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.

APPENDIX

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

$$\dot {\varepsilon } = {{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}^{{ - 1}}}\left[ { - K\varepsilon - \mu \operatorname{sgn} (\varepsilon )\left\| {LG} \right\|\left\| C \right\|\left| x \right| + LG\phi + \psi } \right].$$

(A.1)

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

$$\dot {V} = \varepsilon \dot {\varepsilon } = \varepsilon {{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}^{{ - 1}}}\left[ { - K\varepsilon - \mu \operatorname{sgn} (\varepsilon )\left\| {LG} \right\|\left\| C \right\|\left| x \right| + LG\phi + \psi } \right].$$

(A.2)

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

$$\begin{gathered} ( - K + \alpha ){{\varepsilon }^{2}} + \varepsilon \psi \;\leqslant \;0\;\forall (\varepsilon ,\;\psi ): \hfill \\ 0.5{{\varepsilon }^{2}}\; \geqslant \;c,\;\,{{\psi }^{2}}\;\leqslant \;{{\kappa }^{2}}. \hfill \\ \end{gathered} $$

(A.3)

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

$$\begin{gathered} {{z}^{{\text{T}}}}\left[ {\begin{array}{*{20}{c}} { - K + \alpha }&{0.5} \\ \star &0 \end{array}} \right]z\;\leqslant \;0, \\ {{z}^{{\text{T}}}}\left[ {\begin{array}{*{20}{c}} { - 0.5}&0 \\ \star &0 \end{array}} \right]z\;\leqslant \; - {\kern 1pt} c,\quad {{z}^{{\text{T}}}}\left[ {\begin{array}{*{20}{c}} 0&0 \\ \star &1 \end{array}} \right]z\;\leqslant \;{{\kappa }^{2}}. \\ \end{gathered} $$

(A.4)

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

$$x_{i}^{{\text{T}}}NMN{{x}_{i}} = {{\lambda }_{i}}x_{i}^{{\text{T}}}N{{x}_{i}}.$$

Hence, the values λ_i can be expressed as

$${{\lambda }_{i}} = \frac{{x_{i}^{{\text{T}}}NMN{{x}_{i}}}}{{x_{i}^{{\text{T}}}N{{x}_{i}}}}.$$

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

$$\begin{aligned} \dot {V} = {{\varepsilon }^{{\text{T}}}}\dot {\varepsilon } = {{\varepsilon }^{{\text{T}}}}{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}^{{ - 1}}}\left[ { - K{{\varepsilon }^{{^{{^{{^{{^{{}}}}}}}}}}}} \right. \\ - \;\left. {\bar {\sigma }\mu \operatorname{sgn} (\varepsilon )\left\| {{\kern 1pt} {{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}}^{{ - 1}}}} \right\|\left\| {LG} \right\|\left\| C \right\|\left| x \right| + LG\phi + \psi } \right]. \\ \end{aligned} $$

(A.5)

Formula (A.5) can be written as

$$\dot {V} = {{\dot {V}}_{1}} + {{\dot {V}}_{2}},$$

(A.6)

where

$${{\dot {V}}_{1}} = - {{\varepsilon }^{{\text{T}}}}{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}^{{ - 1}}}K\varepsilon + {{\varepsilon }^{{\text{T}}}}{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}^{{ - 1}}}\psi ,$$

$$\begin{aligned} {{{\dot {V}}}_{2}} = - {{\varepsilon }^{{\text{T}}}}{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}^{{ - 1}}}{\text{sgn}}(\varepsilon )\bar {\sigma }\mu \left\| {{{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}}^{{ - 1}}}} \right\|\left\| {LG} \right\|\left\| C \right\|\left| x \right| \\ \quad \quad + \;{{\varepsilon }^{{\text{T}}}}{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}^{{ - 1}}}LG\phi . \\ \end{aligned} $$

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

$$\begin{gathered} {{{\dot {V}}}_{2}}\;\leqslant \; - \left( {\sum\limits_{i = 1}^{v} {\left| {{{\varepsilon }_{i}}} \right|} } \right){{{\bar {\sigma }}}^{{ - 1}}}\bar {\sigma }\mu \left\| {{\kern 1pt} {{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}}^{{ - 1}}}} \right\|\left\| {LG} \right\|\left\| C \right\|\left| x \right| \\ + \;\mu \left| \varepsilon \right|\left\| {{\kern 1pt} {{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}}^{{ - 1}}}} \right\|\left\| {LG} \right\|\left\| C \right\|\left| x \right|\;\leqslant \;0. \\ \end{gathered} $$

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

$$\begin{gathered} - {{\varepsilon }^{{\text{T}}}}\left[ {{{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}}^{{ - 1}}}K + \alpha I} \right]\varepsilon + {{\varepsilon }^{{\text{T}}}}{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}^{{ - 1}}}\psi \;\leqslant \;0 \\ \forall (\varepsilon ,\psi ):0.5{{\varepsilon }^{{\text{T}}}}\varepsilon \; \geqslant \;c,\;\,{{\psi }^{{\text{T}}}}\psi \;\leqslant \;{{\kappa }^{2}}. \\ \end{gathered} $$

(A.7)

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

$$\begin{gathered} {{z}^{{\text{T}}}}\left[ {\begin{array}{*{20}{c}} { - {{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}}^{{ - 1}}}K - \alpha I}&{0.5{{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}}^{{ - 1}}}} \\ \star &0 \end{array}} \right]z\;\leqslant \;0, \\ {{z}^{{\text{T}}}}\left[ {\begin{array}{*{20}{c}} { - 0.5I}&0 \\ \star &0 \end{array}} \right]z\;\leqslant \; - {\kern 1pt} c,\quad {{z}^{{\text{T}}}}\left[ {\begin{array}{*{20}{c}} 0&0 \\ \star &I \end{array}} \right]z\;\leqslant \;{{\kappa }^{2}}. \\ \end{gathered} $$

(A.8)

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

$$\begin{gathered} \left[ {\begin{array}{*{20}{c}} { - {{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}}^{{ - 1}}}K - \alpha I + 0.5{{\tau }_{1}}I}&{0.5{{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}}^{{ - 1}}}} \\ \star &{ - {{\tau }_{2}}I} \end{array}} \right] \prec 0, \\ - c{{\tau }_{1}} + {{\kappa }^{2}}{{\tau }_{2}}\;\leqslant \;0. \\ \end{gathered} $$

(A.9)

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

$$\begin{gathered} \left[ {\begin{array}{*{20}{c}} {{{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}}^{{ - 1}}}}&0 \\ \star &{{{{\left( {\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \right)}}^{{ - 1}}}} \end{array}} \right] \\ \times \;\left[ {\begin{array}{*{20}{c}} { - K + (0.5{{\tau }_{1}} - \alpha )\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}}&{0.5I} \\ \star &{ - {{\tau }_{2}}\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \end{array}} \right] \prec 0. \\ \end{gathered} $$

(A.10)

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,

$$\left[ {\begin{array}{*{20}{c}} { - K + (0.5{{\tau }_{1}} - \alpha )\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}}&{0.5I} \\ \star &{ - {{\tau }_{2}}\frac{{\partial {{\Phi }^{{ - 1}}}(\varepsilon ,\;t)}}{{\partial \varepsilon }}} \end{array}} \right] \preceq - \beta I \prec 0.$$

(A.11)

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

$$\begin{gathered} - {{\tau }_{2}}\sigma + \beta < 0, \\ - K + [(0.5{{\tau }_{1}} - \alpha )\sigma + \beta ]I + \frac{1}{{{{\tau }_{2}}\sigma - \beta }}I \preceq 0, \\ - c{{\tau }_{1}} + {{\kappa }^{2}}{{\tau }_{2}}\;\leqslant \;0, \\ {{\tau }_{1}} > 0,\quad {{\tau }_{2}} > 0, \\ \alpha > 0,\quad \beta > 0,\quad 0 < \sigma \;\leqslant \;\bar {\sigma }. \\ \end{gathered} $$

(A.12)

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.