Keywords

1 Introduction

The foundation of continuous time system tries to exhibit stability in response. It is very cumbersome task to find out the uncontrolled parameters in time-varying nonlinear system. Linear Quadratic Regulator (LQR) is generally used for Linear Dynamical System [1]. In Nonlinear system the LQR is applied by pretending the system parameters are linear [2]. The development of nonlinear Optimum Control around the year 1950s and 1960s has been addressed in various theoretical and practical aspects where the objective is to minimize the cost given by performance index [3].

The Dynamic Programming Principle (DPP) by Bellman in USA was going ahead towards some kind of Partial Differential Equation (PDE) [4]. The solution of this Differential Equation was done by Renowned Scientists Hamilton, Jacobi, and Bellman (HJB) and the process was notified as HJB equations [5, 6].

The Performance analyzer, i.e., performance index in time-varying system has moving initial and terminal time but the time interval between these two times is finite [7]. Here the oscillations of responses of this system, converge to nonlinear system is at a marginal level. So though the system is not in stable condition, the characteristic of system response may not be hampered. The LTV system constructs an approximation solution named as Approximating Sequence of Riccati Equation (ASRE) [8, 9, 10].

The nonlinear system with linear input (affine system) is controlled with this method. In Sect. 2, The Linearization of Nonlinear System is analyzed.

In Sect. 3, the development of Non-Quadratic Performance Index is discussed which shows a form to represent the cost function in nonlinear system. The stabilization of the system by ASRE method is derived in this Section. Section 5, The Quadratic Control of a Time-Varying System is applied.

2 Linearization of Output to the Non-linear System

The deviation in between actual response (y(t)) and desired tracking response (r(t)) is known as Synchronization deviation. This is obtained as:

$$ \varepsilon = \left| {r(t)\sim \,y(t)} \right| $$
(1)

For closed-loop the dynamical error,

$$ \varepsilon = \left| {y(t)} \right| $$
(2)

Now, consider the synchronization output with controller gain ‘K’ is expressed as:

$$ \left| {\mathop {y(t)}\limits^{ \bullet } } \right| = F(x(t) + G(x(t),u(t)) $$
(3)

is a nonlinear system with affine input.

Here, F € n x n and G € n x m are two positive definite matrices.

So from Eq. (1) and Eq. (2)

$$ \begin{aligned} \mathop \varepsilon \limits^{ \bullet } & = F(y(t)) + G(y(t),u(t)) - F(r(t)) \\ & = F(\varepsilon (t)) + G((x_{1} (t) + x_{2} (t)u(t)) - F(r(t)) \\ & = F(\varepsilon (t)) + GK\varepsilon (t) - F(r(t)) \\ & = F_{K} (t)\varepsilon (t) - F(r(t))[F_{K} = (F + GK)\varepsilon (t)] \\ & \quad ({\text{put}},((x_{1} (t) + x_{2} (t)u(t)) = K\varepsilon (t)) \\ \end{aligned} $$
(4)

In Eq. (4) the representation of error is in a linear form where actual output follows the input by putting r(t) = 0.

3 Example: A Nonlinear Self Explanatory System

Consider the optimal tracking state response of nonlinear system as:

$$ \begin{aligned} & \frac{{{\text{d}}x_{1} (t)}}{{{\text{d}}t}} = x_{2} \\ & \frac{{{\text{d}}x_{2} (t)}}{{{\text{d}}t}} = - a \cdot \cos (x_{1} ) + \tau , \\ \end{aligned} $$
(5)

where ‘τ’ is constant.

Now the optimal desired response set the value as: \( \left[ \begin{aligned} r_{1} (t) \hfill \\ r_{2} (t) \hfill \\ \end{aligned} \right] = \left[ \begin{aligned} \pi /4 \hfill \\ 0 \hfill \\ \end{aligned} \right] \)

To explore this problem, we set,

$$ \mathop y\limits^{ \bullet } = F(x_{1} (t)) + G(x_{2} (t))u(t) $$
(6)

where \( F = \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill \\ \end{array} } \right] \) and \( G = \left[ \begin{aligned} 0 \hfill \\ 1 \hfill \\ \end{aligned} \right] \)

and \( y_{1} (t) - a*\cos (x_{1} ) \) and y2(t) = 1. By pole placement technology at −1 and −2, the controller K is set as K = [−2 −3], then the optimal input is written as

$$ u = a \cdot \cos (x_{1} ) - 2(x_{1} - \pi /4) - 3x_{2} $$
(7)

Using Matlab, the characteristics of system state and controller input are shown in Figs. 1 and 2.

Fig. 1
figure 1

Characteristic of state of the system

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Fig. 2
figure 2

The characteristic of control input

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4 Stabilization of the System Using ASRE Methodology

Approximating Sequence of Riccati Equation (ASRE) control strategy keeps the deviation between desired output and actual output at sustainable level by Non-Quadratic and Time-Varying Performance Regulator.

Consider, x(t), u(t) represent the state vector and the plan input, respectively, and y be the system output. Now, the state and the plant input are regulated by nonlinear valued function ‘M € n × n’ and ‘N € n × m’.

For the Non-Quadratic problem, the LTV is introduced as

$$ \frac{{{\text{d}}x^{\Delta } (t)}}{{{\text{d}}t}} = M(x^{\Delta } (t) - \Delta x(t))x^{\Delta } (t) + N(x^{\Delta } (t),u^{\Delta } (t)) $$
$$ [ {\text{where}},\left. {\left( {x^{\Delta } (t) = x(t) + \Delta x(t)} \right){\text{ and }}\left( {u^{\Delta } (t) = u(t) + \Delta u(t)} \right)} \right] $$
$$ y^{\Delta } (t) = \gamma \left[ {x^{\Delta } (t)} \right] = \gamma (x(t) + \Delta x(t)) $$
(8)

where the elementary value of the state \( x^{\Delta } (t_{0} ) = x_{0} \),

The performance measuring cost function is given by

$$ J^{\Delta } (u) = U_{1} \left| {\varepsilon^{\Delta } (t_{f} )} \right|^{2} + U_{2} \int\limits_{{t_{0} }}^{{t_{f} }} {[\varepsilon^{{\Delta^{T} }} (t)} C\varepsilon^{\Delta } (t) + u^{{\Delta^{T} }} (t)Du^{\Delta } (t)]{\text{d}}t $$
(9)

where \( u^{\Delta } (t) = - D^{ - 1} N^{T} (x^{\Delta } (t) - \Delta x(t))[P^{\Delta } (t)x^{\Delta } (t) - s^{\Delta } (t)] \) and \( \varepsilon^{\Delta } (t) = r(t) - \gamma (x^{\Delta } (t))x(t) \)

Here, in Eq. (9) C € n × n positive semi-definite matrix, D € m × m positive definite matrix, and PΔ € n × n positive definite matrix,

In the above Eq. (9) s € n is transformation vector forward direction.

The vector \( s^{\Delta } (t_{f} ) \) € n is the solution of terminated point of nonlinear differential equation and it is expressed as

$$ s^{\Delta } (t) = \gamma (x^{\Delta } (t_{f} ) - \Delta x(t_{f} ))Ur(t_{f} ) $$
(10)

The positive definite matrix, \( P^{\Delta } (t_{f} ) = \gamma^{T} (x^{\Delta } (t_{f} ) - \Delta x(t_{f} ))U\gamma (x^{\Delta } (t_{f} ) - \Delta x(t_{f} )) \)

Now, The change of state in an optimal tracking system stabilizes the state which is expressed by Linear Differential Equation as

$$ \begin{aligned} \frac{{{\text{d}}x^{\Delta } (t)}}{{{\text{d}}t}} & = N(x^{\Delta } (t) - \Delta x(t))D^{ - 1} N^{T} (x^{\Delta } (t) - \Delta x(t))s^{\Delta } (t) + M(x^{\Delta } (t) \\ & \quad - \Delta x(t)) - N(x^{\Delta } (t) - \Delta x(t)) \\ & \quad \times [D^{ - 1} N^{T} (x^{\Delta } (t) - \Delta x(t))P^{\Delta } (t)] \\ \end{aligned} $$
(11)

5 Quadratic Control of a Time-Varying System

Consider a state variable time-varying system and output of this system are written as

$$ \begin{array}{*{20}l} {\frac{{{\text{d}}x_{1} (t)}}{{{\text{d}}t}} = x_{2} (t)} \hfill \\ {\frac{{{\text{d}}x_{2} (t)}}{{{\text{d}}t}} = - x_{1} (t) + x_{2} (t).(1.5 - 0.2x_{2}^{2} (t)) + u(t)} \hfill \\ \end{array} {\text{and}}\quad y(t) = f\left( {Cx(t) + Du(t)} \right) $$
(12)

The initial value of the state is x1(0) = x2(0) = −5 and C is Identity matrix and D set as zero. The system is represented in state matrix form as

$$ \frac{{{\text{d}}x(t)}}{{{\text{d}}t}} = \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill \\ 1 \hfill & {1.5 - 0.2x_{2} (t)} \hfill \\ \end{array} } \right] $$
(13)

From the below Fig. 3, it is shown that the given states are oscillated with the propagation of time.

Fig. 3
figure 3

The characteristic of the given two states

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According to Matrix Differential Equation of the Algebraic Riccati Equation is

$$ \frac{{{\text{d}}P^{\Delta } }}{{{\text{d}}t}} = P^{\Delta } (t)M( \cdot ) - M^{T} ( \cdot )P^{\Delta } (t) + P^{\Delta } (t)\gamma D^{ - 1} \gamma^{T} P^{\Delta } (t) - Q $$
(14)

Thus Eq. (14) is called Differential Matrix Riccati Equation. The Characteristic of “\( P^{\Delta } (t) \)” is shown in the below Fig. 4 on the basis of the value of the Controller parameter (K) and “P”. (Here, \( P^{\Delta } (t) \) = in MATLAB syntax—P).

The value of the parameters—K and P is evaluated from MATLAB as (Table 1 and Fig. 4).

Table 1 Value of controller parameter and definite matrix
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Fig. 4
figure 4

The characteristic of “P”

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Using Quadratic Regulator the state and the output response are controlled. The following figures (Fig. 5a, b) show the regulable response of state and output and corresponding Optimum Input. The Characteristic of Optimum Input from Fig. 5b shows the optimum value of the input at initial point and the output and state response are saturated within the optimal point which is nearest to zero value. The state response starts with its initial value |−5|. The output also follows the characteristic of state. So after getting the optimum point from the Fig. 4 by ASRE solution, the saturation of States and Output is controlled through the system is oscillated at the early stages (Fig. 3).

Fig. 5
figure 5

a The controlled response of states and output of the system; b the characteristic of optimum input

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6 Conclusion

In this paper, a new method to evaluate a performance analysis of nonlinear system using quadratic regulator is explained. By which the closed-loop dynamical error is minimized with desired optimal tracking response. The evaluation of ASRE control design is implemented in nonlinear affine system. The superiority of ASRE controller provides significant performance in real time application of nonlinear system. The extension of this method in non affine system is yet to be developed in future.