H∞ Robust LMI-Based Nonlinear State Feedback Controller of Uncertain Nonlinear Systems with External Disturbances

In this paper, we propose a nonlinear state feedback controller based on linear matrix inequality (LMI) for a class of nonlinear systems with parametric uncertainties and external disturbances. The primary goals of the proposed controller are to guarantee system stability and performance in the presence of system uncertainties and time-dependent disturbances. To meet the specified objectives, the LMI form is calculated as a hierarchical control structure. Using the Lyapunov stability function, the asymptotic stability of the nominal system obtained from the nonlinear state feedback is proven, and the LMI condition is attained. After applying the nonlinear state feedback controller, asymptotic stability conditions for the nominal system are constructed using the Lyapunov function, and the nonlinear state-feedback control mechanism is determined accordingly. Considering the external disturbance as input, the terms of the state matrices are substituted in the obtained LMI, and the LMI condition for a nominal system is achieved in the presence of disturbances. The asymptotic stability condition of the uncertain system in the presence of external disturbances is determined by adding uncertainties to the system. The proposed approach yields a simple control mechanism representing an independent of system order. The performance of the proposed approach was assessed using a simulation study of a ball and beam system.

Keywords: state feedback stability; linear matrix inequality; parametric uncertainty; external disturbance; robust control

The prevalence of nonlinearities in dynamic systems makes control design challenging. This is further exacerbated when disturbances are added to the system. To date, various approaches have been developed and implemented to control and/or stabilize nonlinear disturbed systems [[1], [3], [5]]. Because system states are affected by disturbances and nonlinearities, errors emerge in dynamical systems, and disregarding their impact, may lead to extreme deterioration in system performance and even instability [[6], [8]]. As a result, it is well-established that controlling nonlinear systems in the presence of disturbances has significant practical implications [[9], [11]]. In recent years, there has been a noticeable increase in the focus on developing suitable control laws for nonlinear dynamical systems in the presence of parameter uncertainties [[12], [14]]. Yu et al. [[15]] proposed a command-filtering backstepping control approach based on neural network approximation for nonlinear SISO systems with unknown disturbances. Using active disturbance rejection control, Ran et al. [[16]] investigated the stabilization problem of a class of nonlinear systems with actuator saturation. Celentano and Basin [[17]] proposed an approach to design robust controllers to track a reference signal with a small error norm for uncertain nonlinear systems with bounded disturbances. In [[18]], a robust $H_{\infty }$ controller was designed to maintain the states on a predefined sliding surface and guarantee the discrete time-reaching condition. In [[19]], by incorporating a linear feedback control technique, a dynamic feedback method, and an uncertainty and disturbance estimator (UDE)-based control approach, two UDE-based control methods were designed to stabilize fractional order systems with both model uncertainties and external disturbances.

The majority of the problems encountered by engineers in practical applications are subject some level of uncertainty due to inaccurate modeling, including robotic manipulators [[20]], non-holonomic plants [[21]], power systems [[22]], and flexible space configurations [[23]]. Thus, system uncertainties should always be considered when designing a control system for stability and performance. Over the years, significant research has been conducted on the implementation of robust nonlinear state-feedback control schemes that handle system uncertainties. One of the methods described is linear matrix inequality (LMI), which is a practical tool used in the analysis and design of control systems [[24], [26]]. The LMI approach is a practical and powerful tool for dealing with system uncertainties, such as parametric [[27]] or unstructured uncertainties [[28]]. The use of state and output feedback control methods to achieve robust linear system stabilization was investigated in [[30]]. Uncertainty is a polytope studied in both continuous and discrete time systems. It proposes synthesis conditions that directly address the control gain as an optimization variable, in contrast to other LMI-based robust stabilization approaches. The LMI approach was used in [[31]] to solve the problem of static output feedback (SOF) stabilization for uncertain linear systems. This problem frequently results in the possibility of bilinear matrix inequality (BMI). As a result, it has been proposed to transform the BMI into a new LMI with a line search over a scalar variable using various technical lemmas, resulting in an improved and less conservative LMI condition with a line search over two scalar variables. The design of state and output feedback controllers for uncertain linear parameter varying (LPV) systems is considered to guarantee some desired bounds on $H_2$ and $H_{\infty }$ performances and satisfy some desired constraints on the location of closed-loop poles [[32]]. In [[34]], by using a quadratic Lyapunov function, adequate conditions for a state feed-back-based $H_{\infty }$ controller and an observer-based $H_{\infty }$ controller were proposed in the form of non-convex matrix inequalities that consider actuator saturation. Furthermore, undesirable inputs, such as disturbances, should be considered; Oh et al. [[35]] focused on reducing the impact of disturbances on specific applications in this manner. In [[36], [38]], the authors focused on the design of robust controllers for linear systems with uncertainty and disturbance. These controllers considered stabilizing the plant for all conceivable parameter uncertainties, as well as providing a preferred reduction level for uncontrollable external disturbances. Nonlinearity, in addition to uncertainty and disturbance, is a challenge [[40]]. LMI conditions are presented for robust asymptotic stability of rational uncertain nonlinear systems. The uncertainties are modeled as polytope-bounded, real-time varying parameters, and the system vector field is a rational function of the states and uncertain parameters. A rational Lyapunov function of states and uncertain parameters provides enough LMI conditions for the asymptotic stability of the origin. LMI was used to stabilize uncertain nonlinear systems with Lipschitz nonlinearities in [[41]]. The proposed controller improves both transient and steady-state performance. A nonlinear function was optimized using a modified random search approach in the control law. For the stability criteria [[42], [44]], the S-procedure can be used first, followed by the Schur complement and the matrix inversion lemma to convert BMIs into LMIs using these formulas and by implementing various mechatronics configurations. The control law described in [[45]] incorporates a new nonlinear function that optimized using a modified random search technique. Furthermore, Mobayen and Pujol-Vázquez [[46]] investigated the challenge of robust $H_{\infty }$ performance analyses for a category of nonlinear systems with exterior disturbances. Jennawasin et al. [[47]] introduced new variables that divide the system and Lyapunov matrices to avoid the limitations of conventional state-feedback techniques; thus, state-feedback controller parameterization is unrelated to Lyapunov matrices. Because the suggested design condition is bilinear in the choice variables, an iterative approach is also suggested. In [[48]], the input to state-stability Lyapunov functions was used to develop an LMI-framed linear-state variable feedback control scheme for MIMO Lipschitz nonlinear systems with bounded parameter uncertainty. The authors of [[49]] addressed the problem of observer-based robust control for nonlinear uncertain systems. Amini et al. [[51]] investigated the robust static output feedback problem for nonlinear systems with uncertainty. An output feedback controller architecture using LMI ensures pole-positioning $H_2$ and $H_{\infty }$ limitations. A one-sided Lipschitz nonlinear system was considered in [[52]], and LMI approaches were used to create state feedback, with static output feedback controllers as the suggested controllers ensure finite, time-bounded, closed-loop systems with a desired $H_{\infty }$ performance index.

The aforementioned articles proposed and implemented a wide range of LMI-based controllers to solve different problems. However, in addition to suffering from some drawbacks, most of the available controllers do not simultaneously address nonlinearities, uncertainties, and disturbances, often overlooking at least one of the noted objectives. For example, the work proposed in [[30], [32], [34]] satisfies the requirement of uncertain linear systems, whereas [[35]] considers nominal linear systems in the presence of disturbances. Uncertain nonlinear systems were investigated in [[47], [49]]. Additionally, most of the above approaches make some assumptions and impose limitations on the system, such as Lipschitzian nonlinearities.

In this paper, we propose a simple and less restrictive LMI-based hierarchical control design procedure for nonlinear systems subject to disturbances and uncertainties. The main contributions of this work are as follows:

• Development of a simple and easy-to-implement nonlinear state feedback controller for a class of nonlinear and uncertain systems;
• Design of a hierarchical $H_{\infty }$ controller to ensure system stability and mitigate the effect of disturbances through Lyapunov stability theory; and
• Determination of the feedback gains and upper bound of ${\Vert .\Vert }_{\infty }$ of the transfer function using linear matrix inequalities.

The remainder of this paper is organized as follows. In Section 2, we introduce the problem definition and preliminaries. The proposed LMI-based state feedback control design is derived in Section 3. The effectiveness of the proposed approach is illustrated using a simulation study of a ball and beam system in Section 4. Finally, conclusions are presented in Section 5.

Consider the following uncertain nonlinear dynamical system with exerted disturbances:

(1) $$\{\begin{array}{l}\dot{x}\left(t\right)=f\left(x\right)+\left(A+\mathsf{\Delta }A\right)x\left(t\right)+B{u}_n\left(t\right)+D_i\omega \left(t\right)\hfill \\ y\left(t\right)=Cx\left(t\right)+D_o\omega \left(t\right)\hfill \end{array}$$

where $x\left(t\right)\in R^n,y\left(t\right)\in R^m,u_n\left(t\right),\mathrm{and}\omega \left(t\right)\in R^{\omega }$ are the system states, outputs, nonlinear input control, and disturbance, respectively. The nominal matrices, $A,B,C,D_i,D_o$ , are constant known matrices with appropriate dimensions. $\mathsf{\Delta }A$ Represents system uncertainties, $M\in {ℝ}^{n\times n},N\in {ℝ}^{n\times n}$ are known matrices, and $\mathsf{\Delta }A=M\mathsf{\Delta }N$ [[53]] $∆$ is a matrix that contains the uncertain parameters and satisfies $ǁ{\mathsf{\Delta }}^T\mathsf{\Delta }ǁ\le I$ [[53]].

The following nonlinear state feedback control law is proposed:

(2) $$u_n\left(t\right)=u\left(t\right)-B^{-1}f\left(x\right)$$

(3) $$u\left(t\right)=Kx\left(t\right),f\left(0\right)\ne 0$$

where $u\left(t\right)\in R^u$ is the input controller for the linear system.

$f\left(x\right)$ is a continuously differentiable function.

For the non-square or rank deficient $B$ matrix, instead of the formal inverse matrix, the pseudo-inverse will be used.

For any scalar and real $\alpha >0$ , the following inequality exists [[53]]:

(4) $$p^{T}q+q^{T}p\le \alpha p^{T}P+{\alpha }^{-1}{q}^{T}q$$

where the matrices $p$ and $q$ are of consistent dimensions.

First, consider the nominal dynamical system without disturbances, which is defined as:

(5) $$\{\begin{array}{c}\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)\\ y\left(t\right)=Cx\left(t\right)+Du\left(t\right)\end{array}$$

With the transfer function of control input ( $u\left(t\right)$ ) to output ( $y\left(t\right)$ ) [[55]]:

(6) $$G_{yu}=C{\left(SI-A\right)}^{-1}B+D=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$$

According to inequality, as defined in [[56]]:

(7) $$\Vert y\left(t\right){\Vert }_2\le \Vert G_{yu}{\left(s\right)\Vert }_{\infty }\Vert u\left(t\right){\Vert }_2$$

The aim is to design a robust $H_{\infty }$ controller to attenuate the effect of the inputs on the outputs, which means that the bounded term $\Vert G_{yu}{\left(s\right)\Vert }_{\infty }$ should be minimized.

Let $G_{yu}$ be provided in (6), and assume $\Vert G_{yu}\Vert {}_{\infty }\le \gamma$ , where $\gamma$ is an unknown positive parameter. If there exists a symmetric positive matrix ( $P$ ), such that the subsequent linear matrix inequalities hold:

(8) $$\left[\begin{array}{ccc}{A}^{T}P+PA& PB& C^T\\ B^{T}P& -\gamma I& D^T\\ C& D& -\gamma I\end{array}\right]<0$$

then, the nominal dynamical system (5) is asymptotically stable.

Suppose the system is controllable from the input to the states with $\Vert G_{yu}\Vert {}_{\infty }\le {\beta }^{-1}$ ( $\beta$ is an unknown positive parameter), which is equivalent to $\sup \frac{\Vert y_2\Vert }{\Vert u_2\Vert }u\ne 0<{\beta }^{-1},$ or ${\Vert y\Vert }_2<{\beta }^{-1}{\Vert u\Vert }_2,$ which can be expressed as:

(9) $${{\displaystyle \int }}_0^{t\to \infty }\left(\beta y^T\left(\tau \right)y\left(\tau \right)-{\beta }^{-1}{u}^T\left(\tau \right)u\left(\tau \right)d\tau \right)<0$$

The purpose is to represent the above robust condition in the form of LMI. Consider a scalar Lyapunov function in the form of:

(10) $$V\left(x\left(t\right)\right)=x^{T}Px,P>0$$

Suppose the derivative of Lyapunov's function holds in the following relation:

(11) $$\dot{V}\left(x\left(t\right)\right)+\beta y^T\left(\tau \right)y\left(\tau \right)-{\beta }^{-1}{u}^T\left(\tau \right)u\left(\tau \right)<0$$

Integrating (11) yields:

(12) $${{\displaystyle \int }}_0^{t\to \infty }\dot{V}\left(x\left(\tau \right)\right)d\tau +{{\displaystyle \int }}_0^{t\to \infty }\left[\beta y^T\left(\tau \right)y\left(\tau \right)-{\beta }^{-1}{u}^T\left(\tau \right)u\left(\tau \right)\right]d\tau <0$$

or

(13) $$V\left(t\to \infty \right)-V\left(0\right)+{{\displaystyle \int }}_0^{t\to \infty }\left[\beta y^T\left(\tau \right)y\left(\tau \right)-{\beta }^{-1}{u}^T\left(\tau \right)u\left(\tau \right)\right]d\tau <0$$

Assuming:

(14) $$x\left(0\right)=0\Rightarrow V\left(0\right)=x^{T}Px=0$$

As a consequence,

(15) $$V\left(t\to \infty \right)+{{\displaystyle \int }}_0^{t\to \infty }\left[\beta y^T\left(\tau \right)y\left(\tau \right)-{\beta }^{-1}{u}^T\left(\tau \right)u\left(\tau \right)\right]d\tau <0$$

Because the function $V\left(x\left(t\right)\right)$ is greater than zero, Equation (9) is obtained. Therefore, if relation (11) is established, then relation (9) will also be proven. The following investigates under which conditions relationship (11) should be obtained.

When Equation (5) substitutes the derivative of (10), the result is:

(16) $$\dot{V}\left(x\left(t\right)\right)={\dot{x}}^{T}Px+x^{T}P\dot{x}={\left(Ax+Bu\right)}^{T}Px+x^{T}P\left(Ax+Bu\right)$$

Substituting the equation obtained from (16) into Equation (11) yields:

(17) $$\begin{gathered} \dot{V}(x(t))+\beta y^T(\tau )y(\tau )-{\beta }^{-1}{u}^T(\tau )u(\tau ) \\ =(x^T{A}^T+u{B}^T)Px+x^{T}P(Ax+Bu) \\ +\beta (x^T{C}^T+u^T{D}^T)(Cx+Du)-{\beta }^{-1}{u}^{T}u<0 \end{gathered}$$

The right-hand side of (17) can be simplified as follows:

(18) $$\begin{gathered} =x^T\left[A^{T}P+PA+\beta C^{T}C\right]x+u^T\left[\beta D^{T}D-{\beta }^{-1}I\right]u \\ +x^T[PB+\beta C^{T}D]u+u^T[B^{T}P+\beta D^{T}C]x<0 \end{gathered}$$

Then, Equation (17) is converted to the following matrix form:

(19) $$\Rightarrow {\left[\begin{array}{c}x\\ u\end{array}\right]}^T\left[\begin{array}{cc}{A}^{T}P+PA+\beta C^{T}C& PB+\beta C^{T}D\\ B^{T}P+\beta D^{T}C& \beta D^{T}D-{\beta }^{-1}I\end{array}\right]⏟H\left[\begin{array}{c}x\\ u\end{array}\right]<0$$

If the matrix ( $H$ ) is negative and definite, then relation (11) and, consequently, relation (9) will be established. Assuming $\left(A,B\right)$ is controllable, it can be shown that $H<0$ is also a necessary condition to fulfill Equation (9). Relation $H$ can be considered in the form of an LMI with the variable $P$ , where $P>0$ ; hence, if LMI $\left(P\right)$ has a solution, condition $\Vert G_{yu}\Vert {}_{\infty }\le {\beta }^{-1}$ is met. Using Schur's complement lemma [[57]], matrix $H$ results in:

(20) $$\mathrm{LMI}\left(P\right):\{\begin{array}{c}\left[\begin{array}{ccc}{A}^{T}P+PA& PB& C^T\\ B^{T}P& -{\beta }^{-1}I& D^T\\ C& D& -{\beta }^{-1}I\end{array}\right]\\ P>0\end{array}$$

If ${\beta }^{-1}$ is unknown, by defining the new variable ( $\gamma ={\beta }^{-1}$ ), the problem is defined in the new LMI $\left(\gamma ,P\right)$ form:

(21) $$\mathrm{LMI}\left(\gamma ,P\right):\{\begin{array}{c}\left[\begin{array}{ccc}{A}^{T}P+PA& PB& C^T\\ B^{T}P& -\gamma I& D^T\\ C& D& -\gamma I\end{array}\right]<0\\ \begin{array}{c}P>0\\ \gamma >0\end{array}\end{array}$$

Therefore, the nominal dynamical system (5) is asymptotically stable if the LMI (20) or (21) is feasible. □

In the second step, if the nonlinear controller (2) is added to the nominal dynamical system with disturbances (1), the following equation is achieved:

(22) $$\{\begin{array}{l}\dot{x}=\left(A+BK\right)x+D_i\omega \hfill \\ y=Cx+D_o\omega \hfill \end{array}$$

The purpose is to determine the gain ( $K$ ) so that the closed-loop system in the presence of disturbances is stable, while minimizing the effect of input ( $\omega \left(t\right)$ ) on system output ( $y\left(t\right)$ ). The transfer function of input $\omega \left(t\right)$ to output $y\left(t\right)$ is [[55]]:

(23) $$G_{y\omega }=C{\left(SI-A+BK\right)}^{-1}{D}_i+D_o=\left(\begin{array}{cc}A+BK& D_i\\ C& D_o\end{array}\right)$$

Let $G_{y\omega }$ be provided in (23), and assume $\Vert G_{y\omega }\Vert {}_{\infty }\le \gamma$ , where $\gamma$ is a positive variable. If there exist symmetric positive matrices ( $X$ and $W$ ) with suitable dimensions such that the following linear matrix inequalities are satisfied:

(24) $$\left[\begin{array}{ccc}{\left(AX+BW\right)}^{\mathrm{T}}+AX+BW& D_i& {\left(CX\right)}^{\mathrm{T}}\\ D_i{}^T& -\gamma I& D_o{}^T\\ CX& D_o& -\gamma I\end{array}\right]<0$$

then, the nominal dynamical system with disturbances (22) is asymptotically stable.

Placing the state matrices of the dynamic system (22) in the LMI elements resulting from Equation (8) yields:

(25) $$\left[\begin{array}{ccc}{\left(A+BK\right)}^{T}P+P\left(A+BK\right)& P{D}_i& C^T\\ D_i{}^{T}P& -\gamma I& D_o{}^T\\ C& D_o& -\gamma I\end{array}\right]$$

Using a congruence transformation, such as $T=diag\left(P^{-1},I,I\right)$ , and pre-post multiplying $T$ in Equation (25) yields:

(26) $$\left[\begin{array}{ccc}{P}^{-1}{A}^T+P^{-1}{K}^T{B}^T+A{P}^{-1}+BK{P}^{-1}& D_i& P^{-1}{C}^T\\ D_i{}^T& -\gamma I& D_o{}^T\\ C{P}^{-1}& D_o& -\gamma I\end{array}\right]$$

That is,

(27) $$\left[\begin{array}{ccc}{\left(AX+BKX\right)}^{\mathrm{T}}+\left(AX+BKX\right)& D_i& {\left(CX\right)}^{\mathrm{T}}\\ D_i{}^T& -\gamma I& D_o{}^T\\ CX& D_o& -\gamma I\end{array}\right]<0$$

where $K=W{X}^{-1}$ .

The case is recast as an LMI feasibility problem in the manner of (24); therefore, if there exist matrices $W$ and $X>0$ , the disturbed dynamical system (22) is asymptotically stable. □

The aforementioned theorems ensure the asymptotic stability of the nominal systems under both non-perturbation and perturbation conditions using the robust control approach. We will continue our investigation into the problem of robust state feedback control of nonlinear systems with parameter uncertainties in the following sections. Consider the nonlinear dynamical system (1), the transfer function of which is given below [[55]]:

(28) $$G_{y\omega }=C{\left(SI-A-\Delta A-BK\right)}^{-1}{D}_i+D_o=\left(\begin{array}{cc}A+\Delta A+BK& D_i\\ C& D_o\end{array}\right)$$

According to input–output relationships defined in [[56]]:

(29) $$\Vert y\left(t\right){\Vert }_2\le \Vert G_{y\omega }{\left(s\right)\Vert }_{\infty }\Vert \omega \left(t\right){\Vert }_2$$

The purpose is to implement a robust $H_{\infty }$ controller to attenuate the perturbations, while ensuring asymptotic stability of the dynamical system (1).

Let $G_{y\omega }$ be provided in (28), and assume $\Vert G_{y\omega }\Vert {}_{\infty }\le \gamma$ , where $\gamma$ is a positive variable. If there exists a scalar $\alpha$ , a symmetric positive matrix ( $X$ ), and a matrix ( $W$ ) with appropriate dimensions such that the following linear matrix inequalities are satisfied:

(30) $$\begin{gathered} \left[\begin{array}{cccc}\theta \left(X,W\right)& D_i& {\left(CX\right)}^{\mathrm{T}}& {\left(NX\right)}^{\mathrm{T}}\\ D_i{}^T& -\gamma I& D_o{}^T& 0\\ CX& D_o& -\gamma I& 0\\ NX& 0& 0& -\alpha I\end{array}\right]<0 \\ \Theta \left(X,W\right)=\left(AX+BW\right)+{\left(AX+BW\right)}^T+\alpha M{M}^T \end{gathered}$$

where an $H_{\infty }$ state feedback control law is provided by:

(31) $$u\left(t\right)=W{X}^{-1}x\left(t\right)$$

then, the perturbed nonlinear dynamical system with uncertainty (1) is asymptotically stable.

Substituting the nonlinear controller (2) in Equation (1) yields the following closed-loop system:

(32) $$\{\begin{array}{l}\dot{x}\left(t\right)=\left(A+\Delta A+BK\right)x\left(t\right)+D_i\omega \left(t\right)\hfill \\ y\left(t\right)=Cx\left(t\right)+D_o\omega \left(t\right)\hfill \end{array}$$

The state matrices in Equation (32) are inserted in the LMI derived from Theorem 2. Therefore:

(33) $$\left[\begin{array}{ccc}\left(\left(A+\Delta A\right)X+BW\right)+{\left(\left(A+\Delta A\right)X+BW\right)}^T& D_i& {\left(CX\right)}^{\mathrm{T}}\\ D_i{}^T& -\gamma I& D_o{}^T\\ CX& D_o& -\gamma I\end{array}\right]<0$$

Separating the uncertain terms yields:

(34) $$\left[\begin{array}{ccc}\left(AX+BW\right)+{\left(AX+BW\right)}^T+\Delta AX+{\left(\Delta AX\right)}^T& D_i& {\left(CX\right)}^{\mathrm{T}}\\ D_i{}^T& -\gamma I& D_o{}^T\\ CX& D_o& -\gamma I\end{array}\right]<0$$

The uncertainty term of (34) can be expressed as [[53]]:

(35) $$\Delta AX=M\Delta NX$$

Remark 2 implies that:

(36) $$=\left[\begin{array}{ccc}\left(AX+BW\right)+{\left(AX+BW\right)}^T+\alpha M{M}^T+{\alpha }^{-1}X{N}^{T}NX& D_i& {\left(CX\right)}^T\\ D_i{}^T& -\gamma I& D_o{}^T\\ CX& D_o& -\gamma I\end{array}\right]<0$$

By applying the Schur complement lemma [[57]]:

$$\begin{gathered} \left[\begin{array}{cccc}\Theta \left(X,W\right)& D_i& {\left(CX\right)}^{\mathrm{T}}& {\left(NX\right)}^{\mathrm{T}}\\ D_i{}^T& -\gamma I& D_o{}^T& 0\\ CX& D_o& -\gamma I& 0\\ NX& 0& 0& -\alpha I\end{array}\right]<0 \\ \Theta \left(X,W\right)=\left(AX+BW\right)+{\left(AX+BW\right)}^T+\alpha M{M}^T \end{gathered}$$

Finally, if the LMI defined in Theorem 3 has a solution, the system is asymptotically stable. □

To evaluate the performance of the proposed approach, we applied it to a ball and beam system and compared the obtained dynamics to that of the system proposed in [[1]]. In a ball and beam configuration, the objective is typically to balance a ball in an unstable equilibrium position. As shown in Figure 1, the ball and beam configuration includes a rotating servo-motor actuator, a rotary beam, and sensors at the beam's two ends. In this paper, we considers three scenarios. In the first scenario, we investigate the stability and controller design of a nominal nonlinear system with no disturbances using LMIs. Then, to overcome the system's disturbances, a perturbed nonlinear system is studied in the second scenario, and an LMI-based controller is developed. The third scenario simultaneously considers nonlinearities, perturbations, and uncertainty terms. According to Figure 2, the ball position coordinate is the beam angle coordinate.

To achieve kinetic energy, first, it is necessary to obtain the linear and angular velocities of the ball and beam.

(37) $$\{\begin{array}{c}{x}_{\mathrm{ball}}=r\cos \left(\theta \right)\\ y_{\mathrm{bal}}=r\sin \left(\theta \right)\end{array}\Rightarrow \begin{array}{c}{\dot{x}}_{\mathrm{ball}}=\dot{r}\cos \left(\theta \right)-r\dot{\theta }\sin \left(\theta \right)\\ {\dot{y}}_{\mathrm{ball}}=\dot{r}\sin \left(\theta \right)+r\dot{\theta }\cos \left(\theta \right)\end{array}$$

Therefore, the angular velocity of the ball and beam is:

(38) $$\{\begin{array}{l}{\omega }_{\mathrm{ball}}=-\frac{\dot{r}}{R}+\dot{\theta }\hfill \\ {\omega }_{\mathrm{beam}}=\dot{\theta }\hfill \end{array}$$

where $R$ is the ball radius. Consequently, the total kinetic and potential energy of the system are as follows:

(39) $$\{\begin{array}{l}K=\frac{1}{2}m\left({\dot{r}}^2+r^2{\dot{\theta }}^2\right)+\frac{1}{2}J{\dot{\theta }}^2+\frac{1}{2}{J}_R{\left(-\frac{\dot{r}}{R}+\dot{\theta }\right)}^2\hfill \\ P=mgr\sin \theta \hfill \\ L=K-P\hfill \end{array}$$

where $m$ is the ball's mass, and $J$ and $J_R$ are the moments of inertia of the ball and beam, respectively. By using the Lagrange relation:

(40) $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)-\left(\frac{\partial L}{\partial q}\right)=Q_i$$

The generalized coordinates of the system are $r$ and $\theta$ , yielding the following:

(41) $$\begin{array}{l}\frac{d}{dt}\left(m\dot{r}+\left(J_R/R^2\right)\dot{r}-\left(J_R/R\right)\dot{\theta }\right)+mg\sin \theta -mr{\dot{\theta }}^2=0\\ \Rightarrow \left(m+J_R/R^2\right)\ddot{r}-\left(\frac{J_R}{R}\right)\ddot{\theta }+mg\sin \theta -mr{\dot{\theta }}^2=0\end{array}$$

and the equation in $\theta$ direction is:

(42) $$\begin{array}{l}\frac{d}{dt}\left(J\dot{\theta }+m{r}^2\dot{\theta }+J_R\dot{\theta }-\left(J_R/R\right)\dot{r}\right)+mgr\cos \theta ={\tau }_u\\ \Rightarrow -\left(J_R/R\right)\ddot{r}+\left(J+m{r}^2+J_R\right)\ddot{\theta }+mgr\cos \theta +2mr\dot{r}\dot{\theta }={\tau }_u\end{array}$$

Therefore:

(43) $$\left[\begin{array}{cc}m+J_R/R^2& -J_R/R\\ -J_R/R& m{r}^2+J+J_R\end{array}\right]\left[\begin{array}{c}\ddot{r}\\ \ddot{\theta }\end{array}\right]+\left[\begin{array}{c}mg\sin \theta -mr{\dot{\theta }}^2\\ mgr\cos \theta +2mr\dot{r}\dot{\theta }\end{array}\right]=\left[\begin{array}{c}0\\ {\tau }_u\end{array}\right]$$

Finally, the equation can be expressed as follows:

(44) $$\left[\begin{array}{c}\ddot{r}\\ \ddot{\theta }\end{array}\right]=\frac{1}{\Delta }\left[\begin{array}{c}\left(m{r}^2+J+J_R\right)\left(mr{\dot{\theta }}^2-mg\sin \theta \right)+\left(J_R/R\right)\left({\tau }_u-mgr\cos \theta -2mr\dot{r}\dot{\theta }\right)\\ \left(J_R/R\right)\left(mr{\dot{\theta }}^2-mg\sin \theta \right)+\left(m+J_R/R^2\right)\left({\tau }_u-mgr\cos \theta -2mr\dot{r}\dot{\theta }\right)\end{array}\right]$$

where:

(45) $$\Delta =m\left(J+J_R\right)+m\left(m+J_R/R^2\right)r^2+J{J}_R/R^2$$

Therefore, the state equation of the system is:

(46) $$\frac{d}{dt}\left[\begin{array}{c}r\\ v\\ \theta \\ \omega \end{array}\right]=\left[\begin{array}{c}v\\ \frac{1}{\Delta }\left[\left(m{r}^2+J+J_R\right)\left(mr{\omega }^2-mg\sin \theta \right)+\left(\frac{J_R}{R}\right)\left({\tau }_u-mgr\cos \theta -2mrv\omega \right)\right]\\ \omega \\ \frac{1}{\Delta }\left[\left(J_R/R\right)\left(mr{\omega }^2-mg\sin \theta \right)+\left(m+J_R/R^2\right)\left({\tau }_u-mgr\cos \theta -2mrv\omega \right)\right]\end{array}\right]$$

The nonlinear term of the system in Equation (46) is extracted as follows:

(47) $$f\left(x\right)=\left[\begin{array}{c}0\\ \frac{1}{\Delta }\left[m\left(m{r}^2+J+J_R\right)r{\omega }^2-2m\left(\frac{J_R}{R}\right)rv\omega \right]\\ 0\\ \frac{1}{\Delta }\left[m\left(J_R/R\right)r{\omega }^2-2m\left(m+J_R/R^2\right)rv\omega \right]\end{array}\right]$$

Linearizing the system Equation (46) around the origin yields:

(48) $$\begin{array}{l}A=\left[\begin{array}{cccc}0& 1& 0& 0\\ -\frac{J_R/R\cdot mg}{m\left(J+J_R\right)+J{J}_R/R^2}& 0& -\frac{\left(J+J_R\right)\cdot mg}{m\left(J+J_R\right)+J{J}_R/R^2}& 0\\ 0& 0& 0& 1\\ -\frac{\left(m+J_R/R^2\right)\cdot mg}{m\left(J+J_R\right)+J{J}_R/R^2}& 0& -\frac{J_R/R\cdot mg}{m\left(J+J_R\right)+J{J}_R/R^2}& 0\end{array}\right]\\ B=\left[\begin{array}{c}0\\ \frac{J_R/R}{m\left(J+J_R\right)+J{J}_R/R^2}\\ 0\\ \frac{m+J_R/R^2}{m\left(J+J_R\right)+J{J}_R/R^2}\end{array}\right]\end{array}$$

where $x=[\stackrel{.}{}\stackrel{.}{}{]}^t$ . Eventually, using the obtained Equations (47) and (48), the dynamic equation of the system (44) can be converted to the form of relation (4). Then, the following equations are obtained after substituting the values in Table 1.

(49) $$\begin{array}{ll}\dot{x}\left(t\right)& =\left[\begin{array}{c}0\\ \frac{1}{\Delta }\left[m\left(m{x}_1{}^2+J+J_R\right)x_1{x}_4{}^2-2m\left(\frac{J_R}{R}\right)x_1{x}_2{x}_4\right]\\ 0\\ \frac{1}{\Delta }\left[m\left(J_R/R\right)x_1{x}_4{}^2-2m\left(m+J_R/R^2\right)x_1{x}_2{x}_4\right]\end{array}\right]\\ & +\left[\begin{array}{cccc}0& 1& 0& 0\\ -2.3207& 0& -6.997& 0\\ 0& 0& 0& 1\\ -537.6354& 0& -2.3207& 0\end{array}\right]x\left(t\right)+\left[\begin{array}{c}0\\ 2.1506\\ 0\\ 498.2257\end{array}\right]u\left(t\right)+\left[\begin{array}{c}0.2963\\ 0.7447\\ 0.189\\ 0.6868\end{array}\right]\sin \left(t\right)\\ y\left(t\right)& =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]x\left(t\right)+\left[\begin{array}{c}0.1835\\ 0.3685\end{array}\right]sin\left(t\right)\end{array}$$

The following mathematical analysis and proof of the stability and convergence properties of the proposed controller combined with the ball and beam system is presented. Consider the nominal Equation (49) without disturbances by replacing Equations (2) and (3) into system equations as follows:

(50) $$\begin{array}{l}\dot{x}\left(t\right)=A\left(x\right)+Bu\left(t\right)\hfill \\ y\left(t\right)=C\left(x\right)\hfill \end{array}$$

The aim is to design a robust $H_{\infty }$ controller to reduce the effect of input on the output; in particular, the bounded term $\Vert G_{yu}{\left(s\right)\Vert }_{\infty }$ should be minimized. Suppose the system is controllable from the input to the states with the following relation [[55]]:

(51) $$\Vert G_{yu}\Vert {}_{\infty }\le {\beta }^{-1}=\sup \frac{{\Vert y\Vert }_2}{{\Vert u\Vert }_2}u\ne 0<{\beta }^{-1}={{\displaystyle \int }}_0^{t\to \infty }\beta y^T\left(\tau \right)y\left(\tau \right)-{\beta }^{-1}{u}^T\left(\tau \right)u\left(\tau \right)d\tau <0$$

Then, consider a scalar Lyapunov function:

(52) $$\begin{array}{c}\mathrm{V}=x^{T}Px,P>0\\ x\left(0\right)=0\Rightarrow V\left(0\right)=x^{T}Px=0\end{array}$$

Based on Equation (51), suppose the derivative of the Lyapunov's function holds in the following relation:

(53) $$\dot{V}\left(x\left(t\right)\right)+\beta y^T\left(\tau \right)y\left(\tau \right)-{\beta }^{-1}{u}^T\left(\tau \right)u\left(\tau \right)<0$$

Integrating (53) yields:

(54) $$V\left(t\to \infty \right)+{{\displaystyle \int }}_0^{t\to \infty }\left[\beta y^T\left(\tau \right)y\left(\tau \right)-{\beta }^{-1}{u}^T\left(\tau \right)u\left(\tau \right)\right]d\tau <0$$

Because the function $V\left(x\right)$ is greater than zero, Equation (51) is obtained. Therefore, if relation (53) is established, then Equation (51) will also be proven. Substituting Equation (50) in $\dot{V}\left(x\right)$ and substituting in Equation (53) yields:

(55) $$\begin{gathered} \dot{V}(x(t))+\beta y^T(\tau )y(\tau )-{\beta }^{-1}{u}^T(\tau )u(\tau ) \\ ={\left[\begin{array}{c}x\\ u\end{array}\right]}^T\left[\begin{array}{cc}{A}^{T}P+PA+\beta C^{T}C& PB+\beta C^{T}D\\ B^{T}P+\beta D^{T}C& \beta D^{T}D-{\beta }^{-1}I\end{array}\right]⏟H\left[\begin{array}{c}x\\ u\end{array}\right]<0 \end{gathered}$$

If the matrix ( $H$ ) is negative and definite, then relation (53) and, consequently, relation (51) will be established. Assuming the pair $\left(A,B\right)$ is controllable, it can be shown that $H<0$ is also a necessary condition to fulfill Equation (51). The relation $H$ can be considered in the form of an LMI with the variable $P$ , where $P>0$ ; hence, if LMI $\left(P\right)$ has a solution, the condition $\Vert G_{yu}\Vert {}_{\infty }\le {\beta }^{-1}$ is met. Using Schur's complement lemma [[57]], the matrix $H$ results in:

(56) $$\mathrm{LMI}\left(P\right):\{\begin{array}{c}\left[\begin{array}{ccc}{A}^{T}P+PA& PB& C^T\\ B^{T}P& -{\beta }^{-1}I& D^T\\ C& D& -{\beta }^{-1}I\end{array}\right]\\ P>0\end{array}$$

If ${\beta }^{-1}$ is unknown, by defining the new variable, $\gamma ={\beta }^{-1}$ , the problem is defined in the new LMI $\left(\gamma ,P\right)$ in the following form:

(57) $$\mathrm{LMI}\left(\gamma ,P\right):\{\begin{array}{c}\left[\begin{array}{ccc}{A}^{T}P+PA& PB& C^T\\ B^{T}P& -\gamma I& D^T\\ C& D& -\gamma I\end{array}\right]<0\\ \begin{array}{c}P>0\\ \gamma >0\end{array}\end{array}$$

Therefore, the nominal dynamical system (50) is asymptotically stable if the LMI (56) or (57) is feasible. Next, by replacing the disturbances and uncertainties as inputs, such as (7), and implementing procedures (25) through (36), we can obtain relation (30), in which the system would be stable if the LMI has a solution.

For the first scenario, a nominal system without any perturbations is considered. The initial condition is $x\left(0\right)={\left[\begin{array}{cccc}0.5& 0& -0.2& -0.25\end{array}\right]}^T$ ; then, using MATLAB® YALMIP®, the variables $\gamma ,W,X,\mathrm{and}K$ will be obtained:

$$\begin{array}{l}\gamma =0.7843\\ W=\left[\begin{array}{ccc}0.7748& -0.684& \begin{array}{cc}0.0469& -0.336\end{array}\end{array}\right]\\ X=\left[\begin{array}{cccc}0.7187& -0.6294& 0.0521& -0.3067\\ -0.6294& 1.5083& 0.241& 0.8276\\ 0.0521& 0.241& 0.2137& -0.4017\\ -0.3067& 0.8276& -0.4017& 5.3009\end{array}\right]\\ K=[\begin{array}{cccc}1.1166& 0.0408& -0.1266& -0.0148\end{array}]\end{array}$$

The simulation results are shown in the following figures. The main goal of the ball and beam system is to keep the ball in a specific position. As shown in Figure 3, the proposed approach achieves asymptotic stability for the system, and the nonlinear system's states reach steady states after 2.8 s. Figure 4 shows that the proposed controller effort is bounded in both transient and steady states. Figure 1 depicts a comparison of the proposed approach and the method applied in [[1]], demonstrating that the proposed approach (illustrated using Theorem 1) has a faster response, particularly for output one.

For the perturbed nominal system, the initial condition is considered as $x\left(0\right)={\left[\begin{array}{cccc}0.5& 0& -0.2& -0.25\end{array}\right]}^T$ . Moreover, the applied disturbance type is $\omega \left(t\right)=\sin \left(t\right)$ , which varies over time; its related constants that affect the state derivatives and system outputs are: $D_i={\left[\begin{array}{cccc}0.2963& -0.7447& 0.189& 0.6868\end{array}\right]}^T$ and $D_o={\left[\begin{array}{cc}0.1835& 0.3685\end{array}\right]}^T$ , respectively. Using YALMIP®, the following variables will be obtained:

$$\begin{gathered} \gamma =0.8405 \\ W=\left[\begin{array}{cccc}2.0016& -3.7359& -0.3365& -2.9301\end{array}\right] \\ X=\left[\begin{array}{cccc}1.8556& -3.3983& -0.0744& -2.5043\\ -3.3983& 11.4914& 2.5947& 7.7637\\ -0.0744& 2.5947& 2.5743& -7.6734\\ -2.5043& 7.7637& -7.6734& 162.4374\end{array}\right] \\ K=\left[\begin{array}{cccc}1.2673& 0.1213& -0.2667& -0.0169\end{array}\right] \end{gathered}$$

As illustrated in Figure 5 and Figure 6, the proposed approach (Theorem 2) yields adequate performance with acceptable disturbance reduction. Despite the perturbation, states and control efforts would remain constrained and reach a steady state around 3.25 s. Furthermore, when compared to method applied in [[1]], the ball position converges to the origin faster, achieving better performance.

To assess the robustness of the proposed control, we assume the ball and beam system as described, with an added uncertainty term of a magnitude of 10% in matrix A. Consider the following initial condition and constant matrices:

$$\begin{gathered} x\left(0\right)=\left[\begin{array}{c}0.5\\ 0\\ -0.2\\ -0.25\end{array}\right]N=\left[\begin{array}{cccc}0.002& 0.0003& 0.0074& 0.005\\ 0.1005& 0.0048& -0.3665& 0.009\\ 0.0061& 0.0062& 0.0086& 0.0081\\ 0.1003& 0.0058& 0.1005& 0.0018\end{array}\right] \\ {D}_i=\left[\begin{array}{c}0.2963\\ -0.7447\\ 0.189\\ 0.6868\end{array}\right]D_o={\left[\begin{array}{cc}0.1835& 0.3685\end{array}\right]}^{T}M=I_4 \end{gathered}$$

Using the YALMIP® solver, the LMI condition (30) is solved as:

$$\begin{gathered} \gamma =0.8575 \\ \alpha =0.9169 \\ W=\left[\begin{array}{cccc}1.4106& -2.0595& -0.0271& -0.9381\end{array}\right] \\ X=\left[\begin{array}{cccc}1.3095& -1.895& 0.0109& -0.8543\\ -1.895& 4.5306& 0.801& 2.6356\\ 0.0109& 0.801& 0.575& -1.2041\\ -0.8543& 2.6356& -1.2041& 21.7883\end{array}\right] \\ K=\left[\begin{array}{cccc}1.7033& 0.4645& -0.8985& -0.0821\end{array}\right] \end{gathered}$$

The state trajectories and control effort in this case are depicted in Figure 7 and Figure 8, respectively.

Figure 7 and Figure 8 show that the proposed approach performs adequately. The nonlinear system's states and control efforts are always limited, and the provided controller reduces system disturbances to a reasonable extent. According to the results presented in Figure 7 and Figure 8, the proposed approach (Theorem 3) provides a faster and bounded answer with a lower amplitude of oscillations in steady states than the method presented in [[1]].

In this paper, we proposed an LMI-based nonlinear state feedback-stabilizing controller for a class of nonlinear systems with parametric uncertainties and external disturbances. The LMI stability criteria were first formulated for the nominal system; then, the disturbances and uncertainties were added. The LMI form was determined using the asymptotic stability criterion, and the nonlinear state-feedback control technique was determined. Among the advantages of the proposed approach are its low computational cost and its simple formulation, which is independent of the number of inputs and outputs in the system. The performance of the approach was assessed using a ball and beam system. The obtained results confirm the boundedness of the controller effort and states. Additionally, the proposed approach was shown to produce satisfactory results in the presence of varying disturbances. In steady state, the obtained results showed the ball reaching the origin in 3.6 s and oscillating with low amplitude. Our future work will focus on extending the proposed control approach by incorporating linear parametric variation (LPV) and input saturation.

Graph: Figure 1 Three-dimensional graphic of ball and beam system DS SolidWorks®, 2020 SP5.0 Premium x64.

Graph: Figure 2 Two-dimensional graphic of a ball and beam system Microsoft Visio®, Pro 2019 v2005 Build 14026.20302 Retail.

Graph: Figure 3 (a) State x1 trajectoriy of a nominal nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method applied in [[1]]). (b) State x2 trajectoriy of a nominal nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method applied in [[1]]). (c) State x3 trajectory of a nominal nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method applied in [[1]]). (d) State x4 trajectory of a nominal nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method applied in [[1]]).

Graph: Figure 4 Controller effort of the suggested approach in comparison to the method applied in [[1]].

Graph: Figure 5 (a) State x1 trajectory of a perturbed nominal nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method of Golestani et al., 2020, applied in [[1]]). (b) State x2 trajectory of a perturbed nominal nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method of Golestani et al., 2020, applied in [[1]]). (c) State x3 trajectory of a perturbed nominal nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method of Golestani et al., 2020, applied in [[1]]). (d) State x4 trajectory of a perturbed nominal nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method of Golestani et al., 2020, applied in [[1]]).

Graph: Figure 6 Control effort of suggested approach compared to the method of Golestani et al., 2020, applied in [[1]].

Graph: Figure 7 (a) State x1 trajectory of a perturbed and uncertain nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method of Golestani et al., 2020, applied in [[1]]). (b) State x2 trajectory of a perturbed and uncertain nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method of Golestani et al., 2020, applied in [[1]]). (c) State x3 trajectory of a perturbed and uncertain nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method applied in [[1]]). (d) State x4 trajectory of a perturbed and uncertain nonlinear system (indicating the suggested approach's stabilizing efficiency in comparison to the method of Golestani et al., 2020, applied in [[1]]).

Graph: Figure 8 Controller effort of proposed technique in comparison with the method of Golestani et al., 2020 [[1]].

Table 1 Physical parameter values for the ball and beam system.

Conceptualization, M.C., M.T.V. and S.M.; formal analysis, S.M., A.F., M.C. and M.T.V.; funding acquisition, A.F.; investigation, S.M., M.C. and M.T.V.; methodology, M.C. and S.M.; writing—original draft, M.T.V. and S.M.; writing—review and editing and supervision, S.M., M.C., A.F. and M.T.V. All authors have read and agreed to the published version of the manuscript.

The data that support the findings of this study are available within the article.

The authors declare no conflict of interest.

The authors appreciate the respected reviewers, the associate editor, and the editor-in-chief for their invaluable time.