About Unsolved Problems in Stabilization of Two Coupled Controlled Inverted Pendulums under Stochastic Perturbations

1. Introduction

The proposed paper continues a series of papers devoted to unsolved problems in the theory of stability and optimal control for stochastic systems (see [1,2,3,4,5]). Here the problem of the controlled inverted pendulum is considered, which appears many years ago [6] and has a long history (see [7] and the references therein). The mathematical model of the controlled inverted pendulum has the form of nonlinear differential equation of the second order

$$\ddot{x}\left(t\right)-asinx\left(t\right)=u\left(t\right),a>0,t\ge 0,$$

(1)

where $x\left(t\right)$ measures the angle between the rod and the upward vertical (Figure 1).

The classical way of stabilization for the equation (1) uses [6] the control $u\left(t\right)$, which is a linear combination of the state and velocity of the pendulum, i.e.,

$$u\left(t\right)=-b_{1}x\left(t\right)-b_2\dot{x}\left(t\right),b_1>a,b_2>0.$$

But this type of control, which represents instantaneous feedback, is quite difficult to realize because usually it is necessary to have some finite time to make measurements of the coordinates and velocities, to treat the results of the measurements and to implement them in the control action.

Unlike of the classical way of stabilization another way of stabilization is considered in [8]. It is supposed that only the trajectory of the pendulum is observed and the control $u\left(t\right)$ does not depend on the velocity, but depends on the all previous values of the trajectory $x\left(s\right)$, $s\le t$, and is given in the form

$$u\left(t\right)={\int }_0^{\infty }dK\left(\tau \right)x(t-\tau ),$$

(2)

where the kernel $K\left(\tau \right)$ is a continuous from the right function of bounded variation on $[0,\infty )$ and the integral is understood in the Stieltjes sense. It means, in particular, that both distributed and discrete delays can be used depending on the concrete choice of the kernel $K\left(\tau \right)$.

2. Zero and Nonzero Equilibria

It is clear that (1), (2) define the delay differential equation

$$\ddot{x}\left(t\right)-asinx\left(t\right)={\int }_0^{\infty }dK\left(\tau \right)x(t-\tau ),a>0,t\ge 0,$$

(3)

which has the zero solution. However, this equation has not the zero solution only. Put

$$k_i={\int }_0^{\infty }{\tau }^{i}dK\left(\tau \right),i=0,1,k_2={\int }_0^{\infty }{\tau }^2\left|dK\left(\tau \right)\right|,$$

(4)

and note that the nonzero equilibrium $x\left(t\right)=\widehat{x}$ of the equation (3) is a root of the equation

$$asin\widehat{x}+k_0\widehat{x}=0$$

(5)

or $S\left(\widehat{x}\right)=0$, where

$$S\left(x\right)={\displaystyle \frac{sinx}{x}}+{\displaystyle \frac{k_0}{a}}.$$

Remark 1.

As it is noted in [7] for all $x\ne 0$

$$-\alpha \le {\displaystyle \frac{sinx}{x}}<1,$$

where $0.217233<\alpha <0.217234$. Therefore, if

$$-\alpha \le -{\displaystyle \frac{k_0}{a}}<1$$

or

$$0<a+k_0\le (1+\alpha )a$$

then there exists at least one nonzero root of the equation (5).

3. Two Types of Stochastic Perturbations

In addition, it is supposed that the pendulum is under influence of stochastic perturbations, so, the considered stabilization problem is a problem of the theory of stochastic functional differential equations [8,9,10].

Let $\{\Omega ,\mathfrak{F},\mathbf{P}\}$ be a complete probability space, $\{{\mathfrak{F}}_t,t\ge 0\}$ be a nondecreasing family of sub-$\sigma$-algebras of $\mathfrak{F}$, i.e., ${\mathfrak{F}}_s\subset {\mathfrak{F}}_t$ for $s<t$, $\mathbf{E}$ be the mathematical expectation with respect to the probability $\mathbf{P}$. Let $w\left(t\right)$ and $\nu \left(t\right)$ be respectively ${\mathfrak{F}}_t$-measurable the Wiener and the Poisson processes, $\mathbf{E}\nu \left(t\right)=\lambda t$, $\lambda >0$ , $\tilde{\nu }\left(t\right)=\nu \left(t\right)-\lambda t$ [9,10],

$$\xi \left(t\right)=\sigma w\left(t\right)+\gamma \tilde{\nu }\left(t\right).$$

(6)

Supposing for the beginning that the parameter a in (1) is influenced by additive stochastic perturbations, i.e., $a\to a+\dot{\xi }\left(t\right)$, where $\xi \left(t\right)$ is defined in (6), and using the control (2), rewrite the equation (1) as

$$\ddot{x}\left(t\right)-(a+\dot{\xi }\left(t\right))sinx\left(t\right)={\int }_0^{\infty }dK\left(\tau \right)x(t-\tau ),a>0,t\ge 0,$$

(7)

or, using $x_1\left(t\right)=x\left(t\right)$, $x_2=\dot{x}\left(t\right)$ and (6), in the form of the system of stochastic differential equations [9,10] with the zero solution

$$\begin{gathered} d{x}_1\left(t\right)=x_2\left(t\right)dt, \\ d{x}_2\left(t\right)=\left(asin{x}_1\left(t\right)+{\int }_0^{\infty }dK\left(\tau \right)x_1(t-\tau )\right)dt \\ +sin{x}_1\left(t\right)(\sigma dw\left(t\right)+\gamma d\tilde{\nu }\left(t\right)). \end{gathered}$$

(8)

From the other hand let us suppose that the equation (3) is influenced by additive stochastic perturbations of the form $(x_1\left(t\right)-\widehat{x})\dot{\xi }\left(t\right)$, where $\widehat{x}$ is a nonzero root of the equation (5) and $\xi \left(t\right)$ is defined in (6). Then similarly to (8) we obtain the system of stochastic differential equations with the solution $x_1\left(t\right)=\widehat{x}$, $x_2\left(t\right)=0$:

$$\begin{gathered} d{x}_1\left(t\right)=x_2\left(t\right)dt, \\ d{x}_2\left(t\right)=\left(asin{x}_1\left(t\right)+{\int }_0^{\infty }dK\left(\tau \right)x_1(t-\tau )\right)dt \\ +(x_1\left(t\right)-\widehat{x})(\sigma dw\left(t\right)+\gamma d\tilde{\nu }\left(t\right)). \end{gathered}$$

(9)

Using the general method of Lyapunov functionals construction [8], in [7] the following two theorems about stability in probability (see definition in [7,8]) have been proven

Theorem 1.

Let be

$$\begin{gathered} {\alpha }_1=-(a+k_0)>0,k_1>0,k_2<2, \\ {\sigma }^2+\lambda {\gamma }^2<2{\alpha }_1\left(k_1-k_2\sqrt{{\displaystyle \frac{{\alpha }_1}{2(2-k_2)}}}\right), \end{gathered}$$

(10)

where $k_i$, $i=0,1,2$, are defined in (4). Then the zero solution of the equation (7) (or the system (8)) is stable in probability.

Theorem 2.

Let be

$$\begin{gathered} {\alpha }_2=-(acos\left(\widehat{x}\right)+k_0)>0,k_1>0,k_2<2, \\ {\sigma }^2+\lambda {\gamma }^2<2{\alpha }_2\left(k_1-k_2\sqrt{{\displaystyle \frac{{\alpha }_2}{2(2-k_2)}}}\right), \end{gathered}$$

(11)

where $k_i$, $i=0,1,2$, are defined in (4). Then the nonzero equilibrium $\widehat{x}$ of the system (9) is stable in probability.

4. Unsolved Problems

Let us consider the mathematical model of two coupled controlled inverted pendulums in the form of the system of two nonlinear differential equations of the second order:

$$\begin{gathered} {\ddot{x}}_1\left(t\right)-a_{1}sin{x}_1\left(t\right)-b_1{x}_2\left(t\right)=u_1\left(t\right), \\ {\ddot{x}}_2\left(t\right)-a_{2}sin{x}_2\left(t\right)-b_2{x}_1\left(t\right)=u_2\left(t\right), \\ {a}_1>0,a_2>0,t\ge 0. \end{gathered}$$

(12)

It is assumed that only the trajectories of both pendulums are observed, the controls $u_1\left(t\right)$ and $u_2\left(t\right)$ do not depend on the velocity, but depend respectively on all previous values of the trajectories $x_1\left(s\right)$ and $x_2\left(s\right)$, $s\le t$, and are given in the form

$$u_1\left(t\right)={\int }_0^{\infty }d{K}_1\left(\tau \right)x_1(t-\tau ),u_2\left(t\right)={\int }_0^{\infty }d{K}_2\left(\tau \right)x_2(t-\tau ).$$

(13)

Similarly to (2) here it is supposed that $K_1\left(\tau \right)$ and $K_2\left(\tau \right)$ are continuous from the right functions of bounded variation on $[0,\infty )$ and the integrals are understood in the Stieltjes sense.

If $b_1=b_2=0$ then the system (12), (13) splits into two independent equations, for each from which the statements of both Theorems 1 and 2 can be formulated.

Putting

$$x_{11}\left(t\right)=x_1\left(t\right),x_{12}\left(t\right)={\dot{x}}_1\left(t\right),x_{21}\left(t\right)=x_2\left(t\right),x_{22}\left(t\right)={\dot{x}}_2\left(t\right),$$

we obtain the following two problems.

4.1. The Problem 1

Supposing that the parameters $a_i$, $i=1,2$, in (12) are influenced by stochastic perturbations

$$a_i\to a_i+{\dot{\xi }}_i\left(t\right),{\xi }_i\left(t\right)={\sigma }_i{w}_i\left(t\right)+{\gamma }_i{\tilde{\nu }}_i\left(t\right),i=1,2,$$

where ${\tilde{\nu }}_i\left(t\right)={\nu }_i\left(t\right)-{\lambda }_{i}t$, $w_i\left(t\right)$ and ${\nu }_i\left(t\right)$ are mutually independent the Wiener and the Poisson processes, we obtain

$$\begin{gathered} {\dot{x}}_{11}\left(t\right)=x_{12}\left(t\right), \\ {\dot{x}}_{12}\left(t\right)=(a_1+{\dot{\xi }}_1\left(t\right))sin{x}_{11}\left(t\right)+b_1{x}_{21}\left(t\right)+{\int }_0^{\infty }d{K}_1\left(\tau \right)x_{11}(t-\tau ), \\ {\dot{x}}_{21}\left(t\right)=x_{22}\left(t\right), \\ {\dot{x}}_{22}\left(t\right)=(a_2+{\dot{\xi }}_2\left(t\right))sin{x}_{21}\left(t\right)+b_2{x}_{11}\left(t\right)+{\int }_0^{\infty }d{K}_2\left(\tau \right)x_{21}(t-\tau ). \end{gathered}$$

or

$$\begin{gathered} d{x}_{11}\left(t\right)=x_{12}\left(t\right)dt, \\ d{x}_{12}\left(t\right)=\left(a_{1}sin{x}_{11}\left(t\right)+b_1{x}_{21}\left(t\right)+{\int }_0^{\infty }d{K}_1\left(\tau \right)x_{11}(t-\tau )\right)dt \\ +sin{x}_{11}\left(t\right)({\sigma }_{1}d{w}_1\left(t\right)+{\gamma }_{1}d{\tilde{\nu }}_1\left(t\right)), \\ d{x}_{21}\left(t\right)=x_{22}\left(t\right)dt, \\ d{x}_{22}\left(t\right)=\left(a_{2}sin{x}_{21}\left(t\right)+b_2{x}_{11}\left(t\right)+{\int }_0^{\infty }d{K}_2\left(\tau \right)x_{21}(t-\tau )\right)dt \\ +sin{x}_{21}\left(t\right)({\sigma }_{2}d{w}_2\left(t\right)+{\gamma }_{2}d{\tilde{\nu }}_2\left(t\right)). \end{gathered}$$

(14)

Problem 1: Formulate and prove for the system (14) an analogue of the Theorem 1.

4.2. The Problem 2

The nonzero equilibrium $({\widehat{x}}_1,{\widehat{x}}_2)$ of the system (12), (13) is a solution of the system of algebraic equations

$$\begin{gathered} a_{1}sin{\widehat{x}}_1+b_1{\widehat{x}}_2+k_{01}{\widehat{x}}_1=0, \\ {a}_{2}sin{\widehat{x}}_2+b_2{\widehat{x}}_1+k_{02}{\widehat{x}}_2=0, \end{gathered}$$

(15)

where $k_{0i}={\int }_0^{\infty }d{K}_i\left(\tau \right)$, $i=1,2$, and is also the equilibrium of the system

$$\begin{gathered} d{x}_{11}\left(t\right)=x_{12}\left(t\right)dt, \\ d{x}_{12}\left(t\right)=\left(a_{1}sin{x}_{11}\left(t\right)+b_1{x}_{21}\left(t\right)+{\int }_0^{\infty }d{K}_1\left(\tau \right)x_{11}(t-\tau )\right)dt+(x_{11}\left(t\right)-{\widehat{x}}_1)d{\xi }_1\left(t\right), \\ d{x}_{21}\left(t\right)=x_{22}\left(t\right)dt, \\ d{x}_{22}\left(t\right)=\left(a_{2}sin{x}_{21}\left(t\right)+b_2{x}_{11}\left(t\right)+{\int }_0^{\infty }d{K}_2\left(\tau \right)x_{21}(t-\tau )\right)dt+(x_{21}\left(t\right)-{\widehat{x}}_2)d{\xi }_2\left(t\right). \end{gathered}$$

(16)

Problem 2: Formulate and prove for the system (16) an analogue of the Theorem 2.

5. Conclusions

The list of the author’s unsolved problems continues to increase. But the author continues to hope that his unsolved problems will be solved someday, and invites interested readers to help him with this.