Polynomial control design for polynomial systems: A non-iterative sum of squares approach

Abstract

This paper proposes a non-iterative state feedback design approach for polynomial systems using polynomial Lyapunov function based on the sum of squares (SOS) decomposition. The polynomial Lyapunov matrix consists of states of the system leading to the non-convex problem. A lower bound on the time derivative of the Lyapunov matrix is considered to turn the non-convex problem into a convex one; and hence, the solutions are computed through semi-definite programming methods in a non-iterative fashion. Furthermore, we show that the proposed approach can be applied to a wide range of practical and industrial systems that their controller design is challenging, such as different chaotic systems, chemical continuous stirred tank reactor, and power permanent magnet synchronous machine. Finally, software-in-the-loop (SiL) real-time simulations are presented to prove the practical application of the proposed approach.

Keywords

Polynomial chaotic system Polynomial CSTR plant polynomial PMSM system sum of squares (SOS) decomposition Polynomial Lyapunov function software-in-the-loop (SiL)

Introduction

Stability analysis of nonlinear systems is a challenging problem attracting researchers’ attention in recent decades. Based on the Lyapunov function and storage function techniques, different approaches are proposed to handle the nonlinear system stability analysis and performance issues (Khalil, 2002). However, these approaches are generally analytically complicated, motivating numerical stability analysis approaches (Rakhshan et al., 2016; Tanaka, 2001).

Recently, linear matrix inequality (LMI) technique is widely used as a numerical approach which can be solved by convex optimization (Boyd et al., 1994). To derive the controller design of nonlinear systems in terms of LMIs, Takagi-Sugeno (TS) fuzzy or linear parameter varying (LPV) models are commonly used (Tanaka et al., 1998; Vafamand and Rakhshan, 2017). In these models, the original nonlinear system is described by a polyhedron region whose vertices are the local representation of the original system. Owing to the convex property of the obtained polyhedron, the controller design conditions are then formulated based on the vertices (Jaadari, 2013). However, in the majority of the existing proposed approaches, the information of fuzzy membership functions and the varying parameters are not considered in the controller design conditions, which yields to conservative results.

Recently, the sum of squares (SOS)-based decomposition approaches provide a method of designing controllers in a less conservative and less complex fashion through the semi-definite programming and defining polynomial matrix inequalities (PMIs) (Papachristodoulou and Prajna, 2005). Comparing with the LMI conditions, PMIs can be applied to a wider class of static and dynamic systems, particularly, polynomial systems. Furthermore, the TS fuzzy or LPV models that are obtained by sector nonlinearity concept or first-order approximation of Taylor series (Tanaka, 2001) only represent the original system in a local region. Meanwhile, the SOS-based approaches can be directly applied to such polynomial systems in a global fashion.

Polynomial models can effectively describe the behavior of a wide range of industrial applications, process plants, chaotic and hyperchaotic phenomena (Mardani et al., 2017; Pitarch et al., 2017, 2016; Rakhshan et al., 2017; Vafamand et al., 2017c). However, there are limitations in the controller design process; for example, in order to be able to exploit the convex optimization framework, some researchers try to constrain the Lyapunov function to be only a function of the states whose corresponding rows in the control matrix are zero, and its inverse has a specific form (Prajna et al., 2004; Tanaka et al., 2009). Furthermore, Papachristodoulou and Prajna (2005) proposed an iterative SOS approach in order to convert the non-convex optimization problem to a convex one. Although these approaches change the non-convex optimization problem into a convex problem, they increase the conservativeness of the solution.

In this paper, we present a non-iterative algorithm based on the SOS decomposition to solve the polynomial matrix inequalities in order to find a proper controller while reducing the conservativeness. Furthermore, we show that one can achieve a tradeoff between computational cost and controller structure simplicity using the polynomial static state feedback control law stated in terms of the polynomial matrix inequalities.

The main contributions of this paper can be summarized as follows:

Despite Prajna et al. (2004) and Tanaka et al. (2009), the Lyapunov function in the proposed control design does not require to be a function of those states whose corresponding rows in control matrix are zeroes.

This paper solves the problem of non-singularity of polynomial matrices that cannot be guaranteed in Ma and Yang (2009).

The proposed approach reduces the number of SOS variables and avoids the curse of dimensionality for high-order systems, which was encountered in Hindi (2002), Ma and Yang (2009) and Zhao and Wang (2009).

It is shown through numerical simulation that many systems with challenging stability analysis and controller design such as chaotic systems, chemical reactor systems, and electrical motors, can be controlled with a polynomial controller.

Finally, software-in-the-loop (SiL) real-time simulations are presented to prove the practical application of the proposed controller.

The rest of the paper is organized as follows: Section 2 provides the main concepts of the SOS decomposition and the system description. Stability conditions are discussed in Section 3. Simulation and comparison results are given in Section 4. Finally, conclusions are drawn in Section 5.

Preliminaries

In this section, a brief review of SOS decomposition and system description are provided. The reader can refer to Papachristodoulo and Prajna (2002) and Parrilo (2000) for more detail.

SOS programming

Generally, a SOS problem is to examine the non-negativity of a polynomial $f (x),$ indicated as powers of x and its associated coefficients. The idea is to alternate the non-negativity by the counterpart condition of being SOS polynomials and pursue to find such decomposition. The fundamental concepts of the SOS decomposition are briefly discussed in the following.

A monomial in $x = {[x_{1} x_{2} \dots x_{n}]}^{T}$ is a function of the form $x_{1}^{α_{1}} x_{2}^{α_{2}} \dots x_{n}^{α_{n}}$ , where $α_{1}, α_{2}, \dots, α_{n}$ are non-negative integers (Tanaka et al., 2009). The integer value d denotes the degree of monomial, which is calculated as $d_{m} = \sum_{i = 1}^{n} α_{i}$ . A polynomial $f (x)$ is described by a finite linear combination of monomials, where their linear coefficients are real. The degree of polynomial $f (x)$ is defined as the maximum of the monomial degrees. The multivariate polynomial $f (x)$ for $x \in ℛ^{n \times 1}$ is SOS if there exist polynomials $f_{i} (x)$ , where $i = 1, 2, \dots, m,$ such that

f (x) = \sum_{i = 1}^{m} f_{i}^{2} (x)

(1)

Consequently, the scalar polynomial function $f (x)$ is nonnegative. Equation (1) shows that the set of SOS polynomials for n variables is a proper convex cone (Hindi, 2002; Papachristodoulou and Prajna, 2005). The polynomial function in (1) can be viewed as a special quadratic form specified in the following proposition.

Proposition 2.1: Assume that $f (x)$ is a polynomial in $x \in R^{n}$ of degree $2 d$ . Let $Z (x)$ be a column vector whose entries are all monomials in x with the degree not bigger than d. Then, $f (x)$ is SOS if and only if there exists a positive semi-definite matrix Q such that the following equation is satisfied (Parrilo, 2000)

f (x) = Z {(x)}^{T} QZ (x) .

(2)

However, in many problems, we encounter with polynomial matrices and it is necessary to investigate the positivity of such matrix terms. Let $L (x) \in R^{n \times n}$ be a symmetric matrix of degree $2 d$ , $M (x) \in R^{s \times n}$ be a polynomial matrix of degree less than or equal to d, and $z (x) \in R^{n \times 1}$ be a column vector whose entries are monomials in x with degree no greater than d. The following expressions are equivalent (Papachristodoulou et al., 2016):

$L (x) \geq 0$ for $\forall x$ .

$v^{T} L (x) v$ is SOS, where $v \in R^{n}$ is a monomial vector of any arbitrary variable independent of x.

There exists a polynomial matrix $M (x) \in R^{s \times n}$ such that $L (x) = M^{T} (x) M (x)$ .

System description

Consider the following polynomial state space representation

\overset{\cdot}{x} (t) = A (x) x (t) + B (x) u (t),

(3)

where $x = {[x_{1}^{T} x_{2}^{T} \dots x_{n}^{T}]}^{T} \in R^{n \times 1}$ and $u \in R^{m \times 1}$ are state and control input vector, $A (x) \in R^{n \times n}$ and $B (x) \in R^{n \times m}$ are the polynomial system and polynomial input matrices, respectively. Based on the nonlinear plant (3), a nonlinear static state feedback controller is

u (t) = K (x) x (t),

(4)

where $K (x) \in R^{m \times n}$ is a polynomial matrix in x. By substituting the control law (4) in the open-loop system (3), the closed-loop polynomial system is

\overset{\cdot}{x} (t) = (A (x) + B (x) K (x)) x (t) .

(5)

The goal is to derive stabilization conditions in terms of polynomial matrix inequalities, which can be solved effectively by SOS techniques.

Notation: For the sake of brevity and ease of notations, x, $\overset{\cdot}{x}$ , and u stand for $x (t)$ , $\overset{\cdot}{x} (t)$ , and $u (t)$ . Furthermore, we drop the x in $A (x)$ , $B (x)$ , $K (x)$ and other polynomial matrices as we face them in the rest of the paper.

Stability conditions

To obtain stability conditions, the following polynomial quadratic Lyapunov function candidate is employed

V = x^{T} P^{- 1} x,

(6)

where $P \in R^{n \times n}$ is a symmetric positive definite polynomial matrix in $x$ . Following, the goal is to derive the sufficient SOS-based conditions that assure the stability of the closed-loop system (5) through the Lyapunov stability theory.

Theorem 1: Consider the time derivative of Lyapunov matrix has a lower bound such that

- α I \leq \overset{\cdot}{P},

(7)

where $α$ is a given positive scalar. If there exist polynomial matrices $P = P^{T}$ and N, such that the following SOS-decomposition conditions are satisfied

ν^{T} (P - ρ_{1} (x) I) ν is SOS,

(8)

- ν^{T} (P A^{T} T^{T} + N B^{T} T^{T} + TAP + TB N^{T} + α I - ρ_{2} (x) I) ν is SOS,

(9)

where $ρ_{1} (x)$ and $ρ_{2} (x)$ are any arbitrary small nonnegative polynomial functions and $ν$ is a monomial vector that is independent of x, then the polynomial system (3) is feedback stabilizable with the feedback controller (4) and the state feedback matrix is computed by $K = N^{T} P^{- 1}$ .

Proof: Considering the quadratic polynomial Lyapunov function (6), one has:

\overset{\cdot}{V} = {\overset{\cdot}{x}}^{T} P^{- 1} x + x^{T} {\overset{\cdot}{P}}^{- 1} x + x^{T} P^{- 1} \overset{\cdot}{x}

(10)

Substituting the closed-loop system (5) into (10), results in:

\overset{\cdot}{V} = x^{T} (A^{T} + K^{T} B^{T}) P^{- 1} x + x^{T} {\overset{\cdot}{P}}^{- 1} x + x^{T} P^{- 1} (A + BK) x

(11)

Since $P P^{- 1} = I \Rightarrow \overset{\cdot}{P} P^{- 1} + P {\overset{\cdot}{P}}^{- 1} = 0,$ therefore:

{\overset{\cdot}{P}}^{- 1} = - P^{- 1} \overset{\cdot}{P} P^{- 1}

(12)

Substituting (12) into (11), one obtains:

\begin{matrix} \overset{\cdot}{V} = x^{T} (A^{T} + K^{T} B^{T}) P^{- 1} x - x^{T} P^{- 1} \overset{\cdot}{P} P^{- 1} x + {\hat{x}}^{T} P^{- 1} (A + BK) x \\ = x^{T} P^{- 1} {P (A^{T} + K^{T} B^{T}) + (A + BK) P - \overset{\cdot}{P}} P^{- 1} x \end{matrix}

(13)

By employing the congruence transformation (Vafamand et al., 2017a), the negative definiteness of $\overset{\cdot}{V}$ is guaranteed if $Γ_{1} < 0$ , where $Γ_{1} is$

\begin{matrix} Γ_{1} = P (A^{T} + K^{T} B^{T}) + (A + BK) P - \overset{\cdot}{P} \\ = P A^{T} + P K^{T} B^{T} + AP + BKP - \overset{\cdot}{P} < 0 . \end{matrix}

(14)

By defining $N \overset{Δ}{=} P K^{T}$ , $Γ_{1}$ is equal to

Γ_{1} = P A^{T} + N B^{T} + AP + B N^{T} - \overset{\cdot}{P} < 0 .

(15)

Since P is a polynomial matrix in x, therefore, based on the chain derivation law, one has

\overset{\cdot}{P} = \frac{\partial P}{\partial x} \overset{\cdot}{x} = \frac{\partial P}{\partial x} (A + BK) x .

(16)

As a result, the product of unknown term $\frac{\partial P}{\partial x}$ by the polynomial feedback gain matrix K leads to a non-convex condition. To handle this problem, different approaches are proposed (Vafamand and Sha Sadeghi, 2015; Vafamand et al., 2017b). One of the most effective solutions is to consider upper and/or lower bounds for time derivative terms. Recalling the lower bound (7), one concludes $Γ_{1} \leq Γ_{2}$ , where $Γ_{2}$ is

Γ_{2} = P A^{T} + N B^{T} + AP + B N^{T} + α I < 0 .

(17)

Therefore, the SOS decomposition condition (9) is obtained. Also, (8) is a sufficient condition to guarantee the positive definiteness of polynomial Lyapunov function (6).

Remark 1: The SOS decomposition “ $ν^{T} Σ ν is SOS$ ” means that “ $Σ \geq 0$ ”. We must distinguish the SOS decomposition conditions from the well-known LMIs. Therefore, in this paper, we use the notation “ $ν^{T} Σ ν is SOS$ ” to clarify the differences between the SOS and LMI conditions. Note that, the designer does not need to select the vector $ν$ as this is handled internally by the existing SOS decomposition solvers (Papachristodoulou et al., 2016).

Remark 2: Since P is computed as a polynomial matrix with a degree of higher than 2, the Lyapunov candidate is not radially bounded in general. Also, the solution of (7), has a local nature. Thus, Theorem 1 with polynomial Lyapunov function provides the local stabilization. However, if the degree of the Lyapunov function is chosen as zero, then the constraint (7) is removed and the Lyapunov function $x^{T} P^{- 1} x$ is radially unbound. So, the global stability is assured. However, the degree of K can be deliberately chosen to design a polynomial controller, which guarantees the global closed-loop stability of a polynomial system.

Remark 3: In the conditions (8) and (9), two small positive polynomial scalars $ρ_{1} (x)$ and $ρ_{2} (x)$ appear. Existence of a positive polynomial function in the SOS decompositions guarantees the positive/negative definiteness of SOS decompositions (i.e. “ $ν^{T} (Σ - ρ (x) I) ν is SOS$ ” means that “ $Σ > 0$ ”). Consequently, the positive definiteness of the Lyapunov candidate (6) and the negative definiteness of its time derivative are assured. The scalars $ρ_{1} (x)$ and $ρ_{2} (x)$ are selected by the designer and the unknown polynomial matrices P and N are computed by semi-definite programming techniques such that satisfy the conditions of Theorem 1. The $ρ_{1} (x)$ and $ρ_{2} (x)$ can be constructed of the form $β + \sum_{i = 1}^{n} a_{i} x_{i}^{2}$ , where $β$ and $α_{i}$ s are arbitrary small scalars.

Figure 1 summarizes the general approach of the proposed control strategy for polynomial systems. As is shown in this flowchart, first the dynamics of the desired system are considered based on (3). By applying Theorem 1 and using SOSTOOLS, we obtain $P (x)$ and $N (x)$ . Then, the controller’s gain could be calculated using $K (x) = N^{T} (x) P^{- 1} (x)$ . Finally, we apply the designed controller on the polynomial system.

Figure 1.

The design procedure of the proposed approach.

Simulation study

In this section, four control design examples for a chemical reactor, two chaotic systems, and an electric motor are discussed to demonstrate the effectiveness of the proposed methodology in the Software-In-the Loop (SiL) simulation technique. The structure of the SiL is shown in Figure 2, which consists of (1) OPAL-RT as a Real-Time Simulator (RTS) that simulates the proposed controller depicted in Figure 1; (2) a PC as the command station (programming host) in which Matlab/Simulink based code that is executed on the OPAL-RT is generated; and (3) a router used as a connector of all the setup devices in the same sub-network. The OPAL-RT is also connected to the DK60 board through Ethernet ports (Khooban, 2017, 2018).

Figure 2.

The real-time setup.

The model-to-data workflow of the system model under the test is shown in Figure 3. To make the Simulink model of the complete system, including the proposed method compatible with the Opal-RT, the model is discretized and compiled with the help of MATLAB and the Opal RT-Lab library. After editing, the complete system model is split into three subsystems as master, slave and console for RT-Lab simulation. In the master subsystem, the system’s model excluding the controllers and the scope is kept. The controllers are kept in the slave subsystem and the visual output devices such as scopes kept in the console subsystem. After compilation, the complete model including all three subsystems is loaded to the Opal-RT server for converting to the equivalent ‘C’ code of the model under the test.

Figure 3.

The real-time setup.

Example 1 (CSTR system): Consider a well-mixed continuous stirred tank reactor (CSTR) in which an isothermal multi-component chemical reaction A⇄B→C is being carried out. The dynamics of the CSTR is given by the following mathematical model (Ghaffari et al., 2013)

{\begin{matrix} {\overset{\cdot}{\bar{x}}}_{1} = 1 - {\bar{x}}_{1} - D a_{1} {\bar{x}}_{1} + D a_{2} {\bar{x}}_{2}^{2} \\ {\overset{\cdot}{\bar{x}}}_{2} = D a_{1} {\bar{x}}_{1} - {\bar{x}}_{2} - D a_{2} {\bar{x}}_{2}^{2} - D a_{3} {\bar{x}}_{2}^{2} + \bar{u} \\ {\overset{\cdot}{\bar{x}}}_{3} = - {\bar{x}}_{3} + D a_{3} {\bar{x}}_{2}^{2} \end{matrix}

(18)

where the states ${\bar{x}}_{i}$ for $i = 1, 2, 3$ , are dimensionless concentrations defined by

{\bar{x}}_{1} = \frac{C_{A}}{C_{AF}}; {\bar{x}}_{2} = \frac{C_{B}}{C_{BF}}; {\bar{x}}_{3} = \frac{C_{C}}{C_{CF}} .

The parameters $C_{AF}, C_{BF}, C_{CF}$ indicate the feed concentration of species $A, B$ , and C, respectively. $C_{i}$ for $i = 1, 2, 3$ , are known constant parameters. The dynamic equation (18) can be transformed in a way such that the equilibrium point will be in the origin and can be rearranged as

{\begin{cases} {\overset{\cdot}{x}}_{1} = - (1 + D a_{1}) x_{1} + 2 D A_{2} x_{2 d} + 2 D a_{2} x_{2 d} x_{2} + D a_{2} x_{2}^{2} \\ {\overset{\cdot}{x}}_{2} = D a_{1} x_{1} - (1 + 2 D a_{2} x_{2 d} + 2 D a_{3} x_{2 d}) x_{2} - (D a_{2} + D a_{3}) x_{2}^{2} + u \\ {\overset{\cdot}{x}}_{3} = 2 D a_{3} x_{2 d} x_{2} - x_{3} + D a_{3} x_{2}^{2} \end{cases}

(19)

where $u = \bar{u} - u_{d}$ , $x_{1} = {\bar{x}}_{1} - x_{1 d}$ , $x_{2} = {\bar{x}}_{2} - x_{2 d}$ and $x_{3} = {\bar{x}}_{3} - x_{3 d}$ . The values of the dynamic (19) are $D a_{1} = 0.3$ , $D a_{2} = 0.5$ , $D a_{3} = 0.1$ , $x_{1 d} = 0.3467$ , $x_{2 d} = 0.8796$ , $x_{3 d} = 0.8796$ and $u_{d} = 1$ . Letting $ρ (x) = 0.001 (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})$ and selecting degrees 2 and 4 of the states for the matrices P and K, theorem 1 is then applied to the system (19) considering the feedback gain matrices of the controller (4). In this example, the results of the polynomial controller (4) are compared with Ghaffari et al., (2013) in which a predictive linear controller based on the Lipchitz condition is designed. The computed feedback gains of the controller (4) and that of Ghaffari et al. (2013) are provided in Appendix A.

Simulation is done with the initial condition $x (0) = [1; - 1; 0.5]$ and discretizing step $0.001$ . Figure 4 illustrates the close-loop state evolutions of the CSTR and controller efforts for the proposed approach and Ghaffari et al. (2013).

Figure 4.

Closed-loop CSTR system (the proposed approach “blue dashed line” and Ghaffari et al. (2013)“red solid line”). (a). The state $x_{1}$ . (b). The state $x_{2}$ . (c). The state $x_{3}$ . (d). The control input u.

As it can be seen in Figure 4, the proposed approach provides a better performance and faster closed-loop states convergence and consumes less control signal amplitude compared with Ghaffari et al., 2013). Furthermore, the input signal norm specification and closed-loop states convergence times (2%) based on the proposed approach and Ghaffari et al. (2013) are given in Table 1.

Table 1.

CSTR closed-loop specifications.

Performance	Proposedapproach	The ref.(Ghaffari et al., 2013)
$‖ u ‖_{\infty}$	1.8370	12.1409
$‖ u ‖_{2}$	29.3500	91.3004
Convergence time of $x_{1}$	0.6523	1.6648
Convergence time of $x_{2}$	1.2148	2.0322
Convergence time of $x_{3}$	2.0125	4.7750

Example 2 (Lorenz chaotic system): Consider the nonlinear Lorenz chaotic system

\begin{matrix} {\overset{\cdot}{x}}_{1} = - a x_{1} (t) + a x_{2} (t) + u (t) \\ {\overset{\cdot}{x}}_{2} = c x_{1} (t) - x_{2} (t) - x_{1} (t) x_{3} (t) \\ {\overset{\cdot}{x}}_{3} = x_{1} (t) x_{2} (t) - b x_{3} (t) \end{matrix}

where $a = 10, b = \frac{8}{3} and c = 28$ . Furthermore, $x_{1} (t)$ , $x_{2} (t)$ and $x_{3} (t)$ are the state variables of the system. Meanwhile, $u (t)$ is the control input. The Lorenz chaotic system then can be presented in the following state space polynomial matrix form

A = [\begin{matrix} - a & a & 0 \\ c & - 1 & x_{1} \\ x_{2} & 0 & - b \end{matrix}]; B = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] .

In this simulation, we choose $ρ_{1} (x) = ρ_{2} (x) = 0.001 (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})$ . Moreover, the degree of the polynomial matrix P is set as 2. Note that the open-loop system can be verified to be unstable. By using the proposed method, the closed-loop system has a proper speed response while it is stable. The computed feedback gains of the controller (6) and that of (Tanaka, 2001) in which TS fuzzy model-based controller is designed based on the LMI techniques are provided in Appendix B. Figure 5(a)–(c) denote the three states of the Lorenz system that are stabilized with the initial conditions $[x_{1}, x_{2}, x_{3}] = [1, 1, - 1]$ using the proposed approach and Tanaka (2001). Also, Figure 5(d) shows the control signals evolutions derived based on the proposed method and Tanaka (2001).

Figure 5.

Closed-loop Lorenz system (the proposed approach “blue dashed line” and Tanaka (2001)“red solid line”). (a). The state $x_{1}$ . (b). The state $x_{2}$ . (c). The state $x_{3}$ . (d). The control input u.

As shown in Figure 5, the control signal based on the proposed method has acceptable characteristics. Compared with the similar researches that are using SOS with different approaches like the one in Papachristodoulou and Prajna (2005), the states are stabilized more quickly and the control signal has better norm quantities. Moreover, the system is stabilized using the Lyapunov function with a less degree of complexity. Figure 5 reveals that the close-loop state evolutions and the control signal of Tanaka (2001) experience highly oscillatory behaviors thay deteriorate the performance of Tanaka (2001). This deficiency comes from the concept of the TS fuzzy model that defines a local convex region for the polynomial plant and the fuzzy controller is designed based on the vertices of the convex region. However, the proposed method uses the original polynomial system and fully exploits the polynomial terms to design a polynomial controller. This fact improves the performance of the proposed controller over the TS fuzzy based techniques for the polynomial plants.

Example 3 (Permanent magnet synchronization motor): Consider a power benchmark permanent magnet synchronization motor (PMSM) with the following polynomial dynamic

{\begin{matrix} \frac{d {\overset{\cdot}{i}}_{d}}{dt} = \frac{u_{d} - R_{1} i_{d} + ω L_{q} i_{q}}{L_{d}} \\ \frac{d {\overset{\cdot}{i}}_{q}}{dt} = \frac{u_{q} - R_{1} i_{d} + ω L_{q} i_{q} - ω ψ_{r}}{L_{d}} \\ \frac{d ω}{dt} = \frac{n_{p} ψ_{r} i_{q} + n_{p} (L_{d} - L_{q}) i_{d} i_{q} - β ω}{J} \end{matrix}

(20)

where $i_{d}$ and $i_{q}$ are the direct-axis and quadrature-axis current components, respectively; $ω$ is the motor angular frequency; $u_{d}$ and $u_{q}$ are the direct-axis and quadrature-axis stator voltage components, respectively; $β$ is the viscous damping coefficient, $R_{1}$ is the stator winding resistance, $L_{d}$ and $L_{q}$ are the direct-axis and quadrature-axis stator inductor components, respectively; J is the polar moment of inertia, $n_{p}$ is the number of pole-pairs and $ψ_{r}$ is the permanent magnet flux. The PMSM parameters are chosen as (Li and Park, 2002): $L_{q} = L_{d} = L = 14.25 [mH]$ , $R_{1} = 0.9 [Ω]$ , $n_{p} = 1$ , $ψ_{r} = 0.031 [Nm / A]$ , $J = 4.7 \times 10^{- 5} [Kg m^{2}]$ and $β = 0.0162 [N / rad]$ .

In this example, the effect of selecting the degrees of states for the matrices P and K is studied. The feedback gains of the controller (6) are derived for the degrees 2, 4 and 6 of states, and provided in Appendix C. The initial condition is assumed to be $x (0) = [12 0 23]$ while the control inputs are assumed to saturate by lower and upper limits −100 and 100. Figure 6 illustrates the states evolutions and control efforts for the different controllers corresponding to the degrees 2, 4 and 6 of states.

Figure 6.

Closed-loop PMSM system (degree 6 “blue dashed line”, degree 4 “red solid line”, and degree 2 “green dot dashed line”). (a). The state $x_{1}$ . (b). The state $x_{2}$ . (c). The state $x_{3}$ . (d). The control input $u_{1}$ . (e). The control input $u_{2}$

As it can be seen in Figure 6, all of the controllers with saturation constraint can effectively stabilize the PMSM plant. However, by selecting higher degree for the states of the matrices P and K, the obtained control input experiences the saturation more. The reason is that the matrices P and K comprises higher degrees of states and higher amplitude control signal is achieved. However, the closed-loop PMSM system based on the degree 2 of states, does not reach the saturation limits. Furthermore, the convergence speed of-the PMSM closed-loop system, associated with higher degree of states, is faster than those of smaller degree of states.

Example 4 (Rossler chaotic system): Here, we consider another chaotic system that has a polynomial dynamic (Tanaka, 2001)

\begin{matrix} {\overset{\cdot}{x}}_{1} = - x_{2} (t) - x_{3} (t) \\ {\overset{\cdot}{x}}_{2} = x_{1} (t) + a x_{2} (t) \\ {\overset{\cdot}{x}}_{3} = b x_{1} (t) - {c - x_{1} (t)} x_{3} (t) + u (t) \end{matrix}

where $a = 0.34, b = 0.4 and c = 4.5$ . Moreover, $x_{1} (t)$ , $x_{2} (t)$ and $x_{3} (t)$ are the state variables of the system and $u (t)$ is the control input. Consequently, the Rossler chaotic system will have the following state space polynomial matrix form

A = [\begin{matrix} 0 & - 1 & - 1 \\ 1 & a & 0 \\ b & 0 & - c + x_{1} (t) \end{matrix}]; B = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]

The goal of this example is to show that the stability of the polynomial systems can be also guaranteed by non-polynomial matrices P and/or K. We set $ρ_{1} (x) = ρ_{2} (x) = 0.001$ and the degree of the matrix P is assumed to be zero. Therefore, the Lyapunov matrix is not a function of the system states. In addition, the matrix K is a polynomial matrix whose entries maximum degree is 2. The computed feedback gains of the controller (6) are provided in Appendix D.

The open-loop trajectory of the Rossler system is extremely unstable; however, using the proposed approach, it becomes asymptotically stable. The simulation is done for the initial states of $[x_{1}, x_{2}, x_{3}] = [1, 1, - 1]$ . The closed-loop Rossler system states and control input signals are shown in Figure 7.

Figure 7.

Closed-loop Rossler Chaotic system. (a). The states $x_{1}$ (blue dashed line), $x_{2}$ (red solid line), and $x_{3}$ (green dot dashed line). (b). The controller input u.

Conclusions

In this paper, a polynomial state feedback control design for polynomial systems is proposed. The constraints for stability and the performance of the system are translated in terms of the polynomial matrix inequalities. A non-iterative approach based on the SOS decomposition is presented to solve the polynomial matrix inequalities and convert the non-convex problem to the convex one while reducing the conservativeness. We illustrated that through the suitable selection of the polynomial degrees, we can have a tradeoff between simplicity of the controller and the computational cost of the design. To show the effectiveness and applicability of the proposed approach, different classes of the systems including the chaotic systems, power plant, and chemical process were provided. Simulation results illustrated that the presented approach has superiorities over the other numerical-based methods in reducing the control effort and the closed-loop oscillation behavior. There are some ideas that can be followed in future; the extension to the H∞ problem, robust and/or guaranteed cost controller design, and delayed polynomial systems can be good research areas for this methodology. Furthermore, the proposed method can be further optimized using generalized Eigenvalue problem (GEVP) to find the maximum bound that we set in the theorem for the time derivative of the Lyapunov function. In addition, deriving the sufficient conditions to guarantee the boundedness of the Lyapunov matrix time derivatives and compute the local stability region are of great importance.

Footnotes

Appendix A

The CSTR plant of example 1 is stabilized via the following proposed controller

(21)

u = N (x) P {(x)}^{- 1} x

where the feedback gains are partitioned as

P (x) = [\begin{matrix} P_{11} & P_{12} & P_{13} \\ * & P_{22} & P_{23} \\ * & * & P_{33} \end{matrix}]; N (x) = [\begin{matrix} N_{1} & N_{2} & N_{3} \end{matrix}]

with

\begin{matrix} P_{11} = 25432 x_{1}^{2} - 0.030215 x_{1} x_{2} + 0.36958 x_{1} x_{3} - 0.028089 x_{1} \\ + 24807 x_{2}^{2} + 0.13524 x_{2} x_{3} + 8348.7 x_{2} + 25431 x_{3}^{2} \\ + 0.1567 x_{3} + 33764 \end{matrix}

\begin{matrix} P_{12} = 0.15527 x_{1}^{2} + 0.0000015449 x_{1} x_{2} + 0.00000082455 x_{1} x_{3} \\ + 0.070566 x_{1} + 0.066268 x_{2}^{2} - 0.00000028038 x_{2} x_{3} \\ + 1990.1 x_{2} + 0.15526 x_{3}^{2} - 0.074118 x_{3} + 9969.4 \end{matrix}

\begin{matrix} P_{13} = - 5972.2 x_{1}^{2} + 0.071668 x_{1} x_{2} - 0.064435 x_{1} x_{3} \\ + 0.061769 x_{1} - 2020.6 x_{2}^{2} + 0.048535 x_{2} x_{3} \\ - 3552.9 x_{2} - 5972.2 x_{3}^{2} - 0.095376 x_{3} - 1607.7 \end{matrix}

\begin{matrix} P_{22} = 45312 x_{1}^{2} - 0.10847 x_{1} x_{2} - 0.028418 x_{1} x_{3} \\ + 0.12545 x_{1} + 50131 x_{2}^{2} - 0.047998 x_{2} x_{3} \\ - 23472 x_{2} + 45312 x_{3}^{2} + 0.040737 x_{3} + 58612 \end{matrix}

\begin{matrix} P_{23} = - 0.0061304 x_{1}^{2} - 0.00000022597 x_{1} x_{2} \\ + 0.000000026211 x_{1} x_{3} + 0.021589 x_{1} \\ - 0.057294 x_{2}^{2} + 0.000000051899 x_{2} x_{3} \\ + 2388 x_{2} - 0.0061297 x_{3}^{2} - 0.029677 x_{3} - 23547 \end{matrix}

\begin{matrix} P_{33} = 74848 x_{1}^{2} + 0.25995 x_{1} x_{2} + 0.2285 x_{1} x_{3} \\ + 0.11375 x_{1} + 93562 x_{2}^{2} - 0.2059 x_{2} x_{3} \\ + 2603.2 x_{2} + 74847 x_{3}^{2} - 0.29949 x_{3} + 115160.0 \end{matrix}

\begin{matrix} N_{1} = - 0.013279 x_{1}^{4} - 0.00000062713 x_{1}^{3} x_{2} \\ + 0.00000059856 x_{1}^{3} x_{3} - 0.039403 x_{1}^{3} \\ - 0.025519 x_{1}^{2} x_{2}^{2} + 0.0000023592 x_{1}^{2} x_{2} x_{3} \\ - 26125 x_{1}^{2} x_{2} - 0.03593 x_{1}^{2} x_{3}^{2} - 0.43344 x_{1}^{2} x_{3} - 44554 x_{1}^{2} \\ + 0.0000020577 x_{1} x_{2}^{3} - 0.00000021875 x_{1} x_{2}^{2} x_{3} - 0.45304 x_{1} x_{2}^{2} \\ - 0.00000044061 x_{1} x_{2} x_{3}^{2} + 0.33548 x_{1} x_{2} x_{3} - 0.53077 x_{1} x_{2} \\ + 0.00000061831 x_{1} x_{3}^{3} + 0.19452 x_{1} x_{3}^{2} - 0.75503 x_{1} x_{3} \\ - 0.11538 x_{1} + 0.03051 x_{2}^{4} - 0.0000010262 x_{2}^{3} x_{3} \\ - 29643 x_{2}^{3} - 0.025516 x_{2}^{2} x_{3}^{2} + 0.0058486 x_{2}^{2} x_{3} - 67511 x_{2}^{2} \\ - 0.0000025216 x_{2} x_{3}^{3} - 26126 x_{2} x_{3}^{2} - 0.73301 x_{2} x_{3} \\ + 11354 x_{2} - 0.013281 x_{3}^{4} + 0.32292 x_{3}^{3} - 44554 x_{3}^{2} \\ + 0.087143 x_{3} - 85558 \end{matrix}

\begin{matrix} N_{2} = - 53053 x_{1}^{4} - 0.23929 x_{1}^{3} x_{2} + 0.26765 x_{1}^{3} x_{3} \\ + 0.21917 x_{1}^{3} - 43357 x_{1}^{2} x_{2}^{2} + 0.18239 x_{1}^{2} x_{2} x_{3} \\ + 61301 x_{1}^{2} x_{2} - 45915 x_{1}^{2} x_{3}^{2} - 0.21351 x_{1}^{2} x_{3} \\ + 60550 x_{1}^{2} + 0.14642 x_{1} x_{2}^{3} - 0.077764 x_{1} x_{2}^{2} x_{3} - 0.071773 x_{1} x_{2}^{2} \\ + 0.08221 x_{1} x_{2} x_{3}^{2} + 0.33149 x_{1} x_{2} x_{3} \\ - 0.068416 x_{1} x_{2} + 0.10041 x_{1} x_{3}^{3} - 0.37046 x_{1} x_{3}^{2} \\ - 0.21657 x_{1} x_{3} + 0.2353 x_{1} - 51485 x_{2}^{4} - 0.089209 x_{2}^{3} x_{3} \\ + 73048 x_{2}^{3} - 43358 x_{2}^{2} x_{3}^{2} - 0.21455 x_{2}^{2} x_{3} + 41632 x_{2}^{2} \\ - 0.067694 x_{2} x_{3}^{3} + 61302 x_{2} x_{3}^{2} + 0.056483 x_{2} x_{3} + 4964.6 x_{2} \\ - 53051 x_{3}^{4} - 0.17212 x_{3}^{3} + 60550 x_{3}^{2} + 0.12708 x_{3} + 29358 \end{matrix}

\begin{matrix} N_{3} = - 0.0092943 x_{1}^{4} + 0.0000002314 x_{1}^{3} x_{2} \\ - 0.0000000083002 x_{1}^{3} x_{3} + 0.10269 x_{1}^{3} + 0.023374 x_{1}^{2} x_{2}^{2} \\ - 0.00000060371 x_{1}^{2} x_{2} x_{3} - 54485 x_{1}^{2} x_{2} - 0.059947 x_{1}^{2} x_{3}^{2} \\ - 0.075283 x_{1}^{2} x_{3} - 96003 x_{1}^{2} - 0.0000006264 x_{1} x_{2}^{3} \\ + 0.0000002073 x_{1} x_{2}^{2} x_{3} + 0.026365 x_{1} x_{2}^{2} \\ + 0.00000011671 x_{1} x_{2} x_{3}^{2} + 0.16816 x_{1} x_{2} x_{3} \\ - 0.16113 x_{1} x_{2} - 0.000000066199 x_{1} x_{3}^{3} \\ - 0.015691 x_{1} x_{3}^{2} - 0.088697 x_{1} x_{3} - 0.29078 x_{1} \\ + 0.057102 x_{2}^{4} + 0.00000021446 x_{2}^{3} x_{3} \\ - 49150 x_{2}^{3} + 0.023373 x_{2}^{2} x_{3}^{2} + 0.056482 x_{2}^{2} x_{3} \\ - 62013 x_{2}^{2} + 0.00000017168 x_{2} x_{3}^{3} \\ - 54485 x_{2} x_{3}^{2} - 0.05627 x_{2} x_{3} - 26504 x_{2} \\ - 0.0092938 x_{3}^{4} + 0.012141 x_{3}^{3} - 96001 x_{3}^{2} \\ + 0.18194 x_{3} - 143180.0 \end{matrix}

Also, the controller designed in Ghaffari et al. (2013) is as follows

u = Kx; K = [- 8.24073.8881 - 0.0242]

Appendix B

The Lorenz system of example 2 is stabilized via the following proposed controller

(22)

u = N (x) P {(x)}^{- 1} x

where the feedback gains are partitioned as

P (x) = [\begin{matrix} P_{11} & P_{12} & P_{13} \\ * & P_{22} & P_{23} \\ * & * & P_{33} \end{matrix}]; N (x) = [\begin{matrix} N_{1} & N_{2} & N_{3} \end{matrix}]

with

\begin{matrix} P_{11} = 61.55 x_{1}^{2} + 32.81 x_{1} x_{2} - 4.319 \times 10^{- 5} x_{1} x_{3} - 4.523 e \\ \times 10^{- 5} x_{1} + 75.12 x_{2}^{2} - 4.186 e \times 10^{- 5} x_{2} x_{3} + 5.685 \times 10^{- 5} x_{2} \\ + 70.17 x_{3}^{2} + 0.0001545 x_{3} + 117.4 \end{matrix}

\begin{matrix} P_{12} = 0.03234 x_{1}^{2} - 33.04 x_{1} x_{2} + 7.543 e \times 10^{- 9} x_{1} x_{3} \\ + 5.241 \times 10^{- 6} x_{1} - 0.001567 x_{2}^{2} - 2.747 \\ \times 10^{- 9} x_{2} x_{3} + 3.407 \times 10^{- 6} x_{2} + 0.001135 x_{3}^{2} \\ - 1.822 \times 10^{- 5} x_{3} - 19.32 \end{matrix}

\begin{matrix} P_{13} = - 5.166 \times 10^{- 12} x_{1}^{2} - 4.937 \times 10^{- 12} x_{1} x_{2} \\ + 1.475 \times 10^{- 10} x_{1} x_{3} + 0.06789 x_{1} + 3.531 \\ \times 10^{- 12} x 2^{2} + 2.831 \times 10^{- 12} x_{2} x_{3} + 4.781 x_{2} \\ + 2.292 \times 10^{- 12} x 3^{2} - 1.949 \times 10^{- 8} x_{3} + 1.585 \times 10^{- 5} \end{matrix}

\begin{matrix} P_{22} = 621.9 x_{1}^{2} - 283.9 x_{1} x_{2} - 1.338 \times 10^{- 5} x_{1} x_{3} \\ + 0.0001321 x_{1} + 393.0 x_{2}^{2} + 0.0004954 x_{2} x_{3} \\ - 0.000163 x_{2} + 430.6 x_{3}^{2} - 0.0001017 x_{3} + 247.5 \end{matrix}

\begin{matrix} P_{23} = 1.439 \times 10^{- 9} x_{1}^{2} + 1.712 \times 10^{- 10} x_{1} x_{2} \\ + 4.693 e \times 10^{- 10} x_{1} x_{3} - 27.12 x_{1} - 9.742 \\ \times 10^{- 10} x_{2}^{2} - 3.094 \times 10^{- 9} x_{2} x_{3} + 33.73 x_{2} \\ + 4.131 \times 10^{- 10} x_{3}^{2} + 4.013 \times 10^{- 6} x_{3} - 8.676 \times 10^{- 5} \end{matrix}

\begin{matrix} P_{33} = 0.004794 x_{1}^{2} - 0.0298 x_{1} x_{2} + 2.022 \times 10^{- 9} x_{1} x_{3} \\ + 8.106 \times 10^{- 9} x_{1} + 33.04 x_{2}^{2} - 4.609 \\ \times 10^{- 9} x_{2} x_{3} - 1.66 e \times 10^{- 5} x_{2} + 0.007318 x_{3}^{2} \\ - 1.445 \times 10^{- 9} x_{3} + 390.6 \end{matrix}

\begin{matrix} N_{1} = - 705.1 x_{1}^{4} + 37.9 x_{1}^{3} x_{2} + 0.001194 x_{1}^{3} x_{3} \\ + 0.0003137 x_{1}^{3} - 712.9 x_{1}^{2} x_{2}^{2} - 0.0009672 x_{1}^{2} x_{2} x_{3} \\ + 0.001039 x_{1}^{2} x_{2} - 710.8 x_{1}^{2} x_{3}^{2} + 0.002245 x_{1}^{2} x_{3} - 236.9 x_{1}^{2} \\ + 29.5 x_{1} x_{2}^{3} + 0.001236 x_{1} x_{2}^{2} x_{3} - 0.00414 x_{1} x_{2}^{2} \\ + 28.17 x_{1} x_{2} x_{3}^{2} + 0.003933 x_{1} x_{2} x_{3} + 597.1 x_{1} x_{2} \\ + 0.005048 x_{1} x_{3}^{3} - 0.0008774 x_{1} x_{3}^{2} + 0.00282 x_{1} x_{3} \\ - 0.001039 x_{1} - 672.6 x_{2}^{4} + 0.001263 x_{2}^{3} x_{3} \\ - 0.001527 x_{2}^{3} - 682.9 x_{2}^{2} x_{3}^{2} + 0.0006244 x_{2}^{2} x_{3} \\ - 39.58 x_{2}^{2} + 0.00127 x_{2} x_{3}^{3} + 0.000816 x_{2} x_{3}^{2} \\ - 0.002537 x_{2} x_{3} + 0.001001 x_{2} - 673.7 x_{3}^{4} \\ + 0.000123 x_{3}^{3} - 97.74 x_{3}^{2} + 0.001361 x_{3} + 626.8 \end{matrix}

\begin{matrix} N_{2} = - 4.437 e \times 10^{- 7} x_{1}^{4} + 1.449 \times 10^{- 7} x_{1}^{3} x_{2} \\ + 1.272 \times 10^{- 11} x_{1}^{3} x_{3} - 0.000233 x_{1}^{3} - 2.988 \\ \times 10^{- 7} x_{1}^{2} x_{2}^{2} - 1.267 \times 10^{- 11} x_{1}^{2} x_{2} x_{3} + 0.0006676 x_{1}^{2} x_{2} \\ - 9.459 \times 10^{- 7} x_{1}^{2} x_{3}^{2} - 0.007516 x_{1}^{2} x_{3} - 7001.0 x_{1}^{2} \\ - 4.778 \times 10^{- 7} x_{1} x_{2}^{3} - 4.138 \times 10^{- 12} x_{1} x_{2}^{2} x_{3} \\ + 0.002417 x_{1} x_{2}^{2} - 9.457 \times 10^{- 7} x_{1} x_{2} x_{3}^{2} \\ - 0.008854 x_{1} x_{2} x_{3} + 1656.0 x_{1} x_{2} + 7.338 \\ \times 10^{- 12} x_{1} x_{3}^{3} + 0.0006081 x_{1} x_{3}^{2} + 0.001666 x_{1} x_{3} \\ + 0.001433 x_{1} + 2.347 \times 10^{- 7} x_{2}^{4} + 2.216 \\ \times 10^{- 13} x_{2}^{3} x_{3} + 0.0008256 x_{2}^{3} + 1.371 \times 10^{- 7} x_{2}^{2} x_{3}^{2} \\ - 0.004173 x_{2}^{2} x_{3} - 5434.0 x_{2}^{2} - 3.95 \\ \times 10^{- 12} x_{2} x_{3}^{3} + 0.003097 x_{2} x_{3}^{2} - 0.0007288 x_{2} x_{3} \\ + 0.0004469 x_{2} - 8.493 \times 10^{- 8} x_{3}^{4} \\ - 0.0003167 x_{3}^{3} - 5594.0 x_{3}^{2} - 0.001574 x_{3} - 5492.0 \end{matrix}

\begin{matrix} N_{3} = 2.122 \times 10^{- 9} x_{1}^{4} - 5.359 \times 10^{- 9} x_{1}^{3} x_{2} \\ + 3.908 \times 10^{- 8} x_{1}^{3} x_{3} + 0.00223 x_{1}^{3} - 7.662 \\ \times 10^{- 9} x_{1}^{2} x_{2}^{2} + 2.222 \times 10^{- 8} x_{1}^{2} x_{2} x_{3} - 44.8 x_{1}^{2} x_{2} \\ - 5.061 e \times 10^{- 9} x_{1}^{2} x_{3}^{2} - 5.863 \times 10^{- 8} x_{1}^{2} x_{3} \\ - 0.0006866 x_{1}^{2} + 3.355 \times 10^{- 9} x_{1} x_{2}^{3} - 2.779 \times 10^{- 8} x_{1} x_{2}^{2} x_{3} \\ - 36.11 x_{1} x_{2}^{2} - 3.269 \times 10^{- 8} x_{1} x_{2} x_{3}^{2} + 0.0001337 x_{1} x_{2} x_{3} \\ - 2.941 \times 10^{- 5} x_{1} x_{2} + 2.352 \times 10^{- 8} x_{1} x_{3}^{3} \\ + 0.0006996 x_{1} x_{3}^{2} + 0.003092 x_{1} x_{3} + 86.41 x_{1} \\ + 3.976 \times 10^{- 9} x_{2}^{4} - 1.659 \times 10^{- 8} x_{2}^{3} x_{3} \\ - 61.46 x_{2}^{3} - 6.941 \times 10^{- 9} x_{2}^{2} x_{3}^{2} - 1.069 \times 10^{- 5} x_{2}^{2} x_{3} \\ - 0.0004807 x_{2}^{2} + 1.253 \times 10^{- 8} x_{2} x_{3}^{3} \\ - 56.55 x_{2} x_{3}^{2} - 0.0009155 x_{2} x_{3} - 310.4 x_{2} \\ + 2.589 \times 10^{- 9} x_{3}^{4} + 6.195 \times 10^{- 8} x_{3}^{3} \\ - 0.0005769 x_{3}^{2} + 6.908 \times 10^{- 5} x_{3} - 0.0007163 \end{matrix}

Also, the controller designed in Tanaka (2001) is as follows

u = \sum_{i = 1}^{2} h_{i} K_{i} x

with

h_{1} = 1 - h_{2} = \frac{1}{2} (1 - \frac{x_{1}}{30}); K_{1} = K_{2} = [8.1770 - 140.17161.4175]

Appendix C

The PMSM plant of example 3 is stabilized via the following proposed controller

(23)

u = N (x) P^{- 1} x

where

P = [\begin{matrix} P_{11} & P_{12} & P_{13} \\ * & P_{22} & P_{23} \\ * & * & P_{33} \end{matrix}]; N (x) = [\begin{matrix} N_{11} & N_{12} & N_{13} \\ N_{21} & N_{22} & N_{23} \end{matrix}]

However, for brevity, only the controller characteristics for the degree 2 of states are provided below

\begin{matrix} P_{11} = 257.95 x_{1}^{2} + 3.2476 \times 10^{- 6} x_{1} x_{2} + 2.64935 \times 10^{- 6} x_{1} x_{3} \\ - 6.2795 \times 10^{- 6} x_{1} + 257.95 x_{2}^{2} + 1.13065 \times 10^{- 4} x_{2} x_{3} \\ + 2.84585 \times 10^{- 5} x_{2} + 282.915 x_{3}^{2} + 11.5125 x_{3} \\ + 266.65 \end{matrix}

\begin{matrix} P_{12} = - 2.90355 \times 10^{- 4} x_{1}^{2} - 3.2042 \times 10^{- 11} x_{1} x_{2} \\ - 1.5953 \times 10^{- 10} x_{1} x_{3} + 9.768 \times 10^{- 6} x_{1} \\ - 2.9035 \times 10^{- 4} x_{2}^{2} + 1.2998 \times 10^{- 9} x_{2} x_{3} \\ - 8.023 \times 10^{- 6} x_{2} - 1.3781 \times 10^{- 4} x_{3}^{2} \\ - 105.07 x_{3} - 11.649 \end{matrix}

\begin{matrix} P_{13} = 492.45 x_{1}^{2} + 7.0292 \times 10^{- 6} x_{1} x_{2} + 5.1128 \times 10^{- 6} x_{1} x_{3} \\ - 1.2144 \times 10^{- 5} x_{1} + 492.45 x_{2}^{2} + 2.1638 \times 10^{- 4} x_{2} x_{3} \\ + 5.4296 \times 10^{- 5} x_{2} + 539.78 x_{3}^{2} + 21.840 x_{3} + 508.17 \end{matrix}

\begin{matrix} P_{22} = 9.1771 \times 10^{- 4} x_{1}^{2} + 4.3489 \times 10^{- 10} x_{1} x_{2} \\ - 1.5307 \times 10^{- 10} x_{1} x_{3} - 2.0551 \times 10^{- 9} x_{1} \\ + 9.1771 \times 10^{- 4} x_{2}^{2} + 1.4737 \times 10^{- 10} x_{2} x_{3} \\ + 2.815 \times 10^{- 9} x_{2} + 5.494 \times 10^{- 4} x_{3}^{2} + 3.083 \\ \times 10^{- 2} x_{3} + 267.68 \end{matrix}

\begin{matrix} P_{23} = 1.2440 \times 10^{- 4} x_{1}^{2} + 1.097 \times 10^{- 10} x_{1} x_{2} \\ + 7.3833 \times 10^{- 11} x_{1} x_{3} + 1.8596 \times 10^{- 5} x_{1} \\ + 1.244 \times 10^{- 4} x_{2}^{2} - 1.469 \times 10^{- 12} x_{2} x_{3} \\ - 1.5132 \times 10^{- 5} x_{2} + 1.7386 \times 10^{- 4} x_{3}^{2} \\ - 201.01 x_{3} - 22.437 \end{matrix}

\begin{matrix} P_{33} = 945.68 x_{1}^{2} + 8.916 \times 10^{- 6} x_{1} x_{2} + 9.8882 \times 10^{- 4} x_{1} x_{3} \\ - 2.3601 \times 10^{- 5} x_{1} + 945 . 68_{x 2}^{2} + 4.1413 \times 10^{- 4} x_{2} x_{3} \\ + 1.0403 \times 10^{- 4} x_{2} + 1036.2 x_{3}^{2} + 41.698 x_{3} + 977.13 \end{matrix}

\begin{matrix} N_{11} = - 3157.5 x_{1}^{2} + 7.0616 \times 10^{- 4} x_{1} x_{2} \\ - 1.7126 \times 10^{- 4} x_{1} x_{3} - 3.9835 \times 10^{- 5} x_{1} \\ - 3157.5 x_{2}^{2} + 1.2238 \times 10^{- 4} x_{2} x_{3} \\ + 2.8837 \times 10^{- 5} x_{2} - 2461.7 x_{3}^{2} + 365.26 x_{3} - 3802.6 \end{matrix}

\begin{matrix} N_{12} = 88.766 x_{1}^{2} - 2.1256 \times 10^{- 5} x_{1} x_{2} \\ + 2.8440 \times 10^{- 4} x_{1} x_{3} - 1.3847 \times 10^{- 4} x_{1} \\ + 88.765 x_{2}^{2} + 9.3559 \times 10^{- 5} x_{2} x_{3} + 3.1084 \times 10^{- 5} x_{2} \\ + 459.06 x_{3}^{2} - 848.86 x_{3} - 12.060 \end{matrix}

\begin{matrix} N_{13} = - 558.8 x_{1}^{2} - 4.0300 \times 10^{- 4} x_{1} x_{2} \\ + 8.3413 \times 10^{- 6} x_{1} x_{3} + 4.0059 \times 10^{- 5} x_{1} \\ - 558.8 x_{2}^{2} - 7.0102 \times 10^{- 6} x_{2} x_{3} \\ + 2.4353 \times 10^{- 5} x_{2} - 488.36 x_{3}^{2} + 16.569 x_{3} - 709.49 \end{matrix}

\begin{matrix} N_{21} = 88.766 x_{1}^{2} - 2.1256 \times 10^{- 5} x_{1} x_{2} + 2.8440 \times 10^{- 4} x_{1} x_{3} \\ - 1.3847 \times 10^{- 4} x_{1} + 88.765 x_{2}^{2} + 9.3559 \times 10^{- 5} x_{2} x_{3} \\ + 3.1084 \times 10^{- 5} x 2 + 459.06 x_{3}^{2} - 848.86 x_{3} - 12.060 \end{matrix}

\begin{matrix} N_{22} = - 3126.8 x_{1}^{2} - 1.5296 \times 10^{- 3} x_{1} x_{2} - 1.9273 \times 10^{- 5} x_{1} x_{3} \\ + 2.5772 \times 10^{- 5} x_{1} - 3126.8 x_{2}^{2} + 1.2 \times 10^{- 4} x_{2} x_{3} \\ - 1.0575 \times 10^{- 4} x_{2} - 4124.1 x_{3}^{2} - 1531.7 x_{3} - 3653.7 \end{matrix}

\begin{matrix} N_{23} = 104.85 x_{1}^{2} + 2.6645 \times 10^{- 4} x_{1} x_{2} + 4.0967 \times 10^{- 5} x_{1} x_{3} \\ + 6.1511 \times 10^{- 7} x_{1} + 104.85 x_{2}^{2} - 8.9422 \times 10^{- 5} x_{2} x_{3} \\ - 5.0456 \times 10^{- 6} x_{2} + 401.84 x_{3}^{2} + 1.1617 x_{3} + 0.63016 \end{matrix}

Appendix D

The Rossler system of example 4 is stabilized via the following proposed controller

(24)

u = N (x) P^{- 1} x

where

P = [\begin{matrix} 452.55 & - 169.3 & 284.79 \\ * & 402.39 & - 7.1849 \\ * & * & 378.43 \end{matrix}]; N (x) = [\begin{matrix} N_{1} & N_{2} & N_{3} \end{matrix}]

with

\begin{matrix} N_{1} = - 0.0006721 x_{1}^{2} + 0.0000000001412 x_{1} x_{2} \\ + 0.00000000011638 x_{1} x_{3} - 263.39 x_{1} \\ + 0.0000000000000028997 x_{2}^{2} + 0.000000000000003759 x_{2} x_{3} \\ - 0.00000079692 x_{2} + 0.0000000000000064302 x_{3}^{2} \\ - 0.0000067598 x_{3} + 984.9 \end{matrix}

\begin{matrix} N_{2} = 0.000018032 x_{1}^{2} + 0.000000000067299 x_{1} x_{2} \\ + 0.0000000002249 x_{1} x_{3} + 10.364 x_{1} \\ + 0.0000000000000054283 x_{2}^{2} + 0.0000000000000087935 x_{2} x_{3} \\ + 0.000000018409 x_{2} + 0.0000000000000059891 x_{3}^{2} \\ + 0.0000011625 x_{3} - 228.55 \end{matrix}

\begin{matrix} N_{3} = - 162.63 x_{1}^{2} - 0.000000085302 x_{1} x_{2} \\ - 0.000000034575 x_{1} x_{3} - 433.33 x_{1} - 204.95 x_{2}^{2} \\ + 0.000069914 x_{2} x_{3} - 0.00000014555 x_{2} \\ - 204.95 x_{3}^{2} - 0.0000011904 x_{3} + 814.37 \end{matrix}

The dimensions and number of equations in the resulting semi-definite programing (SDP) and the computation time of each example are given in Table 2. The computational time is obtained through numerical simulations in Matlab 2012a environment in a computer with 5-core CPU processor.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Mohammad-Hassan Khooban

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