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.
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Notes
- 1.
In the spirit of Filippov’s interpretation of differential equations, solutions of (3) are understood as any absolutely continuous function that satisfies the differential inclusion obtained by applying the Filippov regularization to \(\textbf{f}(\bullet , \bullet )\) (See [9, Page 85]), allowing us to consider \(\textbf{f}(\bullet , \bullet )\) discontinuous in the first argument. In the usual Filippov’s interpretation, it is assumed that \(\Vert \textbf{f}(\textbf{x}, t)\Vert \) has an integrable majorant function of time for any \(\textbf{x}\), ensuring existence and uniqueness of solutions in forward time. However, in this work we deal with \(\textbf{f}(\textbf{x}, t)\) for which no majorant function exists, but existence and uniqueness of solutions are still guaranteed by an argument similar to [2]. In particular, existence of solutions follows directly from the equivalence of solutions to a well-posed Filippov system via the time-scale transformation.
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Acknowledgements
Work partially supported by the Christian Doppler Research Association, the Austrian Federal Ministry of Labour and Economy and the National Foundation for Research, Technology and Development, by Agencia I+D+i grant PICT 2018-01385, Argentina and by Consejo Nacional de Ciencia y Tecnología (CONACYT-Mexico) scholarship with grant 739841.
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Appendix
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
where
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
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
where \(\rho \in [0,n]\). Then, the following identities hold:
Proof
A direct calculation yields
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,
which completes the proof. \(\blacksquare \)
1.2 Some Admissible Auxiliary Controllers
Theorem 2
([1, Theorem 3]) Consider a controller
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
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
with \(m_{p}=\frac{1-kp}{q-p}\) and \(m_{q}=\frac{kq-1}{q-p}\). If the control input is selected as
where the sliding variable \(\sigma \) is defined as
then the origin \((x_1,x_2)=(0,0)\) of system (4), with \(n=2\), is fixed-time stable with UBST given by \(T_f=T_{f_1}+T_{f_2}\).
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Aldana-López, R., Seeber, R., Haimovich, H., Gómez-Gutiérrez, D. (2023). Designing Controllers with Predefined Convergence-Time Bound Using Bounded Time-Varying Gains. In: Oliveira, T.R., Fridman, L., Hsu, L. (eds) Sliding-Mode Control and Variable-Structure Systems. Studies in Systems, Decision and Control, vol 490. Springer, Cham. https://doi.org/10.1007/978-3-031-37089-2_3
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