\left( \frac{a_1}{b_1}\right) ^{1/4}, \gamma _2=\frac{\varGamma \left( m_{p}\right) \varGamma \left( m_{q}\right) }{a_2^{k}\varGamma (k) (q-p)}\left( \frac{a_2}{b_2}\right) ^{m_{p}}, $$

with $m_{p}=\frac{1-kp}{q-p}$ and $m_{q}=\frac{kq-1}{q-p}$. If the control input is selected as

$$\begin{aligned} u=-\left[ \frac{\gamma _2}{T_{f_2}}\left( a_2\left|\sigma \right|^{p}+b_2\left|\sigma \right|^{q}\right) ^{k}+\frac{\gamma _1^2}{2T_{f_1}^2}\left( a_1+3b_1x_1^2\right) +\zeta \right] \text{ sign }(\sigma ), \end{aligned}$$

where the sliding variable $\sigma$ is defined as

$$\begin{aligned} \sigma =x_2+\left\lfloor \left\lfloor x_2\right\rceil ^2+\frac{2\gamma _1^2}{T_{f_1}^2}\left( a_1\left\lfloor x_1\right\rceil ^1+b_1\left\lfloor x_1\right\rceil ^3\right) \right\rceil ^{1/2}, \end{aligned}$$

then the origin $(x_1,x_2)=(0,0)$ of system (4), with $n=2$, is fixed-time stable with UBST given by $T_f=T_{f_1}+T_{f_2}$.