Keywords

1 Introduction

Output regulation of control systems is one of the very important problems in control systems theory. Basically, the output regulation problem is to control a fixed linear or nonlinear plant in order to have its output tracking reference signals produced by some external generator (the exosystem).

For linear control systems, the output regulation problem has been solved by Francis and Wonham [12]. For nonlinear control systems, the output regulation problem has been solved by Byrnes and Isidori [5] generalizing the internal model principle obtained by Francis and Wonham [12]. Byrnes and Isidori [5] have made an important assumption in their work which demands that the exosystem dynamics generating reference and/or disturbance signals is a neutrally stable system (Lyapunov stable in both forward and backward time). The class of exosystem signals includes the important particular cases of constant reference signals as well as sinusoidal reference signals. Using Centre Manifold Theory [7], Byrnes and Isidori have derived regulator equations, which completely characterize the solution of the output regulation problem of nonlinear control systems.

The output regulation problem for linear and nonlinear control systems has been the focus of many studies in recent decades [48]. In [21], Mahmoud and Khalil obtained results on the asymptotic regulation of minimum phase nonlinear systems using output feedback. In [13], Fridman solved the output regulation problem for nonlinear control systems with delay using centre manifold theory [7]. In [11], Chen and Huang obtained results on the robust output regulation for output feedback systems with nonlinear exosystems. In [18], Liu and Huang obtained results on the global robust output regulation problem for lower triangular nonlinear systems with unknown control direction. In [88], Yang and Huang obtained new results on the global robust output regulation problem for nonlinear plants subject to nonlinear exosystems.

In [14], Immonen obtained results on the practical output regulation for bounded linear infinite-dimensional state space systems. In [23], Pavlov, Van de Wouw and Nijimeijer obtained results on the global nonlinear output regulation using convergence-based controller design. In [87], Xi and Ding obtained results on the global adaptive output regulation of a class of nonlinear systems with nonlinear exosystems. In [37], Serrani and Isidori obtained results on the global robust output regulation problem for a class of nonlinear systems.

In [39], Sundarapandian obtained results for the output regulation of the Lorenz attractor. In [52], Vaidyanathan obtained results for the output regulation of the unified chaotic system. In [51], Vaidyanathan derived results for the output regulation of the Arneodo-Coullet chaotic system. In [56], Vaidyanathan derived results for the output regulation of the Liu chaotic system.

Chaotic systems are defined as nonlinear dynamical systems which are sensitive to initial conditions, topologically mixing and with dense periodic orbits. Sensitivity to initial conditions of chaotic systems is popularly known as the butterfly effect. Small changes in an initial state will make a very large difference in the behavior of the system at future states. Chaotic behaviour was suspected well over hundred years ago in the study of three bodies problem by Henri Poincaré [4], but chaos was experimentally established by E.N. Lorenz [19] only a few decades ago in the study of 3-D weather models.

Some classical paradigms of 3-D chaotic systems in the literature are Rössler system [31], ACT system [1], Sprott systems [38], Chen system [9], Lü system [20], Liu system [17], Cai system [6], Chen-Lee system [10], Tigan system [49], etc.

Many new chaotic systems have been discovered in the recent years such as Zhou system [89], Zhu system [90], Li system [16], Wei-Yang system [86], Sundarapandian systems [40, 45], Vaidyanathan systems [58, 59, 61–64, 66, 68, 71, 82, 85], Pehlivan system [25], etc.

Synchronization of chaotic systems is a phenomenon that occurs when two or more chaotic systems are coupled or when a chaotic system drives another chaotic system. Because of the butterfly effect which causes exponential divergence of the trajectories of two identical chaotic systems started with nearly the same initial conditions, the synchronization of chaotic systems is a challenging research problem in the chaos literature [2, 3].

Major works on synchronization of chaotic systems deal with the complete synchronization of a pair of chaotic systems called the master and slave systems. The design goal of the complete synchronization is to apply the output of the master system to control the slave system so that the output of the slave system tracks the output of the master system asymptotically with time.

Pecora and Carroll pioneered the research on synchronization of chaotic systems with their seminal papers [8, 24]. The active control method [15, 32, 33, 44, 50, 73, 74, 77] is typically used when the system parameters are available for measurement. Adaptive control method [34–36, 41–43, 54, 60, 67, 72, 75, 76, 81, 84] is typically used when some or all the system parameters are not available for measurement and estimates for the uncertain parameters of the systems.

Backstepping control method [26–30, 47, 78, 83] is also used for the synchronization of chaotic systems, which is a recursive method for stabilizing the origin of a control system in strict-feedback form. Another popular method for the synchronization of chaotic systems is the sliding mode control method [46, 53, 55, 57, 65, 69, 70, 79, 80], which is a nonlinear control method that alters the dynamics of a nonlinear system by application of a discontinuous control signal that forces the system to “slide” along a cross-section of the system’s normal behavior.

In this work, the output regulation problem for the Vaidyanathan jerk chaotic system [82] has been solved using the Byrnes-Isidori regulator equations [5] to derive the state feedback control laws for regulating the output of the Vaidyanathan jerk chaotic system for the important cases of constant reference signals (set-point signals) and periodic reference signals.

This work is organized as follows. In Sect. 2, a review of the solution of the output regulation for nonlinear control systems and Byrnes-Isidori regulator equations has been presented. In Sect. 3, a dynamic analysis of the Vaidyanathan jerk chaotic system [82] is detailed. In Sect. 4, output regulation problem for the Vaidyanathan jerk chaotic system is discussed and new results are derived. In Sect. 5, numerical simulations for the output regulation of the Vaidyanathan jerk chaotic system are detailed. Section 6 summarizes the main results obtained in this work.

2 Review of the Output Regulation for Nonlinear Control Systems

In this section, we consider a multi-variable nonlinear control system modelled by equations of the form

$$\begin{aligned} \dot{x}= & {} f(x) + g(x) u + p(x) \omega \end{aligned}$$
(1)
$$\begin{aligned} \dot{\omega }= & {} s(\omega ) \end{aligned}$$
(2)
$$\begin{aligned} e= & {} h(x) - q(\omega ) \end{aligned}$$
(3)

Here, the differential equation (1) describes the plant dynamics with state x defined in a neighbourhood X of the origin of \({\mathbf R}^n\) and the input u takes values in \({\mathbf R}^m\) subject to the effect of a disturbance represented by the vector field \(p(x) \omega \). The differential equation  (2) describes an autonomous system, known as the exosystem, defined in a neighbourhood W of the origin of \({\mathbf R}^k\), which models the class of disturbance and reference signals taken into consideration. The Eq. (3) defines the error between the actual plant output \(h(x) \in {\mathbf R}^p\) and a reference signal \(q(\omega )\), which models the class of disturbance and reference signals taken into consideration.

We also assume that all the constituent mappings of the system (1), (2) and the error Eq. (3), namely, f, g, p, s, h and q are \(C^1\) mappings vanishing at the origin, i.e.

$$ f(0) = 0, \ g(0) = 0, \ p(0) = 0, \ h(0) = 0 \ \ \text{ and } \ \ q(0) = 0. $$

Thus, for \(u = 0\), the system (1), (2) has an equilibrium state \((x, \omega ) = (0, 0)\) with zero error (3).

A state feedback controller for the composite system (1), (2) has the form

$$\begin{aligned} u = \alpha (x, \omega ) \end{aligned}$$
(4)

where \(\alpha \) is a \(C^1\) mapping defined on \(X \times W\) such that \(\alpha (0, 0) = 0\). Upon substitution of the feedback law (4) in the composite system (1), (2), we get the closed-loop system given by

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x} &{} = &{} f(x) + g(x) \alpha (x, \omega ) + p(x) \omega \\ \dot{\omega } &{} = &{} s(\omega ) \end{array} \right. \end{aligned}$$
(5)

The purpose of designing the state feedback controller (4) is to achieve both internal stability and output regulation. Internal stability means that when the input is disconnected from (5) (i.e. when \(\omega = 0\)), the closed-loop system (5) has an exponentially stable equilibrium at \(x = 0\). Output regulation means that for the closed-loop system (5), for all initial states \((x(0), \omega (0))\) sufficiently close to the origin, \(e(t) \rightarrow 0\) asymptotically as \(t \rightarrow \infty \). Formally, we can summarize the requirements as follows.

State Feedback Regulator Problem [5]:

Find, if possible, a state feedback control law \(u = \alpha (x, \omega )\) such that

(OR1):

(Internal Stability) The equilibrium \(x = 0\) of the dynamics

$$\dot{x} = f(x) + g(x) \alpha (x, 0) $$

is locally asymptotically stable.

(OR2):

(Output Regulation) There exists a neighbourhood \(U \subset X \times W\) of \((x, \omega ) = (0, 0)\) such that for each initial condition \((x(0), \omega (0)) \in U\), the solution \((x(t), \omega (t))\) of the closed-loop system (5) satisfies

$$ \lim \limits _{t \rightarrow \infty } \ [ h(x(t)) - q(\omega (t)) ] = 0. $$

Byrnes and Isidori [5] have solved this problem under the following assumptions.

(H1):

The exosystem dynamics \(\dot{\omega } = s(\omega )\) is neutrally stable at \(\omega = 0\), i.e. the system is Lyapunov stable in both forward and backward time at \(\omega = 0\).

(H2):

The pair (f(x), g(x)) has a stabilizable linear approximation at \(x = 0\), i.e. if

$$ A = \left[ {\partial f \over \partial x} \right] _{x = 0} \ \ \ \text{ and } \ \ \ B = \left[ {\partial g \over \partial x} \right] _{x = 0}, $$

then (A, B) is stabilizable, which means that we can find a gain matrix K so that \(A + B K\) is Hurwitz.    \(\blacksquare \)

Next, we recall the solution of the output regulation problem derived by Byrnes and Isidori [5].

Theorem 1

[5] Under the hypotheses (H1) and (H2), the state feedback regulator problem is solvable if, and only if, there exist \(C^1\) mappings \(x = \pi (\omega )\) with \(\pi (0) = 0\) and \(u = \phi (\omega )\) with \(\phi (0) = 0\), both defined in a neighbourhod of \(W^0 \subset W\) of \(\omega = 0\) such that the following equations (called the Byrnes-Isidori regulator equations) are satisfied:

$$ \left. \begin{array}{ll} \text{(1) } &{} {\partial \pi \over \partial \omega } s(\omega ) = f(\pi (\omega )) + g(\pi (\omega )) \phi (\omega ) + p(\pi (\omega )) \omega \\ \text{(2) } &{} h(\pi (\omega )) - q(\omega ) = 0 \end{array} \right. $$

When the Byrnes-Isidori regulator equations (1) and (2) are satisfied, a control law solving the state feedback regulator problem is given by

$$\begin{aligned} u = \phi (\omega ) + K [ x - \pi (\omega ) ] \end{aligned}$$
(6)

where K is any gain matrix such that \(A + B K\) is Hurwitz.   \(\blacksquare \)

3 Dynamic Analysis of the Vaidyanathan Jerk Chaotic System

The Vaidyanathan jerk chaotic system [82] is described by the 3-D dynamics

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} x_2 \\ \dot{x}_2 &{} = &{} x_3 \\ \dot{x}_3 &{} = &{} x_1 - a [\sinh (x_1) + \sinh (x_2)] - b x_3 \\ \end{array} \right. \end{aligned}$$
(7)

In (7), a and b are constant, positive parameters.

The system (7) exhibits a chaotic attractor when the parameter values are taken as

$$\begin{aligned} a = 0.4, \ \ \ b = 0.8 \end{aligned}$$
(8)

For numerical simulations, we take the initial conditions of the Vaidyanathan jerk system (7) as

$$\begin{aligned} x_1(0) = 0.8, \ \ x_2(0) = 1.2, \ \ x_3(0) = 0.5 \end{aligned}$$
(9)

Figure 1 shows the 3-D phase portrait of the Vaidyanathan jerk chaotic system (7). Figures 2, 3, 4 show the 2-D projections of the Vaidyanathan jerk chaotic system (7) on the \((x_1, x_2), (x_2, x_3)\) and \((x_1, x_3)\) coordinate planes respectively.

Also, the Lyapunov exponents of the Vaidyanathan jerk chaotic system (7) for the parameter values (8) and the initial values (9) are numerically found as

$$\begin{aligned} L_1 = 0.0771, \ \ \ L_2 = 0, \ \ \ L_3 = -0.8791 \end{aligned}$$
(10)

Since \(L_1 + L_2 + L3 = -0.802 < 0\), the Vaidyanathan jerk chaotic system (7) is dissipative and the asymptotic motion of the Vaidyanathan jerk chaotic system (7) settles onto a strange attractor of the system.

Fig. 1
figure 1

3-D phase portrait of the Vaidyanathan jerk chaotic system

Full size image
Fig. 2
figure 2

2-D projection of the Vaidyanathan jerk chaotic system on the \((x_1, x_2)\) plane

Full size image
Fig. 3
figure 3

2-D projection of the Vaidyanathan jerk chaotic system on the \((x_2, x_3)\) plane

Full size image
Fig. 4
figure 4

2-D projection of the Vaidyanathan jerk chaotic system on the \((x_1, x_3)\) plane

Full size image

Also, the Kaplan-Yorke dimension of the Vaidyanathan jerk chaotic system (7) is calculated as

$$\begin{aligned} D_{KY} = 2 + {L_1 + L_2 \over | L_3 |} = 2.0877, \end{aligned}$$
(11)

which is fractional.

4 Output Regulation of Vaidyanathan Jerk Chaotic System

The Vaidyanathan jerk chaotic system [82] is a novel chaotic system described by the dynamics

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} x_2 \\ \dot{x}_2 &{} = &{} x_3 \\ \dot{x}_3 &{} = &{} x_1 - a [ \sinh (x_1) + \sinh (x_2) ] - b x_3 + u \\ \end{array} \right. \end{aligned}$$
(12)

where a, b are positive constants and u is an active feedback control.

In this work, we consider two important cases of output regulation for the Vaidyanathan jerk chaotic system [82]:

  1. (I)

    Tracking of Constant Reference Signals

  2. (II)

    Tracking of Periodic Reference Signals

4.1 Tracking of Constant Reference Signals

In this case, the exosystem is given by the scalar dynamics

$$\begin{aligned} \dot{\omega } = 0 \end{aligned}$$
(13)

It is important to observe that the exosystem (13) is neutrally stable because the solutions of (13) are only constant trajectories, i.e.

$$ \omega (t) \equiv \omega (0) = \omega _0 \ \ \ \text{ for } \text{ all } \ \ t $$

Thus, the assumption (H1) of Theorem 1 (Sect. 2) holds trivially.

Linearizing the dynamics of the Vaidyanathan jerk chaotic system (12) at the origin, we get the system matrices

$$\begin{aligned} A = \left[ \begin{array}{ccccc} 0 &{}&{} 1 &{}&{} 0 \\ 0 &{}&{} 0 &{}&{} 1 \\ 1 - a &{}&{} -a &{}&{} -b \\ \end{array} \right] \ \ \ \text{ and } \ \ \ B = \left[ \begin{array}{r} 0 \\ 0 \\ 1 \\ \end{array} \right] \end{aligned}$$
(14)

Using Kalman’s rank test for controllability [22], it can be easily seen that the pair (A, B) is completely controllable.

Since (A, B) is in Bush companion form, the characteristic equation of \(A + B K\) is given by

$$\begin{aligned} \lambda ^3 + (b - k_3) \lambda ^2 + (a - k_2) \lambda + (a - k_1 - 1) = 0 \end{aligned}$$
(15)

where \(K = \left[ \begin{array}{ccc} k_1&k_2&k_3 \end{array} \right] \).

By the Routh’s stability criterion [22], it can be easily shown that the closed-loop system matrix \( A + B K \) is Hurwitz if and only if

$$\begin{aligned} k_1 < a - 1, \ \ k_2 < a, \ \ k_3 < b, \ \ \ (b - k_3) (a - k_2) > a - k_1 - 1 \end{aligned}$$
(16)

Thus, the assumption (H2) of Theorem 1 (Sect. 2) also holds.

Hence, Theorem 1 can be applied to solve the output regulation problem for the Vaidyanathan jerk chaotic system (12) for the tracking of constant reference signals (set-point signals).

4.1.1 Constant Tracking Problem for \(x_1\)

Here, the tracking problem for the Vaidyanathan jerk chaotic system (12) is given by

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} x_2 \\ \dot{x}_2 &{} = &{} x_3 \\ \dot{x}_3 &{} = &{} x_1 - a [ \sinh (x_1) + \sinh (x_2) ] - b x_3 + u \\ e &{} = &{} x_1 - \omega \\ \end{array} \right. \end{aligned}$$
(17)

By Theorem 1, the regulator equations of (17) are obtained as

$$\begin{aligned} \left\{ \begin{array}{rcc} \pi _2(\omega ) &{} = &{} 0 \\ \pi _3(\omega ) &{} = &{} 0 \\ \pi _1(\omega ) - a [ \sinh (\pi _1(\omega )) + \sinh (\pi _2(\omega )) ] - b \pi _3(\omega ) + \phi (\omega ) &{} = &{} 0 \\ \pi _1(\omega ) - \omega &{} = &{} 0 \\ \end{array} \right. \end{aligned}$$
(18)

Solving the regulator equation (18), we get the unique solution

$$\begin{aligned} \left\{ \begin{array}{ccc} \pi _1(\omega ) &{} = &{} \omega \\ \pi _2(\omega ) &{} = &{} 0 \\ \pi _3(\omega ) &{} = &{} 0 \\ \phi (\omega ) &{} = &{} a \sinh (\omega ) - \omega \\ \end{array} \right. \end{aligned}$$
(19)

By Theorem 1, a state feedback control law solving the output regulation problem is given by

$$\begin{aligned} u = \phi (\omega ) + K [ x - \pi (\omega ) ] = a \sinh (\omega ) - \omega + k_1 (x_1 - \omega ) + k_2 x_2 + k_3 x_3 \end{aligned}$$
(20)

where \(k_1, k_2\) and \(k_3\) satisfy the inequalities (16).

4.1.2 Constant Tracking Problem for \(x_2\)

Here, the tracking problem for the Vaidyanathan jerk chaotic system (12) is given by

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} x_2 \\ \dot{x}_2 &{} = &{} x_3 \\ \dot{x}_3 &{} = &{} x_1 - a [ \sinh (x_1) + \sinh (x_2) ] - b x_3 + u \\ e &{} = &{} x_2 - \omega \\ \end{array} \right. \end{aligned}$$
(21)

By Theorem 1, the regulator equations of (21) are obtained as

$$\begin{aligned} \left\{ \begin{array}{rcc} \pi _2(\omega ) &{} = &{} 0 \\ \pi _3(\omega ) &{} = &{} 0 \\ \pi _1(\omega ) - a [ \sinh (\pi _1(\omega )) + \sinh (\pi _2(\omega )) ] - b \pi _3(\omega ) + \phi (\omega ) &{} = &{} 0 \\ \pi _2(\omega ) - \omega &{} = &{} 0 \\ \end{array} \right. \end{aligned}$$
(22)

The first and fourth equations in (22) contradict each other.

Thus, the regulator equation (22) are not solvable.

Hence, by Theorem 1, we conclude that the output regulation problem is not solvable for this case.

4.1.3 Constant Tracking Problem for \(x_3\)

Here, the tracking problem for the Vaidyanathan jerk chaotic system (12) is given by

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} x_2 \\ \dot{x}_2 &{} = &{} x_3 \\ \dot{x}_3 &{} = &{} x_1 - a [ \sinh (x_1) + \sinh (x_2) ] - b x_3 + u \\ e &{} = &{} x_3 - \omega \\ \end{array} \right. \end{aligned}$$
(23)

By Theorem 1, the regulator equations of (23) are obtained as

$$\begin{aligned} \left\{ \begin{array}{rcc} \pi _2(\omega ) &{} = &{} 0 \\ \pi _3(\omega ) &{} = &{} 0 \\ \pi _1(\omega ) - a [ \sinh (\pi _1(\omega )) + \sinh (\pi _2(\omega )) ] - b \pi _3(\omega ) + \phi (\omega ) &{} = &{} 0 \\ \pi _3(\omega ) - \omega &{} = &{} 0 \\ \end{array} \right. \end{aligned}$$
(24)

The second and fourth equations in (24) contradict each other.

Thus, the regulator equation (24) are not solvable.

Hence, by Theorem 1, we conclude that the output regulation problem is not solvable for this case.

4.2 Tracking of Periodic Reference Signals

In this case, the exosystem is given by the planar dynamics

$$\begin{aligned} \left. \begin{array}{ccl} \dot{\omega }_1 &{} = &{} \nu \, \omega _2 \\ \dot{\omega }_2 &{} = &{} - \nu \, \omega _1 \\ \end{array} \right. \end{aligned}$$
(25)

where \(\nu > 0\) is any fixed constant.

Clearly, the assumption (H1) (Theorem 1) holds. Also, as established in Sect. 4.1, the assumption (H2) of Theorem 1 also holds and that the closed-loop system matrix \(A + B K\) will be Hurwitz if the constants \(k_1, k_2\) and \(k_3\) of the gain matrix K satisfy the inequalities (16).

Hence, Theorem 1 can be applied to solve the output regulation problem for the Vaidyanathan jerk chaotic system (12) for the tracking of periodic reference signals.

4.2.1 Periodic Tracking Problem for \(x_1\)

Here, the tracking problem for the Vaidyanathan jerk chaotic system (12) is given by

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} x_2 \\ \dot{x}_2 &{} = &{} x_3 \\ \dot{x}_3 &{} = &{} x_1 - a [ \sinh (x_1) + \sinh (x_2) ] - b x_3 + u \\ \dot{\omega }_1 &{} = &{} \nu \, \omega _2 \\ \dot{\omega }_2 &{} = &{} - \nu \, \omega _1 \\ e &{} = &{} x_1 - \omega _1 \\ \end{array} \right. \end{aligned}$$
(26)

By Theorem 1, the regulator equations of (26) are obtained as

$$\begin{aligned} \left\{ \begin{array}{rcl} {\partial \pi _1 \over \partial \omega _1} (\nu \omega _2) + {\partial \pi _1 \over \partial \omega _2} (-\nu \omega _1) &{} = &{} \pi _2(\omega ) \\ {\partial \pi _2 \over \partial \omega _1} (\nu \omega _2) + {\partial \pi _2 \over \partial \omega _2} (-\nu \omega _1) &{} = &{} \pi _3(\omega ) \\ {\partial \pi _3 \over \partial \omega _1} (\nu \omega _2) + {\partial \pi _3 \over \partial \omega _2} (-\nu \omega _1) &{} = &{} \pi _1(\omega ) - a [ \sinh (\pi _1(\omega )) + \sinh (\pi _2(\omega ))] \\ &{} &{} - b \pi _3(\omega ) + \phi (\omega ) \\ \pi _1(\omega ) - \omega _1 &{} = &{} 0 \end{array} \right. \end{aligned}$$
(27)

Solving the regulator equation (27), we get the unique solution

$$\begin{aligned} \left\{ \begin{array}{ccc} \pi _1(\omega ) &{} = &{} \omega _1 \\ \pi _2(\omega ) &{} = &{} \nu \omega _2 \\ \pi _3(\omega ) &{} = &{} -\nu ^2 \omega _1 \\ \end{array} \right. \end{aligned}$$
(28)

and

$$\begin{aligned} \phi (\omega ) = -\nu ^3 \omega _2 - \left( b \nu ^2 + 1 \right) \omega _1 + a [ \sinh (\omega _1) + \sinh (\nu \omega _2) \end{aligned}$$
(29)

By Theorem 1, a state feedback control law solving the output regulation problem is given by

$$\begin{aligned} u = \phi (\omega ) + K [ x - \pi (\omega ) ] \end{aligned}$$
(30)

where \(\pi (\omega )\) is given by (28), \(\phi (\omega )\) is given by (29) and \(k_1, k_2\) and \(k_3\) satisfy the inequalities (16).

4.2.2 Periodic Tracking Problem for \(x_2\)

Here, the tracking problem for the Vaidyanathan jerk chaotic system (12) is given by

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} x_2 \\ \dot{x}_2 &{} = &{} x_3 \\ \dot{x}_3 &{} = &{} x_1 - a [ \sinh (x_1) + \sinh (x_2) ] - b x_3 + u \\ \dot{\omega }_1 &{} = &{} \nu \, \omega _2 \\ \dot{\omega }_2 &{} = &{} - \nu \, \omega _1 \\ e &{} = &{} x_2 - \omega _1 \\ \end{array} \right. \end{aligned}$$
(31)

By Theorem 1, the regulator equations of (31) are obtained as

$$\begin{aligned} \left\{ \begin{array}{rcl} {\partial \pi _1 \over \partial \omega _1} (\nu \omega _2) + {\partial \pi _1 \over \partial \omega _2} (-\nu \omega _1) &{} = &{} \pi _2(\omega ) \\ {\partial \pi _2 \over \partial \omega _1} (\nu \omega _2) + {\partial \pi _2 \over \partial \omega _2} (-\nu \omega _1) &{} = &{} \pi _3(\omega ) \\ {\partial \pi _3 \over \partial \omega _1} (\nu \omega _2) + {\partial \pi _3 \over \partial \omega _2} (-\nu \omega _1) &{} = &{} \pi _1(\omega ) - a [ \sinh (\pi _1(\omega )) + \sinh (\pi _2(\omega ))] \\ &{} &{} - b \pi _3(\omega ) + \phi (\omega ) \\ \pi _2(\omega ) - \omega _1 &{} = &{} 0 \end{array} \right. \end{aligned}$$
(32)

Solving the regulator equation (32), we get the unique solution

$$\begin{aligned} \left\{ \begin{array}{ccc} \pi _1(\omega ) &{} = &{} - \nu ^{-1} \omega _2 \\ \pi _2(\omega ) &{} = &{} \omega _1 \\ \pi _3(\omega ) &{} = &{} \nu \omega _2 \\ \end{array} \right. \end{aligned}$$
(33)

and

$$\begin{aligned} \phi (\omega ) = \nu ^2 \omega _1 + \left( \nu ^{-1} + b \nu \right) \omega _2 + a \left[ -\sinh \left( \nu ^{-1} \omega _2 \right) + \sinh (\omega _1) \right] \end{aligned}$$
(34)

By Theorem 1, a state feedback control law solving the output regulation problem is given by

$$\begin{aligned} u = \phi (\omega ) + K [ x - \pi (\omega ) ] \end{aligned}$$
(35)

where \(\pi (\omega )\) is given by (33), \(\phi (\omega )\) is given by (34) and \(k_1, k_2\) and \(k_3\) satisfy the inequalities (16).

4.2.3 Periodic Tracking Problem for \(x_3\)

Here, the tracking problem for the Vaidyanathan jerk chaotic system (12) is given by

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} x_2 \\ \dot{x}_2 &{} = &{} x_3 \\ \dot{x}_3 &{} = &{} x_1 - a [ \sinh (x_1) + \sinh (x_2) ] - b x_3 + u \\ \dot{\omega }_1 &{} = &{} \nu \, \omega _2 \\ \dot{\omega }_2 &{} = &{} - \nu \, \omega _1 \\ e &{} = &{} x_3 - \omega _1 \\ \end{array} \right. \end{aligned}$$
(36)

By Theorem 1, the regulator equations of (36) are obtained as

$$\begin{aligned} \left\{ \begin{array}{rcl} {\partial \pi _1 \over \partial \omega _1} (\nu \omega _2) + {\partial \pi _1 \over \partial \omega _2} (-\nu \omega _1) &{} = &{} \pi _2(\omega ) \\ {\partial \pi _2 \over \partial \omega _1} (\nu \omega _2) + {\partial \pi _2 \over \partial \omega _2} (-\nu \omega _1) &{} = &{} \pi _3(\omega ) \\ {\partial \pi _3 \over \partial \omega _1} (\nu \omega _2) + {\partial \pi _3 \over \partial \omega _2} (-\nu \omega _1) &{} = &{} \pi _1(\omega ) - a [ \sinh (\pi _1(\omega )) + \sinh (\pi _2(\omega ))] \\ &{} &{} - b \pi _3(\omega ) + \phi (\omega ) \\ \pi _3(\omega ) - \omega _1 &{} = &{} 0 \end{array} \right. \end{aligned}$$
(37)

Solving the regulator equation (37), we get the unique solution

$$\begin{aligned} \left\{ \begin{array}{ccc} \pi _1(\omega ) &{} = &{} - \nu ^{-2} \omega _1 \\ \pi _2(\omega ) &{} = &{} - \nu ^{-1} \omega _2 \\ \pi _3(\omega ) &{} = &{} \omega _1 \\ \end{array} \right. \end{aligned}$$
(38)

and

$$\begin{aligned} \phi (\omega ) = \left( b + \nu ^{-2} \right) \omega _1 + \nu \omega _2 - a \left[ \sinh \left( \nu ^{-2} \omega _1 \right) + \sinh \left( \nu ^{-1} \omega _2 \right) \right] \end{aligned}$$
(39)

By Theorem 1, a state feedback control law solving the output regulation problem is given by

$$\begin{aligned} u = \phi (\omega ) + K [ x - \pi (\omega ) ] \end{aligned}$$
(40)

where \(\pi (\omega )\) is given by (38), \(\phi (\omega )\) is given by (39) and \(k_1, k_2\) and \(k_3\) satisfy the inequalities (16).

5 Numerical Simulations

For numerical simulations, we take the parameter values a and b so that the Vaidyanathan jerk system (12) is in the chaotic case, i.e.

$$\begin{aligned} a = 0.4, \ \ \ b = 0.8 \end{aligned}$$
(41)

For achieving the internal stability of the state feedback regulator problem, a gain matrix K which satisfies the inequalities (16) must be used.

With the choice

$$ K = \left[ \begin{array}{ccc} k_1&k_2&k_3 \end{array} \right] = \left[ \begin{array}{ccccc} -125.6&\,&-74.2&\,&-14.2 \end{array} \right] , $$

the matrix \(A + B K\) is Hurwitz with the eigenvalues \(-5, -5, -5\).

In the periodic tracking output regulation problem, the value \(\nu = 1\) is taken in the exosystem dynamics given by (25).

5.1 Tracking of Constant Reference Signals

5.1.1 Constant Tracking Problem for \(x_1\)

Here, the initial conditions are taken as

$$ x_1(0) = 8.1, \ \ x_2(0) = 5.4, \ \ x_3(0) = 6.3, \ \ \ \omega (0) = 2 $$

The simulation graph is depicted in Fig. 5 from which it is clear that the state trajectory \(x_1(t)\) tracks the constant reference signal \(\omega (t) \equiv 2\) in 3 s.

Fig. 5
figure 5

Constant tracking of the state \(x_1\)

Full size image

5.1.2 Constant Tracking Problem for \(x_2\)

As detailed in Sect. 4.1.2, the output regulation problem is not solvable for this case because the Byrnes-Isidori regulator equations do not admit any solution.

5.1.3 Constant Tracking Problem for \(x_3\)

As pointed out in Sect. 4.1.3, the output regulation problem is not solvable for this case because the Byrnes-Isidori regulator equations do not admit any solution.

5.2 Tracking of Periodic Reference Signals

5.2.1 Periodic Tracking Problem for \(x_1\)

Here, the initial conditions are taken as

$$ x_1(0) = 5.1, \ \ x_2(0) = 4.7, \ \ x_3(0) = -2.5, \ \ \ \omega _1(0) = 0, \omega _2(0) = 1 $$

Also, it is assumed that \(\nu = 1\). The simulation graph is depicted in Fig. 6 from which it is clear that the state trajectory \(x_1(t)\) tracks the periodic reference signal \(\omega _1(t) = \sin t\) in 3 s.

Fig. 6
figure 6

Periodic tracking of the state \(x_1\)

Full size image

5.2.2 Periodic Tracking Problem for \(x_2\)

Here, the initial conditions are taken as

$$ x_1(0) = 8.1, \ \ x_2(0) = 3.4, \ \ x_3(0) = -2.7, \ \ \ \omega _1(0) = 0, \omega _2(0) = 1 $$

Also, it is assumed that \(\nu = 1\). The simulation graph is depicted in Fig. 7 from which it is clear that the state trajectory \(x_2(t)\) tracks the periodic reference signal \(\omega _1(t) = \sin t\) in 3 s.

Fig. 7
figure 7

Periodic tracking of the state \(x_2\)

Full size image
Fig. 8
figure 8

Periodic tracking of the state \(x_3\)

Full size image

5.2.3 Periodic Tracking Problem for \(x_3\)

Here, the initial conditions are taken as

$$ x_1(0) = 3.4, \ \ x_2(0) = 2.5, \ \ x_3(0) = -6.9, \ \ \ \omega _1(0) = 0, \omega _2(0) = 1 $$

Also, it is assumed that \(\nu = 1\). The simulation graph is depicted in Fig. 8 from which it is clear that the state trajectory \(x_3(t)\) tracks the periodic reference signal \(\omega _1(t) = \sin t\) in 3 s.

6 Conclusions

Output regulation problem is one of the important problems in the control theory, which aims to control a fixed linear or nonlinear plant in order to have its output tracking reference signals produced by some external generator or the exosystem. Byrnes and Isidori (1990) solved the output regulation problem for a general class of nonlinear systems under some stability assumptions. In this work, the output regulation problem for the Vaidyanathan jerk chaotic system (2014)) has been studied in detail and a complete solution for the output regulation problem for the Vaidyanathan jerk chaotic system has been presented as well. Explicitly, using the Byrnes-Isidori regulator equations (1990), state feedback control laws for regulating the output of the Vaidyanathan jerk chaotic system have been derived. As tracking reference signals, constant and periodic reference signals have been considered and in each case, feedback control laws regulating the output of the Vaidyanathan chaotic system have been derived when the problem is solvable. Numerical simulations using MATLAB are shown to verify the results.