With the development of modern technology, many scien-

tists are studying a wide variety of movements of complex

systems, as well as control of the movements of linear systems

subjected to complex external actions. Similar studies are

carried out both in the ﬁeld of motion control of composite

linear systems, and in the ﬁeld of motion control of systems

with heteronode inﬂuences (hybrid actions). Some of those

problems are addressed as hybrid control problems. In [1],

[2] the authors present a method to achieve such a hybrid

control of position and force. In [3] the hybrid systems as

causal and consistent dynamical systems were discussed, and a

general formulation for an optimal hybrid control problem was

proposed. In [4] the authors survey recent results in the ﬁeld

of optimal control of hybrid and switched systems. They ﬁrst

summarize results that use different problem formulations and

then explore the underlying relations among them. Speciﬁcally,

based on the type of switching, they focus on two important

classes of problems: internally forced switching (IFS) prob-

lems and externally forced switching (EFS) problems. For

IFS problems, they focus on optimal control techniques for

piecewise afﬁne systems. For EFS problems, methodologies

of two-stage optimization, embedding transformation, and

switching LQR design are investigated. Detailed optimization

methods found in the literature are discussed.

Research on hybrid control problems has also been done

for the study of the Covid-19 epidemic. In [5] is being study

epidemics using mathematical modeling, which is crucial for

understanding its dynamics and proposing potential control

measures. A generalized epidemiological model corresponding

to a pandemic is proposed, in which its dynamics are presented

as a new hybrid system obtained by combining a deterministic

model with a stochastic model.

A new hybrid method for construction of control actions of

a linear control system with constant coefﬁcients is considered

in paper [6]. Let as start by presenting some of the main points

of the work.

Assume we have a state space model which have the

following dynamics

˙xi=ai1x1+··· +ainxn+pi1y1+···+

+pikyk+bi1u1+··· +bir ur,(1)

˙yj=cj1y1+··· +cjkyk+dj1x1+··· +djmxm,(2)

where the coefﬁcients ail,bis,cjq ,djf ,piq are real constants

and i= 1, . . . , n,j= 1, . . . , k,r⩽n−m,l= 1, . . . , n,m⩽

k⩽n,s= 1, . . . , r,q= 1, . . . , k. Also x1, . . . , xn, y1...,yk

are the states of the system, and u1, . . . , uris the control

actions applied to the system. We can rewrite the system (1)-

(2) as a system of matrix equations

˙x=Ax +P y +Bu, (3)

˙y=Cy +D¯x. (4)

Control of the Motion of an Inverted Spherical Pendulum on a Moving

Base. Hybrid Impact Approach

ARA AVETISYAN1, SMBAT SHAHINYAN2

1Department on Dynamics of Deformable Systems and Connected Fields

Institute of Mechanics of NAS RA

Yerevan, ARMENIA

2Depatment of Mathematics and Mechanics

Yerevan State University

Yerevan, ARMENIA

Abstract: -A new hybrid method for the construction of control actions of a linear control system with constant coefficients

is considered in this paper. It is assumed in this paper that a part of the discussed system meets some conditions. Some

states of the main system are considered to be control actions for a subsystem for which and LQR stabilizer is acquired.

Then, those control actions of the subsystem are used to construct the control actions for the main system. In the problem

of controlling the motion of a complex linear system of an inverted spherical pendulum on a moving base, a new approach

to the construction of control actions (hybrid action method) was used. It is assumed that a component of the complex

system under discussion satisfies certain conditions. The inertial forces at the center of mass of the base of the composite

system are considered to be the controlling influences on the inversion of the pendulum, for which the LQR stabilizer was

purchased. The determined internal control actions on the inverted pendulum are then used to construct external control

actions on the base of the composite system. In the end, a numerical analysis was carried out.

Key-words: —control problem, hybrid control, linear system, optimal solutions, optimal stabilization, control actions,

numerical example.

Received: March 8, 2024. Revised: September 11, 2024. Accepted: October 7, 2024. Published: November 5, 2024.

1. Introduction

2. Problem Description

EQUATIONS

DOI: 10.37394/232021.2024.4.6

Ara Avetisyan, Smbat Shahinyan

E-ISSN: 2732-9976

Volume 4, 2024

where

A=





a11 ··· a1n

an1··· ann





, P =





p11 ··· p1k

pn1··· pnk





,

B=





b11 ··· b1r

bn1··· bnr





, C =





c11 ··· c1k

ck1··· ckk





,

D=





d11 ··· d1m

dk1··· dkm





.

Here, x= (x1···xn)Tis an ndimensional column vector,

y= (y1···yk)Tis a kdimensional column vector, ¯x=

(¯x1··· ¯xm)Tis an mdimensional column vector that contains

some mstates of xand u= (u1···ur)Tis an rdimensional

column vector.

Let us now deﬁne the following problem.

Problem Deﬁnition 1. We are given the system (1)-(2) (or

(3)-(4), the time period [t0, t1], the initial position of some of

the states (maximum number of the states of (3) can be n/2

and for (4) the number can be kof the system (x(t0); y(t0)) =

(x0;y0)and the desired ﬁnal position of some of the states

(maximum number of the states of (3) can be n/2and for (4)

the number can be kof the system x(t1) = x1. It is required

to ﬁnd the control inputs u(t), (t0⩽t⩽t1) such that it drives

the system from its given initial position to its desired ﬁnal

position.

Assume, the matrices A,B,C,Dare such that

rankK1={D, CD, . . . , Ck−1D}=k(5)

and

rankK2={B1, A1B1, . . . , An+k−1

1B1}=n+k. (6)

Where A1is the following (n+k)×(n+k)matrix

A1=







a1 1 ··· a1ma1m+1 ··· a1np11 ··· p1k

an1··· an m an m+1 ··· an n pn1··· pn k

d1 1 ··· d1m0··· 0c1 1 ··· c1k

dk1··· dk m 0··· 0ck1··· ck k







and B1is an (n+k)×rmatrix as shown below.

B1=







b1 1 ··· b1r

bn1··· bn r

0··· 0







Suppose, also, that there is an additional condition for the

system (2) (or (4)) which assumes that the states y1, . . . , yk

remain close to the point O(0,··· ,0),y(t1) = y1is inﬁnitely

close to zero and there is a constraint given on the system (2)

(or (4)). Now suppose that the constraint is given as

J[•] =

∞





i,j=1

αij yiyj+

i,j=1

βij xixj

dt. (7)

Thus, we can choose x1, . . . , xmto be control actions for

the system (2) (or (4)), and hence, we can deﬁne to the

following problem.

Problem Deﬁnition 2. Assume we are given the dynamics

of the state space model (2) (or (4)) and the constraint (7).

We need to ﬁnd the control actions ¯x0

1[t],...,¯x0

m[t]such that

the system (2) (or (4)) becomes asymptotically stable and the

constraint (7) reaches its minimal value.

Now, because of the assumption (5) the system (2) (or

(4)) becomes fully controllable [1], hence, for any reasonable

initial position y(t0) = y0there exists unique (x0

1···x0

m)T

column vector of control actions which solve the problem 2

[2]. This means that also the states y0

1(t), . . . , y0

k(t)will be

calculated uniquely, moreover

lim

t→∞ ¯x0

i[t]=0,(i= 1, . . . , m)(8)

and

lim

t→∞ y0

i(t)=0,(i= 1, . . . , k).(9)

Now that we solved the second problem, we will discuss the

problem 1. So, by substituting the functions ¯x0

1[t],...,¯x0

m[t]

and y0

1(t), . . . , y0

k(t), which we gained by solving the prob-

lem 2, into the system (1) (or (3)), we can rewrite the system

˙xi=ai m+1xm+1 +. . . +ai nxn+

+bi1u1+. . . +bi nun+fi(t)(10)

where i=m+ 1, . . . , n, and

fi(t) = ai1¯x0

1[t] + . . . +ai m ¯x0

m[t]+

+pi1y0

1(t) + . . . +pi ky0

k(t)(11)

where i= 1, . . . , n. It is obvious that ﬁrst mequations from

the system (10) will become algebraic equations because the

functions x0

1[t], . . . , x0

m[t]will be already known.

According to (6) the system (1)-(2) is fully controllable,

hence, it remains to calculate the control actions u=

(u1(t)···ur(t))Twhich solve the ﬁrst problem, and that

can be done by choosing some known algorithm. Thus, the

problem is solved.

Assume we have a cart with mass Mthat can move freely

on a horizontal plane. Furthermore, assume that there is an

inverted pendulum made of a weightless rod of length land a

3. Differential Equations of Motion and

the Definition of the Problem

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DOI: 10.37394/232021.2024.4.6

Ara Avetisyan, Smbat Shahinyan

E-ISSN: 2732-9976

Volume 4, 2024

small ball of mass m attached to the center of mass Cof the

cart end. Assume the pendulum is attached to the cart with a

ball joint (Fig. 1):

Fig. 1.

Let us introduce a ﬁxed coordinate system Oxyz and a

coordinate system Cx1y1z1attached to the center of mass

Cof the cart to study the movement of the abovementioned

mechanical system. It is also assumed that the axis of Cx1y1z1

remains parallel to corresponding axis of Oxyz while the

mechanical system moves and the control action u which is

in the plane Oxy acts on the cart.

In that case, the coordinates of the ball in Oxyz will be the

following

x=xC+lsin θcos φ, y =yC+lsin θsin φ, (12)

where (xC, yC,0) are the coordinates of the center of mass of

the cart Cin the ﬁxed coordinate system, and (l, θ, φ)are the

spherical coordinates of the ball in Cx1y1z1.

Let us write the differential equations of the mechanical

system. The kinetic energy of the system will be as follows.

T=1

2M˙x2

C+ ˙y2

C+1

2m˙x2

C+ ˙y2

C+l2cos2θ·˙

θ2+

l2sin2θ·˙φ2+ 2l˙xC˙

θcos θcosφ −2l˙xC˙φsin θsin φ+

+ 2l˙yC˙

θcos θsin φ+ 2l˙yCφsin θcos φ,(13)

As for the potential energy, we will have

Π = −mgl(1 −cos θ) : (14)

Using the Lagrangian [7] the mechanical system we will have

the below system as a linear approximation of the differential

equations of the motion.







(M+m)¨xC+ml¨

θ=ux,

(M+m)¨yC=uy,

l¨

θ+ ¨xC=−gθ.

(15)

where uxand uyare the projections of control aciton u in the

coordinate system Oxyz. It is easy to check that the system

(15) is fully controllable. Let us now deﬁne the following

problem.

Problem 1.1. Given the system (15), the time interval

[t0, t1], initial position xC(t0) = xC0,yC(t0) = yC0,

˙yC(t0) = ˙yC0,θ(t0) = θ0,˙

θ(t0) = ˙

θ0and the desired

ﬁnal position xC(t1) = xC1,yC(t1) = yC1,˙yC(t1) = ˙yC1

for the system (15). Find a control action u such that it

drives the system from its given initial position to its ﬁnal

position in [t0, t1]while keeping the pendulum close to its

upper equilibrium point.

We will solve this problem using the hybrid control method

mentioned in part 1. Let us consider only the last equation of

the system (15) which is

l¨

θ+ ¨xC=−gθ. (16)

Let us make the notations x1=θ,x2=˙

θ, and consider ¨xC

as a control action. We will then have

˙x1=x2,

˙x2=−k2x1+u1.(17)

Here k2=g

l,u1=−¨xC

l. It is easy to check that the system

(17) is fully controllable as well.

Now let us state an optimal stabilization problem for the

system (17).

Problem 2.1. Assume we are given the control system (17).

It is required to ﬁnd a control action u0

1(x1, x2, t)such that

the system (17) becomes asymptotically stable when u1=

1(x1, x2, t)and the constraint

J[·] =

∞

(x2

2+u2

1)dt (18)

reaches its minimal value.

We will solve the second problem using Lyapunov-Belman

method [8]. Belman’s expression will have the form

B[·] = x2+∂V

∂x2

(u1−k2x1) + x2

2+u2

Here Vis Lyapunov function. For the optimal control action,

we know ∂B

∂u1



u1=u0

= 0,

Thus

1=−1

∂V

∂x2

.(19)

Hence, to get the optimal Lyapunov’s function we will get

∂V

∂x2

x2−k2∂V

∂x2

x1+x2

2−∂V

∂x22

= 0.(20)

differential equation with partial derivatives. We will look for

a Lyapunov’s function as a quadratic form. From (20) we will

have

V0(x1, x2) = k2x2

1+x2

2,thus u0

1=−x2=−˙

θ.

Substituting into (17) we get

˙x1=x2,

˙x2=−k2x1−x2.(21)

. 6ROXWLRQRIWKH3UREOHP

EQUATIONS

DOI: 10.37394/232021.2024.4.6

Ara Avetisyan, Smbat Shahinyan

E-ISSN: 2732-9976

Volume 4, 2024

Solving the system (21) for k > 0,5we will have for the

spherical pendulum the optimal deﬂection angle from the

vertical position and its angular velocity as functions of time:

θ0(t) = x0

1(t) = e−0,5tAsin p4k2−1t+

+Bcos p4k2−1t,(22)

θ0(t) = x0

1(t) =

=e−0,5t−0,5A−Bp4k2−1sin p4k2−1t+

+Ap4k2−1−0,5Bcos p4k2−1t(23)

For the case when 0< k ⩽0,5the solutions will be

exponentially decreasing functions. We will not show those

solutions here.

Aand Bin (22) and (23) are integration constants and are

calculated using the initial conditions θ(t0) = θ0and ˙

θ(t0) =

θ0:

A=2˙

θ0+θ0

2√4k2−1, B =θ0.(24)

Now let us get back to solving the Problem 1.1. According

to the notation u1=−¨xC

lwe will have

¨xc(t) = −lu0

1(t) = l˙

θ0(t) =

=le−0.5t−0,5A−Bp4k2−1sin p4k2−1t+

+Ap4k2−1−0,5Bcos p4k2−1t,

thus

˙xC(t) = lθ0(t) + C1=

=le−0,5t(Asin p4k2−1t+Bcos p4k2−1t) + C1,

xC(t) = le−0,5t

4k2−0,75 Bp4k2−1−0,5A·

·sin p4k2−1t−Ap4k2−1+0,5B·

·cos p4k2−1t+C1t+C2.(25)

Using the initial and the desired ﬁnal positions xC(t0) =

xC0and xC(t1) = xC1from the system (25) we can calculate

the integration constants C1and C2.

Let us now calculate the uxpart of the control action. From

the ﬁrst equation of (15) we get

ux(t) =(M+m)¨xC+ml¨

θ=le−0,5t·

·(M+m)−Bp4k2−1−0,5A+

+m(1,25 −4k2)A+Bp4k2−1·

·sin p4k2−1t+

+(M+m)Ap4k2−1−0,5B+

+m(1,25 −4k2B−Ap4k2−1·

·cos p4k2−1t(26)

It remains to solve only the second equation of (15) which

(M+m)¨yC=uy(27)

This equation itself is a simple control equation. We can add

a cost function and deﬁne an optimal control problem for (27)

which we can solve by any known method (e.g. the method

of solution may be a problem of momentums). However, we

will not present the problem in this paper.

Let us introduce the below values to calculate the control

actions and the trajectories as functions of time only.

M= 10kg,m= 1kg,l= 0.5m,t0= 0s,t1= 50s,

xC(t0) = 0m,yC(t0) = 0m,θ(t0) = 0.5,θ(t0) = 1s−1,

xC(t1) = 3m,yC(t1) = 2m.

We will have

θ0(t) = e−0.5t(0.1421 sin 8.7977t+ 0.5 cos 8.7977t),

θ0(t) = e−0.5t(−4.4699 sin 8.7977t+ 0.9922 cos 8.7977t),

xc(t)=0.0097 + 0.0598t+ 0.0064e−0.5t(−1.5 cos[8.7977t]+

+ 4.3278 sin[8.7977t]),

˙xC(t)=0.5e−0.5t(0.1421 sin 8.7977t+

+ 0.5 cos 8.7977t)+0.0598,

ux(t) = −0.5e−0.5t(28.825 cos 8.7977t+

+ 55.7317 sin 8.7977t)

State trajectories and graphs of control actions are constructed

and shown in pictures 2-4.

Fig. 2. The graph of the change in the θ0(t)angle of deviation of the

pendulum from the vertical as a function of time.

. 1XPHULFDO([DPSOH

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DOI: 10.37394/232021.2024.4.6

Ara Avetisyan, Smbat Shahinyan

E-ISSN: 2732-9976

Volume 4, 2024

Fig. 3. The graph of the change in the xC(t)coordinate of the center of

mass of the pendulum as a function of time.

Fig. 4. Graph of control action ux(t)

A hybrid control problem of a system of linear differential

equations with constant coefﬁcients is discussed in this paper.

It was assumed that some of the states of the system have to

satisfy some additional conditions. To ensure those conditions

are satisﬁed, some of the states of one subsystem were chosen

to be additional control actions in second subsystem. Then,

an optimal stabilization problem was deﬁned and solved for

the second subsystem using Lyapunov-Bellman method. The

special states which were chosen to be control actions and

the corresponding optimal trajectories were acquired for the

second subsystem. Afterwards, those solutions are substituted

in the ﬁrst subsystem and the main control problem was

solved. An example of a hybrid control problem of an inverted

spherical pendulum with a moving base is studied. The pendu-

lum is chosen as a subsystem the motion of which is controlled

by the moving base (the cart). Analytical representations of

the states and control action are calculated and presented. The

optimal trajectories and the graph of the control action are

constructed for a numerical example.

[1] 1. Marc H. Raibert and John J. Craig. Hybrid position/force control of

Manipulators. Trans ASME Journal of Dynamics, Systems, Measure-

ment, and Control, 1981, vol. 102, pp.˙

126–133.

[2] Hong Zhang and Richard P, Paul, “Hybrid control of robot manipula-

tors”, In Proceedings of the IEEE International Conference on Robotics

and Automation, 1985, pp. 602–607.

[3] P. Riedinger, F. Kratz, “An Optimal Control Approach for Hybrid

Systems,” European Journal of Control, vol. 9 (5), 2003, pp. 449-458.

[4] F. Zhu, P. J. Antsaklis, “Optimal control of hybrid switched systems,”

A brief survey, Discrete Event Dyn Syst, 2015, vol. 25, pp. 345-–364.

DOI 10.1007/s10626-014-0187-5

[5] Sh. Dharmatti, N. Krishnan, “Mathematical modeling and opti-

mal control analysis of pandemic dynamics as a hybrid sys-

tem,” European Journal of Control, vol. 75, January 2024, 100942.

https://doi.org/10.1016/j.ejcon.2023.100942

[6] A. S. Avetisyan, A. S. Shahinyan, “A Hybrid Control Problem for

a Linear System with Constant Coefﬁcients”, Reports of National

Academy of Sciences of Armenia, vol. 121 (2), pp. 91–99.

[7] N. N. Buchholz, The Main Course of Theoretical Mechanics, vol. 2,

Moscow, Nauka, 1972, pp. 332 (in Russian).

[8] E. G. Albrecht, G. S. Shelementyev, Lectures on stabilization theory,

Sverdlovsk, 1972, pp. 274 (in Russian).

. Conclusion

References

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DOI: 10.37394/232021.2024.4.6

Ara Avetisyan, Smbat Shahinyan

E-ISSN: 2732-9976

Volume 4, 2024

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

EQUATIONS

DOI: 10.37394/232021.2024.4.6

Ara Avetisyan, Smbat Shahinyan

E-ISSN: 2732-9976

Volume 4, 2024