Output Stabilization of Lurie-Type Nonlinear Systems in a Given Set

Abstract

This paper considers the problem of stabilizing the output variables of a Lurie-type nonlinear system in a given set at any time instant. A special output transformation is used to reduce the original constrained problem to that of analyzing the input-to-state stability of a new extended system without constraints. For this system, nonlinear control laws are obtained using the technique of linear matrix inequalities. Examples are given to illustrate the effectiveness of the method proposed and confirm the theoretical conclusions.

This work was performed in the Institute for Problems in Mechanical Engineering, the Russian Academy of Sciences, under the support of state order no. 121112500298-6 (The Unified State Information System for Recording Research, Development, Design, and Technological Work for Civilian Purposes).

Authors and Affiliations

1. Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

   B. H. Nguyen

2. ITMO University, St. Petersburg, Russia

   B. H. Nguyen

Corresponding author

Correspondence to B. H. Nguyen.

This paper was recommended for publication by L.B. Rapoport, a member of the Editorial Board

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APPENDIX

Proof of Theorem 2. Substituting (11) into (8) yields the closed loop system

We choose a Lyapunov function of the form V = \(\frac{1}{2}{{\varepsilon }^{2}}\). Its total time derivative along the solutions of (A.1) is given by

For \(V\; \geqslant \;c\), we require the condition \(\dot {V}\;\leqslant \; - {\kern 1pt} 2\alpha V\), where α is any known positive number, i.e., \(\dot {V}\) < 0 ∀ε ∉ Ω. Due to LGϕ \(\leqslant \) |LGϕ| \(\leqslant \) μ||LG|| ||C|| |x| and the constraint |ψ| \(\leqslant \) κ, the above conditions can be written as

Denoting z = col{ε, ψ}, we represent (A.3) in the matrix form

By the S-procedure [13], inequalities (A.4) hold under conditions (12). Hence, system (A.1) is input-to-state stable, and the variable ε(t) is bounded. Owing to the transformation (4), the output y(t) is also bounded, and the state vector x(t) of system (1) possesses the same property accordingly. Therefore, the control variable u(t) in (11) is bounded as well. Due to Theorem 1, the target condition (3) holds.

The proof of Theorem 2 is complete.

Proof of Proposition 1. Obviously, the matrix MN is symmetric. Let λ_i, x_i, i = 1, …, 2n, be the eigenvalues and eigenvectors of the matrix MN, respectively. Then

Hence, the values λ_i can be expressed as

Since M \( \succ \) 0 and N = N^T \( \prec \) 0, we obtain NMN \( \succ \) 0, i.e., x^TNMNx > 0 ∀x ≠ 0. In view of x^TNx < 0 ∀x ≠ 0, it follows that λ_i < 0, i = 1, …, 2n. All eigenvalues of the symmetric matrix MN are negative, so the matrix MN is negative definite.

The proof of Proposition 1 is complete.

Proof of Theorem 3. We choose a Lyapunov function of the form V = \(\frac{1}{2}{{\varepsilon }^{{\text{T}}}}\varepsilon \). Its total time derivative along the solutions of (14) is given by

Formula (A.5) can be written as

where

Considering \(\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)\;\leqslant \;\bar {\sigma }I\), we estimate \({{\dot {V}}_{2}}\) as

Based on this inequality, the condition \(\dot {V}\;\leqslant \;0\) is equivalent to \({{\dot {V}}_{1}}\;\leqslant \;0\). In the case under study, \({{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}^{{ - 1}}}\) is a matrix and cannot be neglected when analyzing the sign definiteness of \(\dot {V}\), in contrast to the previous section. For V \( \geqslant \) c, we require the condition \(\dot {V}\;\leqslant \; - {\kern 1pt} 2\alpha V\), where α is any known positive number. Due to the constraints |ψ| \(\leqslant \) κ, the above conditions can be written as

Denoting z = col{ε, ψ}, z ∈ \({{\mathbb{R}}^{{2m}}}\), we represent (A.7) in the matrix form

By the S-procedure, inequalities (A.8) hold under the conditions

The first inequality in (A.9) is equivalent to

Since \({{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}^{{ - 1}}} \succ 0\), by Proposition 1, the latter inequality holds if, for any β > 0,

Due to condition (A.7), it is required to ensure \(\dot {V}\) < 0 for all ε from the set {ε ∈ \({{\mathbb{R}}^{m}}\) : |ε| \( \geqslant \) \(\sqrt {2c} \), c > 0}. In addition, for all ε from this set, we have an interval uncertainty in (A.11) with 0 \( \prec \) \(\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}\) \( \preceq \) \(\bar {\sigma }I\). Conditions (A.7) will be valid if the LMIs (15) are feasible for any σ ∈ (0, \(\bar {\sigma }\)]. Moreover, obviously, there always exist a matrix K and τ₁, τ₂ > 0 such that (15) are feasible. Indeed, using Schur’s complement lemma [13], we write (15) as

For a given number c > 0 and fixed numbers σ, α, and β, inequalities (A.12) always have finite solutions (K, τ₁, τ₂). Thus, according to Theorem 1, the control law (13) with the gain matrix K satisfying (15) ensures the target condition (3).

The proof of Theorem 3 is complete.

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