State Feedback Optimal L₂-Induced Control of Nonlinear Systems Utilizing Universal Approximation

1. Introduction

The control of nonlinear plants, has been a challenging problem to the control community for many years. A plethora of methods and approaches have been developed over the years, where here we cite just a few. An approach that generalizes linear systems is presented in [1] where a nonlinear plant is considered, but within the class of Lipschitz nonlinear systems. The method utilizes the Lipschitz bounds to derive LMIs that guarantee stability and $H_{\infty }$ performance. Another approach that is confined to another specific class of Lur’e [2] type sector bounded [3] nonlinear systems, where [4] deals with stochastic, in probability, stabilization of the SAR (Stochastic Anti Resonance, see also [5]), whereas [6] deals with the deterministic case. It should be noted that sector bounded systems strongly relate to Hopfield networks [7]. The present paper focuses on the deterministic continuous-time problem, and is aimed at bounded-real-lemma like characterization of the $L_2$-induced norm (actually $H_{\infty }$-norm in linear systems), and its application for control synthesis. A related discrete-time problem has been considered in [8].

We note that the scope of applications of Lur’e type systems control, is much wider from what may seem to be at first sight. This wider scope is enabled by the universal approximation theorem [9] for systems that are not a priori modeled with sector bounded uncertainties. More precisely, when the model is not a priori sector bounded, one may invoke the universal approximation theorem to fit a neural network with a single hidden layer, with, e.g., a tanh activation function and a linear output layer. Since such networks provide an approximation with arbitrarily small error for an wide enough hidden layer. Consequently, the model of system readily is closely approximated as a sector bounded Lur’e system. In the present paper it is shown that using the state feedback optimal control approach developed for the sector-type nonlinearities in combination with the Cybenko’s universal approximation theorem, one can derive $L_2$-induced norm boundedness conditions for dynamic systems with wider classes of nonlinearities. This procedure is illustrated in Section 6 for the optimal $L_2$-induced norm control of a Van der Pol oscillator which nonlinear term is not of bounded-sector type. The model of this system has been intensively studied in the context of the control of nonlinear oscillators. For instance in [10], global stabilization of the Van der Pol system is analysed. Different approaches to control such systems have been proposed, including PID (Proportional-Integral-Derivative) controllers (see e.g., [11]), optimal controllers (e.g., [12,13]), nonlinear adaptive ([14] and predictive controllers ([15]), neural networks based control ([16]), to mention just a few references from the vast literature devoted to this topic.

The main contributions of this paper are the following:

•

Characterization of the $L_2$-induced norm boundedness conditions for a class of systems with multiple sector-type nonlinearities;

•

Sufficient conditions for the existence of a state feedback controller providing an imposed level of the $L_2$ induced norm for systems with sector bounded nonlinearities;

•

A loop transformation procedure to reduce the conservatism of the $L_2$ boundedness conditions;

•

Use of the universal approximation theorem to approximate general nonlinearities with sums of sector bounded nonlinearities;

•

Illustration of the derived theoretical results for the optimal $L_2$ induced norm control of the Van der Pol oscillator.

The remainder of this paper is organized as follows. Section 2 presents the control problem formulation whereas in Section 3 an $L_2$-induced norm characterization for systems with multiple sector bounded nonlinearities is provided. In Section 4, the state feedback optimal $L_2$-induced control problem is solved. In order to reduce the conservatism of the boundedness conditions, a loop transformation procedure is proposed in Section 5. Further, using the universal approximation theorem, an $L_2$-induced norm control problem is considered for the forced Van der Pol oscillator. The paper ends with some final conclusions.

Notation. Throughout the paper the superscript `⊤’ stands for matrix transposition, $\mathcal{R}$ denotes the set of scalar real numbers whereas ${\mathcal{Z}}_{+}$ stands for the non-negative integers. Moreover, ${\mathcal{R}}^n$ denotes the n dimensional Euclidean space, ${\mathcal{R}}^{n\times m}$ is the set of all $n\times m$ real matrices, and the notation $P>0$ ($P\ge 0$), for $P\in {\mathcal{R}}^{n\times n}$ means that P is symmetric and positive definite (positive semi-definite). The trace of a matrix Z is denoted by $Tr\left(Z\right)$, and $\left|v\right|$ denotes the Euclidian norm of an n-dimensional vector v. Finally note that the terms Lyapunov and Riccati equations in this paper, refer to generalised versions of the standard equations appearing in the $H_2$ and $H_{\infty }$ control literature. Further we denote by $vec$ the operation stacks all columns of a matrix into $R^{n^2}$ and by $vech$ the opeator that stacks the lower triangular part of a symmetric matrix into $R^{n(n+1)/2}$. We denote the inverse operation by $vec{h}^{-1}$. Also $col\{x_1,x_2....\}$ with scalars $x_i$, just denotes the column vector, ${[x_1,x_2,...]}^T$.

2. Optimal Control Problem Formulation

Consider the following deterministic nonlinear system

$$\begin{array}{c}\hfill \begin{array}{ccc}\dot{x}\left(t\right)\hfill & =& Ax\left(t\right)+Ff\left(y\left(t\right)\right)+B_{1}w\left(t\right)+B_{2}u\left(t\right)\hfill \\ z\left(t\right)\hfill & =& Mx\left(t\right)+N{u}_2\left(t\right)\hfill \end{array}\end{array}$$

(1)

in which $x\in {\mathbb{R}}^n$ is the state vector, $w\in {\mathbb{R}}^{m_1}$ denotes the exogenous input, $u\in {\mathbb{R}}^{m_2}$ is the control input and $z\in {\mathbb{R}}^p$ is the system output satisfying the condition $M^{\top }N=0$. It is assumed that the nonlinear term $f\left(y\left(t\right)\right)\in {\mathbb{R}}^q$ has the following form

$$\begin{array}{c}\hfill f\left(y\right)=\left[\begin{array}{c}{f}_1\left(y_1\right)\\ ⋮\\ f_q\left(y_q\right)\end{array}\right]\end{array}$$

(2)

in which $f_i:{\mathbb{R}}^n\to \mathbb{R}$ are sector-type nonlinearities satisfying the conditions $f_i\left(y_i\right)(f_i\left(y_i\right)-{\sigma }_i{y}_i)\le 0$ and where $y_i=C_{i}x$ with $C_i\in {\mathbb{R}}^{1\times n}$, $i=1,\dots ,q$.

The problem consists in determining a state feedback control law $u\left(t\right)=Kx\left(t\right)$ such that for a given $\gamma >0$, the resulting system is stable and ${\int }_0^{\infty }\left(z^{\top }\left(t\right)z\left(t\right)-{\gamma }^2{w}^{\top }\left(t\right)w\left(t\right)\right)dt<0$ for all $w\left(t\right)\in L^2([0,\infty ),{\mathbb{R}}^{m_1})$. To this end, we first characterize the induced norm.

3. $L_2$-Induced Norm Characterization

The following result provides sufficient conditions for the boundedness of the $L_2$-induced norm of the system with multiple nonlinearities.

Lemma 1.

Consider the system

$$\begin{array}{c}\hfill \begin{array}{ccc}\dot{x}\left(t\right)\hfill & =& Ax\left(t\right)+Ff\left(y\right(t\left)\right)+Dw\left(t\right)\hfill \\ z\left(t\right)\hfill & =& Lx\left(t\right)\hfill \end{array}\end{array}$$

(3)

where the nonlinearities $f\left(y\right)$ have the form (2). If there exist $P>0$, $\Lambda =\mathrm{diag}\{{\lambda }_1,\dots ,{\lambda }_q\}>0$ and $T=\mathrm{diag}\{{\tau }_1,\dots ,{\tau }_q\}\ge 0$ such that

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}{A}^{\top }P+PA+L^{\top }L& A^{\top }{C}^{\top }\Lambda +PF+{\displaystyle \frac{1}{2}}{C}^{\top }ST& PD\\ {(1,2)}^T& \Lambda CF+F^{\top }{C}^{\top }\Lambda -T& \Lambda CD\\ {(1,3)}^T& {(2,3)}^T& -{\gamma }^{2}I\end{array}\right]<0\end{array}$$

(4)

in which $C\in {\mathbb{R}}^{q\times n}$ has the rows $C_i,i=1,\dots ,q$ and $S:=\mathrm{diag}\{{\sigma }_1,\dots ,{\sigma }_q\}$, then the system (3) is stable and ${\int }_0^{\infty }\left(z^{\top }\left(t\right)z\left(t\right)-{\gamma }^2{w}^{\top }\left(t\right)w\left(t\right)\right)dt<0$ for all $w\left(t\right)\in L^2([0,\infty ),{\mathbb{R}}^{m_1})$.

Proof. 

Consider the Lyapunov function candidate

$$\begin{array}{c}\hfill V\left(x\right)=x^{\top }Px+2\sum _{i=1}^q{\lambda }_i{\int }_0^{y_i}{f}_i\left(s\right)ds.\end{array}$$

Denoting $\Lambda =\mathrm{diag}\{{\lambda }_1,\dots ,{\lambda }_q\}$, direct computations gives that

$$\begin{array}{c}\hfill \begin{array}{c}\dot{V}\left(x\right)=\left(x^{\top }{A}^{\top }+f^{\top }{F}^{\top }+w^{\top }{D}^{\top }\right)\left(Px+C^{\top }\Lambda f\right)\\ +\left(x^{\top }P+F^{\top }\Lambda C\right)\left(Ax+Ff+Dw\right).\end{array}\end{array}$$

(5)

Therefore, $\dot{V}\left(x\right)$ may be expressed as

$$\begin{array}{c}\hfill \dot{V}\left(x\right)={\xi }^{\top }{\mathcal{F}}_0\xi \end{array}$$

(6)

where by notation $\xi :={\left[x^{\top }{f}^{\top }{w}^{\top }\right]}^{\top }$ and

$$\begin{array}{c}\hfill {\mathcal{F}}_0:=\left[\begin{array}{ccc}{A}^{\top }P+PA& A^{\top }{C}^{\top }\Lambda +PF& PD\\ {(1,2)}^{\top }& \Lambda CF+F^{\top }{C}^{\top }\Lambda & \Lambda CD\\ {(1,3)}^{\top }& {(2,3)}^{\top }& 0\end{array}\right]\end{array}$$

(7)

Using the norm condition ${\int }_0^{\infty }\left(z^{\top }\left(t\right)z\left(t\right)-{\gamma }^2{w}^{\top }\left(t\right)w\left(t\right)\right)dt<0$ it is required that $dV/dt+z^{\top }z-{\gamma }^2{w}^{\top }w<0$, where the latter condition can be expressed as ${\xi }^{\top }({\mathcal{F}}_0+{\mathcal{F}}_w)\xi \le 0$, with

$$\begin{array}{c}\hfill {\mathcal{F}}_w:=\left[\begin{array}{ccc}{L}^{\top }L& 0& 0\\ 0& 0& 0\\ 0& 0& -{\gamma }^{2}I\end{array}\right].\end{array}$$

(8)

Define

$$\begin{array}{c}\hfill {\mathcal{F}}_c:=\left[\begin{array}{ccc}0& {\displaystyle \frac{1}{2}}{C}^{\top }ST& 0\\ {\displaystyle \frac{1}{2}}TSC& -T& 0\\ 0& 0& 0\end{array}\right]\end{array}$$

with the diagonal matrix T as introduced in the statement. Noticing that ${\xi }^{\top }{\mathcal{F}}_c\xi =-{\Sigma }_{i=1}^q{\tau }_i{f}_i\left(y_i\right)(f_i\left(y_i\right)-{\sigma }_i{y}_i)\ge 0,$ according with the $\mathcal{S}$ procedure for quadratic terms and strict inequalities (see e.g., [3]), it follows that if there exists ${\tau }_1\ge 0,\dots ,{\tau }_q\ge 0$ such that ${\xi }^{\top }\left({\mathcal{F}}_0+{\mathcal{F}}_w+{\mathcal{F}}_c\right)\xi <0$ for all $\xi :={\left[x^{\top }{f}^{\top }{w}^{\top }\right]}^{\top }$, then ${\xi }^{\top }\left({\mathcal{F}}_0+{\mathcal{F}}_w\right)\xi <0$ for all $\xi$ for which $-f_i\left(y_i\right)(f_i\left(y_i\right)-{\sigma }_i{y}_i)\ge 0,i=1,\dots ,q$ , namely for all $y_i$ satisfying the sector bounded constraints. Then from the condition ${\mathcal{F}}_0+{\mathcal{F}}_w+{\mathcal{F}}_c<0$, one directly obtains the condition (4) from the statement. □

4. State Feedback Optimal $L_2$-Induced Control

Based on the above lemmas one can derive the following result:

Theorem 1.

If the following matrix inequality

$$\begin{array}{c}\hfill \left[\begin{array}{ccccc}{\mathcal{M}}_{11}(X,Y)& {\mathcal{M}}_{12}(X,Y,\Lambda ,T)& B_1& Y^{\top }{N}^{\top }& X{M}^{\top }\\ {(1,2)}^{\top }& \Lambda CF+F^{\top }{C}^{\top }\Lambda -T& \Lambda C{B}_1& 0& 0\\ {(1,3)}^{\top }& {(2,3)}^{\top }& -{\gamma }^2{I}_{m_1}& 0& 0\\ {(1,4)}^{\top }& 0& 0& -I_p& 0\\ {(1,5)}^{\top }& 0& 0& 0& -I_p\end{array}\right]<0\end{array}$$

(9)

where

$$\begin{array}{c}\hfill \begin{array}{ccc}{\mathcal{M}}_{11}(X,Y)\hfill & :=& X{A}^{\top }+AX+Y^{\top }{B}_2^{\top }+B_{2}Y\hfill \\ {\mathcal{M}}_{12}(X,Y,\Lambda ,T)\hfill & :=& X{A}^{\top }{C}^{\top }\Lambda +Y^{\top }{B}_2^{\top }{C}^{\top }\Lambda +F+{\displaystyle \frac{1}{2}}X{C}^{\top }ST,\hfill \end{array}\end{array}$$

is feasible with respect to $Y\in {\mathbb{R}}^{m_2\times n}$, $X>0$, $P>0$, $\Lambda =\mathrm{diag}\{{\lambda }_1,\dots ,{\lambda }_q\}>0$ and $T=\mathrm{diag}\{{\tau }_1,\dots ,{\tau }_q\}>0$, then the state-feedback control law $u_2\left(t\right)=Kx\left(t\right)$ with $K=Y{X}^{-1}$ stabilizes the system (1) and ensures the $L_2$ condition ${\int }_0^{\infty }\left(z^{\top }\left(t\right)z\left(t\right)-{\gamma }^2{w}^{\top }\left(t\right)w\left(t\right)\right)dt<0$ all $w\left(t\right)\in L^2([0,\infty ),{\mathbb{R}}^{m_1})$.

Proof. 

The proof directly results applying the Lemma for $D=B_1$ and for $L=M+N{B}_{2}K$, multiplying the inequality (4) to the left and to the right by $\mathrm{diag}\{P^{-1},I,I,I\}$, denoting $X:=P^{-1}$ and $Y:=K{P}^{-1}$ and using Schur complement arguments. □

5. Loop Transformation and Reduced Conservatism

One can apply the above result for synthesis of controllers for general nonlinear plants, that can be represented by the system (1). We note that far more general models of plants can be represented so, also in cases where system nonlinearities are not a priori sector bounded. In such cases, one may invoke the universal approximation theorem [9] to systems where a single hidden layer, with, e.g., a tanh activation function and a linear output layer, provides an approximation with arbitrarily small error for an arbitrarily wide hidden layer. In such cases, the model of system (1) readily becomes relevant, as the approximate function is now sector bounded. However, the formulation of (1) embeds possible conservatism, as one can express $Ax+Ff=(A+F\mu C)x+F(f-\mu Cx)$. In such a case, ${\sigma }_i$ are replaced by ${\sigma }_i-\mu$ provided $0<\mu <{\sigma }_i$.

We now aim at conservatism reduction. To this end we first consider a non symmetric sector bound. Recall that $Ax+Ff=(A+\mu FC)x+F(f-\mu Cx)$ and note that $\overline{f}=f-\mu Cx$ has shifted sector bounds. Since $f_i={\overline{f}}_i+\mu y_i$, the bounds $0\le y_i{f}_i\le {\sigma }_i{y}_i^2$ are replaced by

$$0\le y_i({\overline{f}}_i+\mu y_i)\le {\sigma }_i{y}_i^2$$

and get the following non-symmetric bounds as expected:

$$-\mu y_i^2\le {\overline{f}}_i{y}_i\le ({\sigma }_i-\mu )y_i^2$$

Defining ${\alpha }_i=-\mu$ and ${\beta }_i={\sigma }_i-\mu$, we finally get

$${\alpha }_i{y}_i^2\le y_i{\overline{f}}_i\le {\beta }_i{y}_i^2$$

Combining the left and right inequality we get

$$y_i^2({\beta }_i{y}_i-{\overline{f}}_i)({\overline{f}}_i-{\alpha }_i{y}_i)\ge 0$$

or in vector form

$${(\overline{f}-\alpha y)}^T(\beta y-\overline{f})\ge 0,$$

for all $y\in {\mathbb{R}}^m$, where $\alpha \le \beta$ and $\alpha :=col\{{\alpha }_1,{\alpha }_2,...,{\alpha }_q\}$ and similarly $\beta :=col\{{\beta }_1,{\beta }_2,...,{\beta }_q\}$.

Substituting $y=Cx$, we readily obtain

$$-{\overline{f}}^{\top }\overline{f}-\alpha \beta x^{\top }{C}^{\top }Cx+\alpha x^{\top }{C}^{\top }\overline{f}+\beta {\overline{f}}^{\top }Cx\ge 0$$

Noting that $\alpha =-\mu I$ and $\beta =S-\mu I$ we get the modified sector condition

$$\begin{array}{c}\hfill -{\overline{f}}^{\mathsf{T}}\overline{f}+(\mu S-{\mu }^{2}I)x^{\mathsf{T}}{C}^{\mathsf{T}}Cx-\mu x^{\mathsf{T}}{C}^{\mathsf{T}}\overline{f}+(S-\mu I){\overline{f}}^{\mathsf{T}}Cx\ge 0.\end{array}$$

(10)

Adding the multipliers ${\tau }_i\ge 0$, $i=1,\dots ,q$ as in the previous section, the following modified form of ${\mathcal{F}}_c$ is introduced

$${\mathcal{F}}_c^{\mu }:=\left[\begin{array}{ccc}\mu C^{\top }(S-\mu I)TC& {\displaystyle \frac{1}{2}}{C}^{\top }(S-2\mu I)T& 0\\ {\displaystyle \frac{1}{2}}T(S-2\mu I)C& -T& 0\\ 0& 0& 0\end{array}\right],$$

(11)

with respect to which the above condition (10) is expressed as ${\overline{\xi }}^{\top }{\mathcal{F}}_c^{\mu }\overline{\xi }\ge 0$, where $\overline{\xi }:={\left[x^T{\overline{f}}^T{w}^T\right]}^T$.

We aim now at obtaining a version of Theorem 1 with the shifted sector bounds.To this end, we consider $dV/dt$ of (6) and re-write it as

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}{x}^{\top }& {\overline{f}}^{\top }+\mu x^{\top }{C}^{\top }& w^{\top }\end{array}\right]{\tilde{\mathcal{F}}}_0\left[\begin{array}{c}x\\ \overline{f}+\mu Cx\\ w\end{array}\right]=\left[\begin{array}{ccc}{x}^{\top }& {\overline{f}}^{\top }& w^{\top }\end{array}\right]\left(T_{\mu }^{\top }{\tilde{\mathcal{F}}}_0{T}_{\mu }\right)\left[\begin{array}{c}x\\ \overline{f}\\ w\end{array}\right],\end{array}$$

where

$$\begin{array}{c}\hfill T_{\mu }:=\left[\begin{array}{ccc}I& 0& 0\\ \mu C& I& 0\\ 0& 0& I\end{array}\right],\end{array}$$

and where ${\tilde{\mathcal{F}}}_0$ is obtained from (9) written for the system (1) with the control $u_2=Kx$, namely replacing the matrices A, L and D by $A+B_{2}K$, $M+NK$ and by $B_1$, respectively. Using again the $\mathcal{S}$ procedure to impose the shifted sector bound constraints associated with ${\tilde{\mathcal{F}}}_0^{\mu }:=T_{\mu }^{\top }{\tilde{\mathcal{F}}}_0{T}_{\mu }$, we readily obtain the loop transformed version of condition (9) of Lema 1,

$$\begin{array}{c}\hfill {\tilde{\mathcal{L}}}_{\mu }<0\end{array}$$

(12)

where

$$\begin{array}{c}\hfill {\tilde{\mathcal{L}}}_{\mu }:=T_{\mu }^{\top }{\tilde{\mathcal{F}}}_0^{\mu }{T}_{\mu }+{\mathcal{F}}_c^{\mu }+{\mathcal{F}}_w\end{array}$$

(13)

The latter inequality can be readily expressed using YALMIP and we intentionally avoid writing the explicit formulate for its blocks, to avoid burdening the reader with un necessary details. Note that similarly to the Lemma of Section 1, the search variables are $P>0$, $\Lambda =\mathrm{diag}\{{\lambda }_1,\dots ,{\lambda }_q\}>0$ and $T=\mathrm{diag}\{{\tau }_1,\dots ,{\tau }_q\}\ge 0$ where $\mu$ is found using line search, within the interval $\mu \in [0,{min}_i{\sigma }_i]$.

Based on Lemma 1, one obtains the following result.

Theorem 2.

If there exist the matrices $P>0$ , the diagonal matrices $\Lambda ,T>0$, the matrix K and a scalar $\mu \in [0,{min}_i{\sigma }_i]$ such that

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}{\mathcal{N}}_{11}(P,\Lambda ,K,\mu )& {\mathcal{N}}_{12}(P,\Lambda ,T,K,\mu )& {\mathcal{N}}_{13}(P,\Lambda ,\mu )\\ {(1,2)}^{\top }& {\mathcal{N}}_{22}(\Lambda ,T)& {\mathcal{N}}_{23}\left(\Lambda \right)\\ {(1,3)}^{\top }& {(2,3)}^{\top }& -{\gamma }^2{I}_{m_1}\end{array}\right]<0\end{array}$$

(14)

in which

$$\begin{array}{c}\hfill \begin{array}{ccc}{\mathcal{N}}_{11}(P,\Lambda ,K,\mu )\hfill & =& {(A+B_{2}K)}^{\top }P+P(A+B_{2}K)+{(M+NK)}^{\top }(M+NK)\hfill \\ & & +\mu C^{\top }[\Lambda C(A+B_{2}K)+F^{\top }P]+\mu [{(A+B_{2}K)}^{\top }{C}^{\top }\Lambda +PF]C\hfill \\ & & +{\mu }^2{C}^{\top }(\Lambda CF+F^{\top }{C}^{\top }\Lambda )C+\mu C^{\top }(S-\mu I)TC\hfill \\ {\mathcal{N}}_{12}(P,\Lambda ,T,K,\mu )\hfill & =& {(A+B_{2}K)}^{\top }{C}^{\top }\Lambda +PF+\mu C^{\top }(\Lambda CF+F^{\top }{C}^{\top }\Lambda )+{\displaystyle \frac{1}{2}}{C}^{\top }(S-2\mu I)T\hfill \\ {\mathcal{N}}_{13}(P,\Lambda ,\mu )\hfill & =& (P+\mu C^{\top }\Lambda C)B_1\hfill \\ {\mathcal{N}}_{22}(\Lambda ,T)\hfill & =& \Lambda CF+F^{\top }{C}^{\top }\Lambda -T\hfill \\ {\mathcal{N}}_{22}\left(\Lambda \right)\hfill & =& \Lambda C{B}_1,\hfill \end{array}\end{array}$$

then K stabilizes the system (1) satisfying the γ-attenuation condition ${\int }_0^{\infty }\left(z^T\left(t\right)z\left(t\right)-{\gamma }^2{w}^T\left(t\right)w\left(t\right)\right)dt<0$ for all $w\left(t\right)\in L^2([0,\infty ),{\mathbb{R}}^{m_1})$.

The above inequality (14) is nonlinear. Further representation of (14) for numerical implementation now follows. Fixing $\Lambda$ and T and multiplying the inequality (14) to the left and to the right by $\mathrm{diag}\{P^{-1},I,I\}$, and denoting $X:=P^{-1}$ and $Y:=K{P}^{-1}$, based on Schur complement arguments and using the inequality $X{C}^{\top }\Lambda C{B}_{2}Y+Y^{\top }{B}_2^{\top }{C}^{\top }\Lambda CX\le X{C}^{\top }\Lambda CX+Y^{\top }{B}_2^{\top }{C}^{\top }\Lambda C{B}_{2}Y$, it results that the condition (14) is accomplished if the following linear matrix inequality (LMI) is feasible with respect to $X>0$ and Y:

$$\begin{array}{c}\hfill \left[\begin{array}{ccccc}{\mathcal{P}}_{11}(X,Y)& {\mathcal{P}}_{12}(X,Y)& {\mathcal{P}}_{13}\left(X\right)& {\mathcal{P}}_{14}\left(X\right)& {\mathcal{P}}_{15}\left(Y\right)\\ {(1,2)}^{\top }& {\mathcal{P}}_{22}& {\mathcal{P}}_{23}& 0& 0\\ {(1,3)}^{\top }& {(2,3)}^{\top }& -{\gamma }^2{I}_{m_1}& 0& 0\\ {(1,4)}^{\top }& 0& 0& -I_n& 0\\ {(1,5)}^{\top }& 0& 0& 0& -I_{m_2}\end{array}\right]<0,\end{array}$$

(15)

where

$$\begin{array}{c}\hfill \begin{array}{ccc}{\mathcal{P}}_{11}(X,Y)\hfill & =& (A+\mu FC)X+X{(A+\mu FC)}^{\top }+B_{2}Y+Y^{\top }{B}_2^{\top }\hfill \\ {\mathcal{P}}_{12}(X,Y)\hfill & =& X[A^{\top }{C}^{\top }\Lambda +\mu C^{\top }(\Lambda CF+F^{\top }{C}^{\top }\Lambda )]+Y^{\top }{B}_2^{\top }{C}^{\top }\Lambda +F+{\displaystyle \frac{1}{2}}X{C}^{\top }(S-2\mu I)T\hfill \\ {\mathcal{P}}_{13}\left(X\right)\hfill & =& (I+\mu X{C}^T\Lambda C)B_1\hfill \\ {\mathcal{P}}_{14}\left(X\right)\hfill & =& X{V}^{\top }\hfill \\ {\mathcal{P}}_{15}\left(Y\right)\hfill & =& Y^{\top }{W}^{\top }\hfill \\ {\mathcal{P}}_{22}\hfill & =& \Lambda CF+F^{\top }{C}^{\top }\Lambda -T\hfill \\ {\mathcal{P}}_{23}\hfill & =& \Lambda C{B}_1\hfill \end{array}\end{array}$$

(16)

and where the matrices V and W satisfy the following conditions:

$$\begin{array}{c}\hfill \begin{array}{ccc}{V}^{\top }V\hfill & =& M^{\top }M+\mu \left[C^{\top }\Lambda CA+A^{\top }{C}^{\top }\Lambda C+C^{\top }(S-\mu I)TC+C^{\top }\Lambda C\right]\hfill \\ & & +{\mu }^2(C^{\top }\Lambda CF+F^{\top }{C}^{\top }\Lambda C)\hfill \\ W^{\top }W\hfill & =& N^{\top }N+\mu B_2^{\top }{C}^{\top }\Lambda C{B}_2.\hfill \end{array}\end{array}$$

(17)

Remark: Note that Theorem 2 involves terms that are bilinear in $\Lambda$ and T that were assumed to be fixed, or in other words, remained for manual choice. One can, however, embed (15) in an optimization scheme, where the diagonal terms in $\Lambda$ and T are vectorized. One can, then minimize a smooth approximation of max(0, ${\lambda }_{max}\left(\mathcal{P}\right)$). While such an approach is free from manual tuning of $\Lambda$ and T, one should keep in mind, that global convergence of such a heuristic approach, does not guarantee global feasibility. Nevertheless the results of Section 6 below, employ such an approach.

6. State Feedback Control with $L_2$-Induced Norm Attenuation for Van der Pol Oscillator

The theoretical results derived in the previous sections are illustrated for a Van der Pol oscillator described by the differential equation:

$$\begin{array}{c}\hfill \ddot{x}\left(t\right)-ϵ(1-x^2\left(t\right))\dot{x}\left(t\right)+x\left(t\right)=0\end{array}$$

(18)

where $ϵ\ge 0$ is a fixed parameter indicating the strength of the nonlinear damping. As it is well-known, this (unforced) equation was introduced a century ago, in the context of oscillations modeling in a vacuum tube electrical circuit. Since then, it was shown that such equation is useful not only in electronics but also in other diverse domains as physics, biology, neurology and more recently, in machine learning and evolutionary algorithms used to represent the real electrocardiographic signals ([17]). The differential Equation (18) may be represented in the state space by letting $x_1:=x$ and $x_2:=\dot{x}$ for which one gets

$$\begin{array}{c}\hfill \begin{array}{ccc}{\dot{x}}_1\left(t\right)\hfill & =& x_2\left(t\right)\hfill \\ {\dot{x}}_2\left(t\right)\hfill & =& ϵ(1-x_1^2\left(t\right))x_2\left(t\right)-x_1\left(t\right).\hfill \end{array}\end{array}$$

(19)

Since the aim of this application is to stabilize (19) ensuring a certain prescribed level of attenuation of the $L_2$-induced norm, one considered the following modified (forced) form of (19):

$$\begin{array}{c}\hfill \begin{array}{ccc}{\dot{x}}_1\left(t\right)\hfill & =& x_2\left(t\right)+w\left(t\right)\hfill \\ {\dot{x}}_2\left(t\right)\hfill & =& ϵ(1-x_1^2\left(t\right))x_2\left(t\right)-x_1\left(t\right)+u\left(t\right),\hfill \end{array}\end{array}$$

(20)

where $w\left(t\right)\in L^2([0,\infty ),\mathbb{R})$ is a disturbance input and $u\left(t\right)\in \mathbb{R}$ denotes the control input.

Since at is seen, it is not a sector-type nonlinearity, it was approximated using the universal approximation theorem proved by Cybenko ([9]), stating that any continuous function can be approximated arbitrarily well by a neural network with at least one hidden layer with a finite number of weights. In the present application, the nonlinear term $ϵ(1-x_1^2)x_2$ was approximated as

$$\begin{array}{c}\hfill ϵ(1-x_1^2)x_2\approx \sum _{i=1}^N\left[W_{2,i}\left(tanh\left(W_{1,i}\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+b_{1,i}\right)\right)+b_{2,i}\right]\end{array}$$

(21)

with N denoting the number of neurons and where the weights and the bias terms were determined by back propagation training (see e.g., [7]). The network architecture is depicted in Figure 1.

For $ϵ=1$ and for $N=10$, the time responses obtained with the approximation (21) of $ϵ(1-x_1^2)x_2$ are presented in Figure 2 (states as a function of time ) and Figure 3 (phase-plane ), comparatively with the time responses of the original Van der Pol oscillator (19).

Using the above approximation of the nonlinearity $ϵ(1-x_1^2)x_2$ and defining

$$\begin{array}{c}\hfill A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right],B_1=\left[\begin{array}{c}1\\ 0\end{array}\right],B_2=\left[\begin{array}{c}0\\ 1\end{array}\right],F=\left[\begin{array}{c}{0}_{1\times N}\\ W_{2,1}\dots W_{2,N}\end{array}\right],\end{array}$$

(22)

it follows that the Var der Pol oscillator (20) may be approximated as the first Equation (1) where $f_i\left(y_i\right)=tanh\left(W_{1,i}\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\right)$ and therefore, $y_i=W_{1,i}\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$, $i=1,\dots ,N$. Following the notations of (2) it results that $C_i=W_{1,i},i=1,\dots ,N$.

As concerns the quality output, one considered $z={\left[\begin{array}{cc}{x}_1& 0.1u\end{array}\right]}^{\top }$, obtaining thus the matrices M and N from the second Equation (1) as

$$\begin{array}{c}\hfill M=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],N=\left[\begin{array}{c}0\\ 0.1\end{array}\right].\end{array}$$

(23)

Note that (15) is convex, only for fixed $\Lambda$ and T for a given $\mu$. Therefore we resorted to using unconstrained minimization of a scalar objective using the derivative-free Nelder–Mead simplex method (i.e., fminsearch from $MATLA{B}^{TM}$) starting from an initial guess provided by solving the LMI of (15) for a guess of $\Lambda$, T and $\mu$ and for $\gamma =500$ for which the LMI was solved. The search variables in this optimization scheme are vectorized version $vec\left(Y\right)$ and $vech\left(X\right)$ respectively of X and Y. Then, $K=Y{X}^{-1}$ it results that $K=\left[\begin{array}{cc}-662.3694& -331.7394\end{array}\right]$. The closed-loop simulation results with this gain are plotted in Figure 4.

7. Conclusions

Boundedness conditions for the $L_2$-induced norm of systems with multiple nonlinearities have been derived in the present paper. Based on these conditions, a state feedback optimal control design was formulated and solved. The solvability conditions were expressed in terms of the feasibility of a specific system of matrix inequalities. A loop transformation procedure was developed in order to reduce the conservatism of the solvability conditions. Although the results were determined under the assumption of sector type nonlinearities, it was shown that in combination with the universal approximation theorem, they may be also applied for systems which nonlinearities do not necessarily satisfy the bounded sector conditions. These developments were illustrated for the $L_2$ control problem of a forced Van der Pol oscillator.