Optimization of Robust LMI-Control Systems for Unstable Vertical Plasma Position in D-Shaped Tokamak

Abstract

The paper is devoted to the synthesis, comparison, and optimization of robust LMI-control systems for the vertical plasma position in a D-shaped tokamak, specifically the T-15MD tokamak (Kurchatov Institute, Moscow, Russia). The novelty of this work is to find out the possibilities of LMI robust control systems, according to the criteria of the robust stability radius and control power peak at the rejection of a minor disruption type disturbance and a reference step signal using a unique unstable first-order plasma model. The plant under control consists of the connected in series plasma model with additive disturbance containing plant uncertainties, horizontal field coil (HFC), and actuator model as a multiphase rectifier. A set of robust controllers was designed by Linear Matrix Inequalities (LMI) method with pole placement in the LMI regions, state H₂/H_∞ performance, and output signal performance. The LMI theorems of the paper are directed to design the robust controllers and study the systems with the aim of eliminating the gap between theory and practice. The main achievement of this work consists in the optimization of robust control systems of the unstable plant with uncertain disturbance on the set of LMI synthesis approaches. The control systems have original quality criteria, such as control power and robust stability radius. The best control system on the basis of two criteria, namely, $D_{\alpha ,r,\vartheta }$ control system provides stabilization of the vertical plasma position on the real-time digital control testbed.

1. Introduction

The most perspective direction of controlled thermonuclear fusion (CTF) are tokamaks with a D-shaped cross-section [1], on the basis of which it is planned to create thermonuclear neutron sources, hybrid thermonuclear reactors, and thermonuclear power plants. Confinement and heating of plasmas in tokamaks for self-sustaining thermonuclear reaction is provided by systems of magnetic and kinetic plasma control with feedback. The plasma in the magnetic field of the tokamak vessel is extremely difficult to diagnose and control as a nonstationary nonlinear dynamic plant with distributed parameters, with structural and parametric uncertainties, subject to the effects of rapid processes of uncontrolled disturbances in the presence of broadband noise. Therefore, one of the main tasks of the CTF has not yet been solved: the prolonged confinement of the high-temperature plasma in tokamaks with given parameters and the prevention of plasma disruptions.

Because of that, the plasma control systems must be developed further. In spite of the extreme complexity of the plasma as a controlled plant and that the plasma physics theory does not coincide very well with the experimental plasma, some facts are known for sure, which make easier plasma magnetic control:

• Plasma equilibrium is subject to the Grad-Shafranov equation [1];
• The plasma is inside feedback loops and its output variations around references are relatively small [2,3];
• Linear plasma models are used for plasma magnetic control systems design [2].

1.1. Tokamak Plasma Vertical Position Control and LMI

The first tokamak was built in 1954 at the Kurchatov Institute of Atomic Energy of the Former Soviet Union. Since then, tokamaks have passed in their evolution from circular in cross-section tokamaks to modern vertically elongated D-shaped tokamaks with an air core, which makes it possible to achieve a significantly higher gas-kinetic pressure of the plasma.

The plasma in a tokamak is a nonlinear system with distributed parameters that is subject to a large number of disturbances. The plasma is a generator of broadband poorly understood noises. The plasma is thermodynamically unstable in the magnetic field of the tokamak. The tokamak plasma is subject to disruptions because of its sensitivity to numerous random disturbances, some of which are dangerous to the device itself. A basic example is vertical instability due to the vertical elongation of the plasma. An increase in the vertical elongation of the tokamak leads to an increase in plasma pressure, but consequently leads to plasma vertical instability, that is axisymmetric and characterized by a predominantly vertical displacement. Thus, it is necessary to develop effective approaches for plasma vertical position stabilization by feedback [4].

A set of toroidal field coils outside the vacuum vessel together with the plasma current creates helical magnetic field lines along which the charged particles (ions and electrons) that form the plasma move forming plasma and rotating along the Larmorian radii (Figure 1). A special subset of the poloidal field (PF) coils, called the central solenoid, is used primarily to generate and control the plasma current, using the transformer principle. The remaining PF-coils are used to shape the plasma, control its position, and, in tokamaks with a vertically elongated plasma cross section, to stabilize the vertical position of the plasma. It is important to note that all the plasma control system loops should be consistent with each other and with all inputs and outputs of the plant under control, namely the plasma in the tokamak.

One of the priority tasks in the modern problem of controlled thermonuclear fusion is the confinement of plasma in the magnetic field of tokamaks by means of feedback control systems with high performance (speed and accuracy) and robust stability margins. In modern tokamaks, engineers strive to implement such regimes in which the triple product of temperature, density, and energy plasma confinement time is maximized within the working range of parameters in order to achieve a self-sustaining thermonuclear reaction. In the plasma of tokamaks operating in these modes, there is a possibility of various instabilities that can lead to disruption of discharge. In this case, mechanical or thermal damage to the tokamak construction can occur during major disruptions. When constructing modern tokamaks, a vertically elongated structure is chosen, as, for example, in the existing tokamaks DIII-D, NSTX (USA), JET (England), JT-60SA (Japan), ASDEX Upgrade (Germany), TCV (Switzerland), EAST (China), KSTAR (South Korea), Globus-M/M2 (Russia), which makes it possible to significantly increase the plasma pressure compared to the plasma pressure in tokamaks with a circular cross section at the same toroidal magnetic field. However, this leads to the emergence of vertical instability of the plasma. It is also known that the best plasma parameters can be achieved when the plasma boundary is located close to the first wall of the tokamak, but it is necessary to accurately stabilize the plasma boundary through feedback control systems so that the hot plasma does not burn through the tokamak vessel. The study of instabilities that arise during a plasma discharge, as well as the development of methods for their suppression, is necessary to ensure the operability of tokamaks. The development of magnetic and kinetic plasma control systems is being actively carried out at all operating tokamaks and at the design stage, which makes it possible to ensure the highest stability margins of systems and obtain a sufficiently high performance of plasma control. This is necessary to ensure high reliability of future thermonuclear reactors and power plants.

In this paper, the plasma vertical position control system is designed, modeled, and optimized for the T-15MD tokamak that has been created in Kurchatov Institute, Moscow, Russia [6,7,8]. In this case, the actuator for the horizontal field coil (HFC) was chosen as a 6-pulse multiphase thyristor rectifier [4]. To design a guaranteed robust control system the linear matrix inequalities (LMIs) [9] were used as a powerful tool in the field of control systems design and analysis. The problems of control system development, such as state feedback synthesis, robustness analysis, and $H_2$/$H_{\infty }$–control as well as output robust controller, can all be reduced to convex or quasi-convex problems that involve LMIs [10]. This approach allows us to develop and optimize robust control systems of plasma unstable vertical positions in the D-shaped tokamak.

The plasma in a tokamak as an unstable plant under control can only be stabilized by feedback. Closed-loop systems with unstable components are only locally stable [11]; therefore, they have a limited controllability area. Controllers for unstable systems are operationally critical. The development of magnetic plasma control systems has to be carefully carried out at the design stage, which makes it possible to ensure a sufficiently high performance of plasma control.

The main novel achievement of the work consists in the optimization of robust control systems of the unstable plant with uncertain disturbance on the set of LMI synthesis approaches. The control systems have original quality criteria, which is a simultaneous optimization of control power and robust stability radius. These two criteria allowed us to compare plasma unstable vertical position control systems with multiphase thyristor rectifier as an actuator designed by LMI for the vertically elongated T-15MD tokamak.

1.2. Paper Structure

The paper is organized as follows. The problem setting is given in Section 2. Section 3 establishes the acceptable disturbance. Section 4 discusses the D-stability control problem. Section 5 and Section 6 present the optimization of LMI conditions and the controller. In Section 7, the robust state feedback control problem was solved. Section 8 concerns the output-feedback stabilization control problem. In Section 9, the robust stability radius for control systems with different controllers is determined and calculated. The comparative analysis of closed-loop vertical position plasma control systems is discussed in Section 10 taking into account robust stability radii and control power. In Section 11, the results of modeling of the control system on the realtime target machines are presented. Section 12, unrolls the main results, explains their meanings and gives some future work ideas. Section 13, summarizes the results.

2. Statement of the Control Problem

2.1. Tokamak T-15MD

The T-15MD plasma has the major radius $R_0$ = 1.48 m, the minor radius a = 0.67 m, the elongation k = 1.7–1.9, the triangularity $\delta$ = 0.3–0.4, the plasma current $I_p$ = 2 MA, and the toroidal magnetic field at the plasma axis B = 2 T [6,7,8]. The HFC (Figure 2 is positioned between a toroidal field (TF) coil and a vacuum vessel in the T-15MD tokamak’s construction (Figure 2) to suppress the vertical plasma instability. The T-15MD tokamak project had this coil moved out of the position between the PF (Poloidal Field) coils to the position depicted in Figure 2 because its original location caused an internal instability of the closed-loop control system for the plasma’s vertical position [12,13].

2.2. Model of the Plant under Control and the Work Goal

The plasma in the tokamak, the vertical displacement diagnostics unit, the HFC, the actuator, and the controller are the components of the open loop control system for the plasma vertical position (Figure 3).

The unstable linear plasma model is described by the first order differential equation with an additive disturbance:

$$T_p\frac{dZ\left(t\right)}{dt}-Z\left(t\right)=K_p(I\left(t\right)+w\left(t\right)),$$

(1)

the control coil model is

$$L\frac{dI\left(t\right)}{dt}+RI\left(t\right)=U\left(t\right),$$

(2)

and the multiphase thyristor rectifier linear model is

$$T_a\frac{dU\left(t\right)}{dt}+U\left(t\right)=K_{a}V\left(t\right).$$

(3)

Equations (1)–(3) were used when designing and studying a stabilizing system in the T-15MD tokamak [4]. Here, Z is the plasma’s vertical displacement, I and U are the current and the voltage of the HFC, $K_p$ and $T_p$ are the gain and time constants of the plasma model, w is the additive disturbance, and $K_a$ and $T_a$ are the gain and time constants of the rectifier model. The numerical values of the model parameters are as follows: $T_a=3.3\times {10}^{-3}$ s, $T_p=20.8\times {10}^{-3}$ s, $T_c=46.7\times {10}^{-3}$ s, $K_a=2000$, $K_p=1.78\times {10}^{-5}$ m/A, $K_c=11.111/\mathrm{\Omega }$.

We determined the inductance L and ohmic resistance R of the control coil in [4] using data from the T-15MD tokamak project. The horizontal magnetic field in T-15MD is mostly produced by the current in this coil. The identification of the nonlinear DINA plasma-physics code modified for the vertically elongated plasma in T-15MD [14] provided an estimate of the time constant in the plasma model, $T_p$ = 20.8 ms. From the linearized DINA-L model, and the gain $K_p$ = 1.78 cm/kA was calculated at a specific point of the T-15MD scenario [12,13]. The gain and time constants for the rectifier model are $K_a$ = 2000 and $T_a$ = 3.3 ms, respectively, and are taken from the rectifier’s technical documentation.

The plasma model unstable pole is equal to $1/T_p$.

The effect of the conductive structures was considered because the plasma vertical displacement diagnostics unit and machine conductive structure are both included in the plasma model.

Since the plasma is an extremely complicated dynamical plant its nonlinear models (codes) describing plasma by means of partial differential equations are very complicated, cumbersome, and slow for control system design and investigation. The simplest plasma model (1) is free from these disadvantages because it is the dynamical unstable unit of the first order and the other plasma part is modeled by the additive disturbance w including uncertainties. On the other hand, this disturbance imitates minor disruptions [1,15]. For the T-15MD tokamak, this model was obtained by the identification procedure [16] of the DINA code [17], which was tuned to the T-15MD tokamak nominal regime and relative to the quasi-stationary phase of the plasma discharge. The model (1) became a basis for the development and installation of plasma vertical position control systems on two Russian operating tokamaks: spherical Globus-M/M2 (Ioffe Institute, St. Petersburg, Russia) and T-11M (The Joint Stock Company State Research Center of the Russian Federation Troitsk Institute for Innovation and Fusion Research) (the actuators of those systems are thyristor current inverters working in self-oscillating mode) [18]. This is the reason to use the model (1) for the design and study of the plasma vertical position control system of the T-15MD tokamak. In this case, a multiphase thyristor rectifier has been adopted as the actuator in this system, similar to the ASDEX Upgrade plasma position control system [19].

The basic original idea of the simplest plasma model for control system design is that the plant output is decomposed into a Taylor series about a reference point and only one unstable mode is included in the control system feedback, and the remaining stable modes are included in the additive perturbation along with the plant uncertainty. This approach makes it possible to design a control system for suppressing vertical plasma instability using only a first-order unstable plant model. Thus, this is the maximum possible simplification of the model of the controlled plant, which is required by engineering approaches when it comes to controlling complex dynamic plants. As a result, the developed control system was designed in the simplest and the most reliable way.

In doing so, there is a chance to take a different way and obtain lower-order models of the plasma in tokamaks. For example, it is possible to obtain models of plants from first-principles equations or by means of identification procedures [20], and then reduce the order of models using traditional approaches [21] or by a machine learning methodology [22]. However, it is impossible to obtain a plasma model of vertical motion in tokamaks of the first order by these approaches, because they do not take into account the physics of the plasma in vertically elongated tokamaks. It is well known that the plasma vertical motion has only one unstable mode and no more. Therefore, only this mode can be taken into account in the plasma model, as the other stable modes decay (vanish). This understanding of plasma physics led to the simplest model of controlled plant, which was used in our development of vertical position plasma control systems.

We have been operating the vertical and horizontal plasma position control systems in the Globus-M2 spherical tokamak and the T-M11 tokamak in Russia for a very long time and have unique experience with these control systems. This gave us an opportunity to develop and investigate in detail an unstable vertical plasma position control system for the T-15MD tokamak, which is planned to be used together with the magnetic control system of horizontal plasma position, current, and shape.

The linear plant model (1)–(3) can be presented in the state-space form

$$\begin{array}{cc}\hfill \dot{x}& =Ax+B_{1}u+Dw,\hfill \\ \hfill y& =Cx,\hfill \end{array}$$

(4)

with the matrices

$$A=\left(\begin{array}{ccc}-\frac{1}{T_a}& 0& 0\\ \frac{K_c}{T_c}& -\frac{1}{T_c}& 0\\ 0& \frac{K_p}{T_p}& \frac{1}{T_p}\end{array}\right),B_1=\left(\begin{array}{c}\frac{K_a}{T_a}\\ 0\\ 0\end{array}\right),D=\left(\begin{array}{c}0\\ 0\\ \frac{K_p}{T_p}\end{array}\right),$$

where $x={\left(\begin{array}{ccc}U& I& Z\end{array}\right)}^T$ is the state, $u\left(t\right)$ is the control input (the voltage on the HFC), $y\left(t\right)$ is the observable output, and $w\left(t\right)$ is the exogenous disturbance input. The pair $(A,B_1)$ is controllable [23].

Generally, it will be assumed that the plant state x is fully available for observation, that is C is equal to the unit matrix.

However, in Section 8, the state x of a system is unavailable for observation and the information about the plant is provided by its scalar output with the matrix

$$C=\left(\begin{array}{ccc}0& 0& 1\end{array}\right).$$

In this case, the pair $(A,C)$ is observable [23].

The linear static state feedback controller

$$u=Kx,K=\left(\begin{array}{ccc}{K}_1& K_2& K_3\end{array}\right)$$

(5)

will be synthesized for the various control problems related to the considered plant. The exception here is Section 8, in which the linear output dynamical controller will be treated.

Using the LMI technique, the objective of this work is to build, examine, and evaluate vertical position stabilization systems on a sample of the simplest plasma vertical movement model of the T-15MD tokamak at minor disruption disturbances during the plasma discharge.

LMI techniques are currently undergoing rapid development, making it possible to create reliable numerical methods for designing control systems as well as to formulate control problems from a common perspective [9,24]. As this takes place, in addition to robustness features and acceptable performance, the control power and robust stability margin were used to compare the control systems designed by the LMI approach. The advantages of the given criterion indexes rest in their focus on practical application of synthesized control systems. The problem of optimization of robust control systems for an unstable plant with uncertain perturbation on the set of synthesis approaches by means of LMI is posed. The control systems have original quality criteria, which are a simultaneous optimization of control power and a robust stability radius. These two criteria allow us to compare plasma unstable vertical position control systems with a multiphase thyristor rectifier as an actuator designed by LMI.

2.3. Dynamical Model of Vertical Plasma Displacement in a Tokamak

It was shown in [25] that plasma as a control plant with distributed parameters in an axially symmetric toroidal magnetic configuration behaves as a first-order dynamic unit when its vertical position is controlled by the magnetic field. All plasma control systems designed using these such simplified linear models of the 2nd order are practically operable. For instance, the control systems for the plasma’s horizontal and vertical positions of the T-11M and the spherical Globus-M tokamaks have been working for more than 10 years [18].

Using the examples of the tokamak project ITER [25], it was shown that the vertical and horizontal plasma displacements model in tokamaks can be modeled as a second-order unit with the control coil taken into account. To describe the vertical plasma displacement in ITER, a multimode numerical model was used to determine 22 current harmonics in the vessel and passive structural elements (author is Y.V. Gribov, Ph.D. in Physics and Mathematics) [26].

The multimode model of vertical plasma displacement in a tokamak [26] describes the behavior of an aperiodic unstable plant, which, in particular, had to be moved from arbitrary initial conditions to the origin. Therefore, a set of phase trajectories was numerically obtained on the plane of dimensionless parameters of vertical displacement and control current exiting from points on the horizontal axis at the negative sign of the maximum voltage on the control coil ($-E$). Figure 4, a shows this set (solid lines) for the case when the active resistance of the control coil is zero. Here, the phase trajectories were separated by varying the initial conditions. It turned out that the phase trajectories do not intersect each other, which makes it possible to approximate the multimode model by an unstable dynamic unit of the second order.

The plasma and the passive stabilization elements can be modeled by the first equation, and the control coil can be represented as an integrator (the second equation) in the following system of equations:

$$\begin{array}{cc}\hfill T\dot{Z}-Z& =K_0(I_c+\alpha T{\dot{I}}_c),\hfill \\ \hfill L\dot{I_c}& =U,\hfill \end{array}$$

(6)

where T is the time constant of the plant model, $K_0$ is the gain constant, $\alpha$ is the constant parameter, and L is the inductance of the control coil. Equations (6) in dimensionless values $\overline{Z}=\frac{Z}{Z_0}$, $\overline{I}=\frac{I}{I_0}$, $\overline{t}=\frac{t}{t_0}$, $\overline{U}=\frac{U}{E}$ at normalizing parameters $Z_0=\frac{E{K}_{0}T(1+\alpha )}{L}$, $I_0=\frac{ET(1+\alpha )}{L}$, $t_0=T$ have the form

$$\begin{array}{cc}\hfill \frac{d\overline{Z}}{d\overline{t}}& =\overline{Z}+\overline{I}+\frac{\alpha }{1+\alpha }\overline{U},\hfill \\ \hfill \frac{d\overline{I}}{d\overline{t}}& =\frac{1}{1+\alpha }\overline{U}.\hfill \end{array}$$

(7)

For this system of equations for $\overline{U}=-1$, the phase trajectories are numerically constructed and are derived from the same initial conditions as the phase trajectories of the approximated model of the plant with distributed parameters. In Figure 4a, the trajectories of the distributed model (7) are shown as dashed lines and correspond to the sufficiently good approximation of the projections of the trajectories of the model onto the phase plane along the time axis.

A similar result on the identification of the plasma model during its vertical displacement was obtained using the DINA plasma-physical code [14] by another method for modeling plasma in the T-15MD tokamak with distributed parameters, which consists in the fact that the plasma vertical displacement was approximated by an unstable first-order dynamic unit by the input–output signals of the DINA code with the sufficiently high accuracy (Figure 4b) [12,13,16].

3. Acceptable Disturbance Estimation

Optimal rejection of the non-random exogenous disturbances in linear control systems is one of the classical problems of control theory. There are numerous approaches to its solution [9,27,28]; here, the concept of invariant (bounding) ellipsoids to characterize the uncertainty in the system state (output) caused by the unknown-but-bounded exogenous disturbances is adopted. The LMI technique is considered as a powerful technical tool in the implementation of this approach.

The core of the invariant ellipsoids approach [29,30] is shortly recalled. Consider the continuous-time dynamical system. It is characterized by simplicity and ease of algorithmization and, as shown by numerous experiments, leads to quite satisfactory results.

$$\begin{array}{cc}\hfill \dot{x}& =Ax+Bw,x\left(0\right)=x_0,\hfill \\ \hfill y& =Cx\hfill \end{array}$$

(8)

where $A\in {\mathbb{R}}^{n\times n}$, $B\in {\mathbb{R}}^{n\times m}$, $C\in {\mathbb{R}}^{l\times n}$ are known matrices, $x\left(t\right)\in {\mathbb{R}}^n$ is the state vector, $y\left(t\right)\in {\mathbb{R}}^l$ is the regulated output, and $w\left(t\right)\in {\mathbb{R}}^m$ is the bounded exogenous disturbance:

$$\parallel w\left(t\right)\parallel \le 1\mathrm{for}\mathrm{all}t\ge 0.$$

(9)

Assume that system (8) is stable (A is Hurwitz matrix), the pair $(A,B)$ is controllable, and C is a full rank matrix. From now onward $\parallel \times \parallel$ denotes the Euclidean vector norm and spectral matrix norm, and all matrix inequalities are understood in the sense of sign-definiteness.

Definition 1

([30]). The ellipsoid

$$\mathcal{E}=\{x\in {\mathbb{R}}^n:x^T{P}^{-1}x\le 1\},P\succ 0,$$

(10)

is said to be invariant for system (8), (9), if the condition $x\left(0\right)\in \mathcal{E}$ implies $x\left(t\right)\in \mathcal{E}$ for all $t\ge 0$.

The following well-known result takes place.

Theorem 1

([9,27]). Ellipsoid (10) is invariant for systems (8), (9) with $x\left(0\right)=0$ if and only if its matrix P satisfies the LMIs

$$AP+P{A}^T+\alpha P+\frac{1}{\alpha }D{D}^T⪯0,P\succ 0,$$

(11)

for some positive α.

This setup could be extended to the case of nonzero initial conditions $x\left(0\right)=x_0$: it is required $x\left(0\right)\in \mathcal{E}$ which, by the Schur lemma [31] is equivalent to the LMI

$$\left(\begin{array}{cc}1& x_0^T\\ x_0& P\end{array}\right)⪰0.$$

This LMI is to be appended to basic LMI constraints (11).

Having an invariant ellipsoid (10) for the system state, it is easy to see that the linear system output y belongs to the so-called bounding ellipsoid defined as

$${\mathcal{E}}_y=\{y\in {\mathbb{R}}^l:y^T{\left(CP{C}^T\right)}^{-1}y\le 1\}.$$

Now, the design problem is considered for the control system

$$\begin{array}{cc}\hfill \dot{x}& =Ax+B_{1}u+B_{2}w,x\left(0\right)=x_0\hfill \\ \hfill y& =Cx,\hfill \end{array}$$

(12)

where $x\left(t\right)\in {\mathbb{R}}^n$ is the state vector, $y\left(t\right)\in {\mathbb{R}}^l$ is the regulated output, $u\left(t\right)\in {\mathbb{R}}^p$ is the control input, and $w\left(t\right)\in {\mathbb{R}}^m$ is the exogenous disturbance satisfied (9). Matrices $A\in {\mathbb{R}}^{n\times n}$, $B_1\in {\mathbb{R}}^{n\times p}$, $B_2\in {\mathbb{R}}^{n\times m}$, $C\in {\mathbb{R}}^{l\times n}$ are known, and the matrix pair $(A,B_1)$ is controllable.

We impose the following constraint on the control input:

$$\parallel u\left(t\right)\parallel \le \overline{u}\mathrm{for}\mathrm{all}t\ge 0.$$

(13)

The problem is to estimate the maximal range $\parallel w\parallel \le \widehat{w}$ of the exogenous disturbances w for which the system output z for the closed-loop system remains in the given ball $\parallel y\parallel \le \overline{y}$ with bounded control input (13) in the form of linear static state feedback

$$u=Kx.$$

The following result holds.

Theorem 2

([30]). Let $\widehat{w}$, $\widehat{P}$, $\widehat{Y}$ be the solution of the optimization problem

$$maxw$$

subject to the constraints

$$\left(\begin{array}{cc}AP+P{A}^T+\alpha P+B_{1}Y+Y^T{B}_1^T& w{B}_2\\ w{B}_2^T& -\alpha I\end{array}\right)⪯0,$$

$$\left(\begin{array}{cc}P& Y^T\\ Y& {\overline{u}}^{2}I\end{array}\right)⪰0,$$

$$\parallel CP{C}^T\parallel \le {\overline{y}}^2,P\succ 0,$$

where the maximization is performed in the matrix variables $P=P^T\in {\mathbb{R}}^{n\times n}$, $Y\in {\mathbb{R}}^{p\times n}$, the scalar variable w, and the parameter α.

Then, the output y of system (12) embraced with the linear static state feedback

$$u=\widehat{Y}{\widehat{P}}^{-1}x$$

remains in the ball $\parallel y\parallel \le \overline{y}$ for bounded control input $\parallel u\left(t\right)\parallel \le \overline{u}$ and for all admissible unknown-but-bounded disturbances $w:$ $\parallel w\parallel \le \widehat{w}$.

Note that, for any fixed $\alpha >0$, the problem obtained supposes minimization of the linear objective function subject to the LMI constraints, i.e., it is SDP. As a consequence, this problem can be easily solved numerically. In particular, the authors used the YALMIP and SeDuMi packages in the Matlab environment. The same remark applies to the statements in Section 7 and Section 8.

Let us turn back to the plasma model. For plant model (4) with the scalar regulated output

$$Z=\underset{C}{\underbrace{\left(\begin{array}{ccc}0& 0& 1\end{array}\right)}}x$$

(14)

and for

$$\overline{z}=0.02\mathrm{m},\overline{u}=1\mathrm{V},$$

Theorem 3 leads to

$$\widehat{w}=1.5461\times {10}^3\mathrm{A}.$$

with the gain matrix

$$\widehat{K}=\left(\begin{array}{ccc}-0.0007& -0.0012& -183.4854\end{array}\right).$$

Therefore, it is guaranteed that the output of closed-loop system (4) embraced with the control $u=\widehat{K}x$ remains within the stripe $\parallel Z\parallel \le 0.02$ m for all admissible $\parallel w\parallel \le 1.5461\times {10}^3$ A and wherein $\parallel u\parallel \le 1$ V.

Then, the various LMI approaches are applied to design and simulate closed-loop control systems by state and only Z-output to obtain the behavior of their values in the feedback: vertical plasma displacement Z, voltage $U\left(t\right)$ and current $I\left(t\right)$ in the HFC, and the control power $P\left(t\right)=U\left(t\right)I\left(t\right)$. These values are used for the comparison of the control systems designed to find out the systems accuracy, speed of response, and maximum of control power. The last parameter is important for the actuator design concerning its generating capacity and cooling conditions.

4. $\mathit{D}$-Stabilization for LMI Region

In this article, we discussed and estimated the acceptable disturbance in Section 3. The constraints were then used when the modeling of the closed-loop systems was carried out. The reference signal was chosen as 3 cm on the basis of the experimental experience on the tokamaks and the size of the vacuum vessel of the T-15MD tokamak.

4.1. $D_{\alpha ,\beta }$ Region

Consider the following strip region (Figure 5)

$$D_{\alpha ,\beta }=\{x+iy|-\beta <x<-\alpha \}.$$

All of the eigenvalues of the matrix of the closed-loop state feedback control system (5) for the plant model (4) are located in this strip region, and the system is stable. The following are the LMI conditions for the D-stability in this case [24]:

$$\begin{array}{cc}\hfill Y& \succ 0,\hfill \\ \hfill AY+Y{A}^T+B_{1}F+F^T{B}_1^T+2\alpha Y& \prec 0,\hfill \\ \hfill AY+Y{A}^T+B_{1}F+F^T{B}_1^T+2\beta Y& \succ 0.\hfill \end{array}$$

If and only if the matrix Y exists and satisfies these LMI criteria, the system is stable.

The auxiliary convex optimization

$$\begin{array}{c}\hfill mint,\\ \hfill s.t.H\left(x\right)\prec Q\left(x\right)+tI\end{array}$$

with LMI constraints (where x and the scalar t are the decision variables) provides the matrices F and Y and by this the gain matrix-row

$$K=F{Y}^{-1}=\left(\begin{array}{ccc}-0.001& -0.004& -1244\end{array}\right).$$

During the numerical simulation, the reference step signal (Figure 6a)

$$Z_{ref}=r\left(1\left(t\right)-1(t-T_r)\right),$$

(15)

where $1\left(t\right)$ is the Heaviside function,

$$r=0.03\mathrm{m},T_r=0.1\mathrm{s},$$

and the disturbance (Figure 6b)

$$w=I_0\left(1\left(t\right)-1(t-T_w)\right),$$

(16)

with

$$I_0=-1500\mathrm{A},T_w=0.1\mathrm{s},$$

have been applied to the plant model. The stable poles of the closed loop system are located in the strip region $D_{250,350}$ and equal $\{-294\pm 595i,-278\}$ (Figure 5). The simulation results are shown in Figure 6.

4.2. $D_{\alpha ,r,\vartheta }$ Region

The $D_{\alpha ,r,\vartheta }$-region to deal with is the intersection of a sector and the left half-plane (Figure 7):

$$D_{\alpha ,r,\vartheta }=\left\{x+iy:x<-\alpha <0,|x+iy|<r,\left|y\right|<xtan\vartheta \right\}.$$

The closed loop system’s poles should be confined within this region by the state feedback controller. The following are the LMI conditions that enable the control problem to be solved:

$$\begin{array}{cc}\hfill S+S^T+2\alpha P& \prec 0,\hfill \\ \hfill \left(\begin{array}{cc}-rP& S\\ S^T& -rP\end{array}\right)& \prec 0,\hfill \\ \hfill \left(\begin{array}{cc}S+S^T& S-S^T\\ -S+S^T& S+S^T\end{array}\right)\circ \left(\begin{array}{cc}sin\vartheta & cos\vartheta \\ cos\vartheta & sin\vartheta \end{array}\right)& \prec 0,\hfill \end{array}$$

(17)

where $S=AP+B_{1}W$; the symbol ∘ stands for the Hadamard product.

The gain matrix is given as

$$K=W{P}^{-1}=\left(\begin{array}{ccc}-0.001& -0.002& -346\end{array}\right).$$

During the numerical simulation the reference step signal (15) with

$$r=0.03\mathrm{m},T_r=0.1\mathrm{s},$$

and disturbance (16) with

$$I_0=-1500\mathrm{A},T_w=0.1\mathrm{s},$$

have been applied to the plant model. The stable poles of the closed loop system are located in the $D_{250,350,30}$ region (Figure 7) and equal $\{-280\pm 138i,-287\}$. The simulation results are shown in Figure 8.

5. ${\mathit{H}}_{\mathbf{2}}$ Stabilization

In this section the $H_2$ state feedback control problem is considered. The closed-loop system must be designed with the controller in such a way that the influence of any disturbance on the system output is prohibited to a desired level.

The effect of the disturbance w to the system output Z is determined by

$$Z\left(s\right)=\underset{G\left(s\right)}{\underbrace{C\left(sI-{(A+B_{1}K)}^{-1}\right)D}}w\left(s\right).$$

(18)

The state feedback control should ensure the condition ${\parallel G\left(s\right)\parallel }_2<\gamma$.

The LMIs

$$\begin{array}{cc}\hfill S+S^T+D{D}^T& \prec 0,\hfill \\ \hfill \left(\begin{array}{cc}-J& CP\\ P{C}^T& -P\end{array}\right)& \prec 0,\hfill \\ \hfill \mathrm{tr}J& <{\gamma }^2,\hfill \end{array}$$

(19)

define the matrices W and P such that the gain matrix is given as

$$K=W{P}^{-1}=\left(\begin{array}{ccc}-62& -81& -8369065\end{array}\right).$$

The solution of the optimization problem $\underset{P,J,W}{min}\rho$ (where $\rho ={\gamma }^2$) provides the minimal attenuation level $\gamma$.

During the numerical simulation the reference step signal (15) with

$$r=0.03\mathrm{m},T_r=0.1\mathrm{s},$$

and the disturbance (16) with

$$w=-1500\mathrm{A},T_w=0.1\mathrm{s},$$

have been applied to the plant model. Simulation results are shown in Figure 9. The corresponding minimal attenuation level is $\gamma =11.86$.

The $H_2$-stabilization problem was posed to minimize the $H_2$-norm of the transfer function $G\left(s\right)$ between the disturbance w and the output Z. The LMI (19) solution gives a chance to minimize that norm by means of a proper controller K placed in the feedback of the control system by the following way:

$$u=Kx.$$

6. ${\mathit{H}}_{\mathbf{2}}$ Design with Desired Pole Region and Triple Pole Controller

The $H_2$ optimization with $D_{\alpha ,r,\vartheta }$ pole location problem is used in this section. The main point is to satisfy the conditions (17) and (19) simultaneously and to seek a gain matrix $K=W{P}^{-1}$ as a mixed LMI controller.

During the numerical simulation, the disturbance (16) with

$$I_0=-1000\mathrm{A},T_w=0.1\mathrm{s},$$

has been applied to the plant model.

The comparison of the designed control system and a modal system with a three-time multiple pole [4] based on the control power criterion is shown in Figure 10. First, the $H_2$ design with the desired pole regions problem is resolved, and the $K=\left(\begin{array}{ccc}-0.001& -0.002& -344\end{array}\right)$ optimal controller is found. After that, the three-time multiple pole modal system was tuned by changing the pole such that both systems had the same peak power, and the systems were then compared. With equal control power peaks, the advantage in Z displacement is roughly 30%, demonstrating that the system with the LMI controller performs better at rejecting disturbances during plasma discharge.

7. Robust State Feedback Control

7.1. Robust $H_2$ Control

The subsection focuses on the mixed $H_2$ robust control system design with desired pole region, specifically in $D_{\alpha ,r,\vartheta }=\{x+iy:x<-\alpha <0,|x+iy|<r,|y|<xtan\vartheta \}$, providing the stability of the plasma vertical position in the tokamak at a possible deviation of plasma parameters from nominal values by 20% and in case of minor disruption disturbances during the plasma discharge. Thus, the resulting controller should minimize the effect of disturbances on the output of the closed-loop control system, and also provide robust stability of the system.

The effect of the disturbance w to the system output Z can be described as shown in (18).

The controller K must ensure that the condition

$${\parallel G\left(s\right)\parallel }_2<\gamma$$

is met. Taking into account the condition of the plasma parameters uncertainty, the matrices A and D of the system (4) can be represented as follows:

$$\begin{array}{cc}\hfill A& =A_0+{\delta }_1{A}_1+{\delta }_2{A}_2,\hfill \\ \hfill D& =D_0+{\delta }_1{D}_1.\hfill \end{array}$$

The LMI conditions for the problem of mixed robust $H_2$ design with the desired LMI pole region are obtained as

$$\begin{array}{cc}\hfill S+S^T+2\alpha P& \prec 0,\hfill \\ \hfill \left(\begin{array}{cc}-rP& S\\ S^T& -rP\end{array}\right)& \prec 0,\hfill \\ \hfill \left(\begin{array}{cc}S+S^T& S-S^T\\ -S+S^T& S+S^T\end{array}\right)\circ \left(\begin{array}{cc}sin\vartheta & cos\vartheta \\ cos\vartheta & sin\vartheta \end{array}\right)& \prec 0,\hfill \\ \hfill S+S^T+D{D}^T& \prec 0,\hfill \\ \hfill \left(\begin{array}{cc}-J& CP\\ P{C}^T& -P\end{array}\right)& \prec 0,\hfill \\ \hfill \mathrm{tr}J& <{\gamma }^2.\hfill \end{array}$$

The minimal attenuation level $\gamma$ is found, and the optimization problem: $\underset{P,J,W}{min}\rho$, $\rho ={\gamma }^2$ is solved. Having the solution W and P of the system of LMIs at the extreme points of the polyhedron, we obtain the gain matrix

$$K=W{P}^{-1}=\left(\begin{array}{ccc}-0.001& -0.0051& -1557.3\end{array}\right).$$

A state feedback control law $u=Kx$ was designed for the tokamak model (4), such that the closed-loop system is stable. Moreover, the solution of the LMI systems designed gives a chance to find the coefficients of the vector-string K, which satisfies all statements for the plant control eventually: stability, robustness, speed of response, accuracy and so on. This methodology is used here in all solutions with the help of the LMIs, that is, their essential advantage over the other well-known methodologies for control systems design for instance modal control design in the state-space of a plant model [32].

During the numerical simulation the reference step signal (15) with

$$r=0.03\mathrm{m},T_r=0.1\mathrm{s},$$

and the disturbance (16) with

$$I_0=-1500\mathrm{A},T_w=0.1\mathrm{s},$$

have been used for the plant model. The simulation results are shown in Figure 11.

7.2. Robust $H_{\infty }$ Control

In this subsection, we present (in a slightly modified form) and discuss the results in the field of $H_{\infty }$ synthesis, which go back to the publication [30] by one of the authors of the present paper; see also the monograph [29].

Consider the following system given by

$$\begin{array}{cc}\hfill \dot{x}& =\left(A+\Delta A\left(t\right)\right)x+\left(B+\Delta B\left(t\right)\right)w,x\left(0\right)=0,\hfill \\ \hfill z& =Cx,\hfill \end{array}$$

(20)

where $A\in {\mathbb{R}}^{n\times n}$, $B\in {\mathbb{R}}^{n\times m}$, $C\in {\mathbb{R}}^{l\times n}$ are fixed known matrices, $x\left(t\right)\in {\mathbb{R}}^n$ is the state vector, $z\left(t\right)\in {\mathbb{R}}^l$ is the output, and $w\left(t\right)\in {\mathbb{R}}^m$ is the exogenous disturbance satisfying the constraint

$$\parallel w\left(t\right)\parallel \le 1\mathrm{for}\mathrm{all}t\ge 0.$$

(21)

The model uncertainty has the form

$$\begin{array}{cc}\hfill \Delta A\left(t\right)& =F_A{\Delta }_A\left(t\right)H_A,\hfill \\ \hfill \Delta B\left(t\right)& =F_B{\Delta }_B\left(t\right)H_B,\hfill \end{array}$$

where $F_A$, $F_B$, $H_A$, and $H_B$ are known "frame" matrices, and the matrix uncertainties ${\Delta }_A\left(t\right)$ and ${\Delta }_B\left(t\right)$ satisfy the constraints

$$\parallel {\Delta }_A\left(t\right)\parallel \le 1,\parallel {\Delta }_B\left(t\right)\parallel \le 1\mathrm{for}\mathrm{all}t\ge 0.$$

(22)

It is assumed that the matrix A is Hurwitz, the pair $(A,B)$ is controllable, and C is a full-rank matrix.

To prove the following theorem, the generalization of the Petersen’s lemma [33] proposed in [34] will be used.

Theorem 3.

Ellipsoid $\mathcal{E}$ is state invariant for system (20)–(22), if its matrix P satisfies the LMIs

$$\left(\begin{array}{cccc}\mathrm{\Xi }& B& P{H}_A^T& 0\\ B^T& -\alpha I& 0& H_B^T\\ H_{A}P& 0& -{\epsilon }_{1}I& 0\\ 0& H_B& 0& -{\epsilon }_{2}I\end{array}\right)⪯0,P\succ 0,$$

(23)

where

$$\mathrm{\Xi }=AP+P{A}^T+\alpha P+{\epsilon }_1{F}_A{F}_A^T+{\epsilon }_2{F}_B{F}_B^T,$$

for some positive $\alpha ,{\epsilon }_1,{\epsilon }_2$.

Proof of Theorem 3.

Let us introduce the quadratic form

$$V\left(x\right)=x^T{P}^{-1}x,$$

considered on the solutions of system (20). Note, that the trajectories $x\left(t\right)$ of system (20) remain in the ellipsoid $\{x\in {\mathbb{R}}^n:V\left(x\right)\le 1\}$ if $\dot{V}\left(x\right)\le 0$ for any x satisfying $V\left(x\right)\ge 1$. Using the S-theorem with two constraints, it can be shown that this condition is equivalent to the existence of positive $\alpha =\alpha \left({\Delta }_A\left(t\right),{\Delta }_B\left(t\right)\right)$ such that

$$\left(\begin{array}{cc}\mathrm{\Lambda }& B+F_B{\Delta }_B\left(t\right)H_B\\ (B+F_B{\Delta }_B\left(t\right)H_B{)}^T& -\alpha (\Delta )I\end{array}\right)⪯0,$$

(24)

where $\mathrm{\Lambda }=P{\left(A+F_A{\Delta }_A\left(t\right)H_A\right)}^T+\left(A+F_A{\Delta }_A\left(t\right)H_A\right)P+\alpha (\Delta )P$. Let there exist $\alpha >0$ such that inequality (24) holds for any appropriate values of the matrix uncertainties. Then,

$$\begin{array}{c}\left(\begin{array}{cc}AP+P{A}^T+\alpha P& B\\ B^T& -\alpha I\end{array}\right)+\left(\begin{array}{c}{F}_A\\ 0\end{array}\right){\Delta }_A\left(t\right)\left(\begin{array}{cc}{H}_{A}P& 0\end{array}\right)+\left(\begin{array}{c}P{H}_A^T\\ 0\end{array}\right){\Delta }_A^T\left(t\right)\left(\begin{array}{cc}{F}_A^T& 0\end{array}\right)\hfill \\ \hfill +\left(\begin{array}{c}{F}_B\\ 0\end{array}\right){\Delta }_B\left(t\right)\left(\begin{array}{cc}0& H_B\end{array}\right)+\left(\begin{array}{c}0\\ H_B^T\end{array}\right){\Delta }_B^T\left(t\right)\left(\begin{array}{cc}{F}_B^T& 0\end{array}\right)⪯0,\end{array}$$

which, by the modification of Petersen’s lemma [34], holds if there exist ${\epsilon }_1,{\epsilon }_2>0$ such that

$$\begin{array}{c}\left(\begin{array}{cc}AP+P{A}^T+\alpha P& B\\ B^T& -\alpha I\end{array}\right)+{\epsilon }_1\left(\begin{array}{cc}{F}_A{F}_A^T& 0\\ 0& 0\end{array}\right)+\frac{1}{{\epsilon }_1}\left(\begin{array}{c}P{H}_A^T\\ 0\end{array}\right)\left(\begin{array}{cc}{H}_{A}P& 0\end{array}\right)\hfill \\ \hfill +{\epsilon }_2\left(\begin{array}{cc}{F}_B{F}_B^T& 0\\ 0& 0\end{array}\right)+\frac{1}{{\epsilon }_2}\left(\begin{array}{c}0\\ H_B^T\end{array}\right)\left(\begin{array}{c}0\\ H_B^T\end{array}\right)⪯0.\end{array}$$

Applying the Schur lemma, we conclude that this matrix inequality is equivalent to (23). The proof is complete. □

Now, incorporate the control term into description and consider the system

$$\begin{array}{cc}\hfill \dot{x}& =\left(A+\Delta A\left(t\right)\right)x+\left(B_1+\Delta B_1\left(t\right)\right)u+\left(B_2+\Delta B_2\left(t\right)\right)w,x\left(0\right)=0,\hfill \\ \hfill z& =Cx+Du,\hfill \end{array}$$

(25)

where $u\in {\mathbb{R}}^p$ is the control input, $z\left(t\right)\in {\mathbb{R}}^l$ is the regulated output, and the model uncertainty has the form

$$\begin{array}{cc}\hfill \Delta A\left(t\right)& =F_A{\Delta }_A\left(t\right)H_A,\hfill \\ \hfill \Delta B_1\left(t\right)& =F_{B_1}{\Delta }_{B_1}\left(t\right)H_{B_1},\hfill \\ \hfill \Delta B_2\left(t\right)& =F_{B_2}{\Delta }_{B_2}\left(t\right)H_{B_2},\hfill \end{array}$$

(26)

with fixed known matrices $F_A$, $F_{B_1}$, $F_{B_2}$, $H_A$, $H_{B_1}$, $H_{B_2}$, and the matrix uncertainties ${\Delta }_A\left(t\right)$, ${\Delta }_{B_1}\left(t\right)$ and ${\Delta }_{B_2}\left(t\right)$ satisfy constraint (22). The other quantities involved have the same meanings as above.

The pair $(A,B_1)$ is controllable and $D^{T}C=0$.

The goal is to find a robustly stabilizing linear static state feedback

$$u=Kx$$

(27)

against all admissible matrix uncertainties, which minimizes the trace of the bounding ellipsoid ${\mathcal{E}}_z$.

The following result holds.

Theorem 4.

Let $\widehat{P}\succ 0$ and $\widehat{Y}$ be the solutions of the minimization problem

$$min\mathrm{tr}\left[CP{C}^T+DR{D}^T\right]$$

(28)

under constraints

$$\left(\begin{array}{ccccc}\mathrm{\Omega }& B_2& P{H}_A^T& Y^T{H}_{B_1}& 0\\ B_2^T& -\alpha I& 0& 0& H_{B_2}^T\\ H_{A}P& 0& -{\epsilon }_{1}I& 0& 0\\ H_{B_1}Y& 0& 0& -{\epsilon }_{2}I& 0\\ 0& H_{B_2}& 0& 0& -{\epsilon }_{3}I\end{array}\right)⪯0,$$

(29)

$$\mathrm{\Omega }=AP+P{A}^T+B_{1}Y+Y^T{B}_1^T+\alpha P+{\epsilon }_1{F}_A{F}_A^T+{\epsilon }_2{F}_{B_1}{F}_{B_1}^T+{\epsilon }_3{F}_{B_2}{F}_{B_2}^T,$$

(30)

$$\left(\begin{array}{cc}R& Y\\ Y^T& P\end{array}\right)⪰0,$$

(31)

with respect to the matrix variables $P=P^T\in {\mathbb{R}}^{n\times n}$, $Y\in {\mathbb{R}}^{p\times n}$, $R=R^T\in {\mathbb{R}}^{p\times p}$, the scalar variables ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3\in \mathbb{R}$, and the scalar parameter α.

Then, controller (27) with the gain matrix $\widehat{K}=\widehat{Y}{\widehat{P}}^{-1}$ robustly stabilizes system (25), (26), and the matrix $\widehat{P}$ defines the invariant ellipsoid for the closed-loop system.

Proof of Theorem 4.

With control (27), closed-loop system (25) takes the form

$$\begin{array}{cc}\hfill \dot{x}& =\left(A+B_{1}K+F_A{\Delta }_A\left(t\right)H_A+F_{B_1}{\Delta }_{B_1}\left(t\right)H_{B_1}K\right)x+\left(B_2+F_{B_2}{\Delta }_{B_2}\left(t\right)H_{B_2}\right)w\left(t\right),\hfill \\ \hfill z& =(C+DK)x.\hfill \end{array}$$

As was shown above, it is easy to obtain the minimization problem

$$min\mathrm{tr}\left[(C+DK)P{(C+DK)}^T\right]$$

(32)

under constraint

$$\left(\begin{array}{cc}\mathrm{\Theta }\left(t\right)& B_2+F_{B_2}{\Delta }_{B_2}\left(t\right)H_{B_2}\\ {\left(B_2+F_{B_2}{\Delta }_{B_2}\left(t\right)H_{B_2}\right)}^T& -\alpha I\end{array}\right)⪯0,$$

(33)

where $\mathrm{\Theta }\left(t\right)=P{\left(A+B_{1}K+\mathrm{\Xi }\left(t\right)\right)}^T+\left(A+B_{1}K+\mathrm{\Xi }\left(t\right)\right)P+\alpha P$ and $\mathrm{\Xi }\left(t\right)=F_A{\Delta }_A\left(t\right)H_A+F_{B_1}{\Delta }_{B_1}\left(t\right)H_{B_1}K.$

Matrix inequality (33) can be rewritten in the form

$$\begin{array}{c}\left(\begin{array}{cc}P{(A+B_{1}K)}^T+(A+B_{1}K)P+\alpha P& B_2\\ B_2^T& -\alpha I\end{array}\right)+\left(\begin{array}{c}{F}_A\\ 0\end{array}\right){\Delta }_A\left(t\right)\left(\begin{array}{cc}{H}_{A}P& 0\end{array}\right)\hfill \\ +\left(\begin{array}{c}{\left(H_{A}P\right)}^T\\ 0\end{array}\right){\Delta }_A^T\left(t\right)\left(\begin{array}{cc}{F}_A^T& 0\end{array}\right)+\left(\begin{array}{c}{F}_{B_1}\\ 0\end{array}\right){\Delta }_{B_1}\left(t\right)\left(\begin{array}{cc}{H}_{B_1}KP& 0\end{array}\right)+\left(\begin{array}{c}{\left(H_{B_1}KP\right)}^T\\ 0\end{array}\right){\Delta }_{B_1}^T\left(t\right)\left(\begin{array}{cc}{F}_{B_1}^T& 0\end{array}\right)\\ \hfill +\left(\begin{array}{c}{F}_{B_2}\\ 0\end{array}\right){\Delta }_{B_2}\left(t\right)\left(\begin{array}{cc}{H}_{B_2}& 0\end{array}\right)+\left(\begin{array}{c}0\\ H_{B_2}^T\end{array}\right){\Delta }_{B_2}^T\left(t\right)\left(\begin{array}{cc}{F}_{B_2}^T& 0\end{array}\right)⪯0.\end{array}$$

By the modification of Petersen’s lemma [34], it holds if there exist ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3>0$ such that

$$\begin{array}{c}\left(\begin{array}{cc}P{(A+B_{1}K)}^T+(A+B_{1}K)P+\alpha P& B_2\\ B_2^T& -\alpha I\end{array}\right)+{\epsilon }_1\left(\begin{array}{cc}{F}_A{F}_A^T& 0\\ 0& 0\end{array}\right)+{\epsilon }_2\left(\begin{array}{cc}{F}_{B_1}{F}_{B_1}^T& 0\\ 0& 0\end{array}\right)\hfill \\ +{\epsilon }_3\left(\begin{array}{cc}{F}_{B_2}{F}_{B_2}^T& 0\\ 0& 0\end{array}\right)+\frac{1}{{\epsilon }_1}\left(\begin{array}{c}{\left(H_{A}P\right)}^T\\ 0\end{array}\right)\left(\begin{array}{cc}{H}_{A}P& 0\end{array}\right)+\frac{1}{{\epsilon }_2}\left(\begin{array}{c}{\left(H_{B_1}KP\right)}^T\\ 0\end{array}\right)\left(\begin{array}{cc}{H}_{B_1}KP& 0\end{array}\right)\\ \hfill +\frac{1}{{\epsilon }_3}\left(\begin{array}{c}0\\ H_{B_2}^T\end{array}\right)\left(\begin{array}{cc}0& H_{B_2}\end{array}\right)⪯0,\end{array}$$

or by Schur lemma

$$\left(\begin{array}{ccccc}\mathrm{\Omega }& B_2& P{H}_A^T& {\left(H_{B_1}KP\right)}^T& 0\\ B_2^T& -\alpha I& 0& 0& H_{B_2}^T\\ H_{A}P& 0& -\epsilon I_1& 0& 0\\ H_{B_1}KP& 0& 0& -{\epsilon }_{2}I& 0\\ 0& H_{B_2}& 0& 0& -{\epsilon }_{3}I\end{array}\right)⪯0,$$

(34)

where

$$\mathrm{\Omega }=(A+B_{1}K)P+P{(A+B_{1}K)}^T+\alpha P+{\epsilon }_1{F}_A{F}_A^T+{\epsilon }_2{F}_{B_1}{F}_{B_1}^T+{\epsilon }_3{F}_{B_2}{F}_{B_2}^T.$$

(35)

Introducing the auxiliary matrix variable $Y=KP$, we give relations (34) and (35) the linear form (29) and (30).

The minimized function in (32) takes the form

$$f(P,Y)=\mathrm{tr}[CP{C}^T+DY{P}^{-1}{Y}^T{D}^T].$$

Introducing the auxiliary matrix $R=R^T$ we obtain by Schur lemma that for $P\succ 0$ the LMI

$$\left(\begin{array}{cc}R& Y\\ Y^T& P\end{array}\right)⪰0$$

is equivalent to

$$R⪰Y{P}^{-1}{Y}^T.$$

Therefore, the minimization of the objective function $f(P,Y)$ is reduced to the minimization of $\mathrm{tr}[CP{C}^T+DR{D}^T]$ under constraint (31). The proof is complete. □

The next investigated problem is the robust $H_{\infty }$-problem for the plant (4) subjected to the unknown-but-bounded exogenous disturbances w with scalar regulated output (14).

Let the parameters $K_p$ and $T_p$ in the plant matrices A, $B_1$, and D in (4) be able to deviate from its nominal by a magnitude of 20%.

At first, the uncertainties in the parameters $K_p$ and $T_p$ can be transformed to the structured matrix uncertainty:

$$A\left(\delta \right)=A_0+F_1{\delta }_1{H}_1+F_2{\delta }_2{H}_2,$$

(36)

where $A_0$ is the nominal value of the matrix A, and

$$F_1=\left(\begin{array}{c}0\\ 0\\ 0.5\frac{K_p}{T_p}\end{array}\right),H_1=\left(\begin{array}{ccc}0& 1& 0\end{array}\right),F_2=\left(\begin{array}{c}0\\ 0\\ 0.25\frac{1}{T_p}\end{array}\right),H_2=\left(\begin{array}{ccc}0& 0& 1\end{array}\right).$$

Indeed, we have

$$T_p\left(\delta \right)=(1+\delta )T_p,$$

where $T_p$ is the nominal value of $T_p\left(\delta \right)$ and $\left|\delta \right|\le 0.2$. Similarly,

$$K_p\left(\delta \right)=(1+\delta )K_p,$$

where $K_p$ is the nominal value of $K_p\left(\delta \right)$ and $\left|\delta \right|\le 0.2$. Therefore, the admissible values of $\frac{K_p\left(\delta \right)}{T_p\left(\delta \right)}$ are within the range

$$(1+{\delta }_1)\frac{K_p}{T_p},$$

where $|{\delta }_1|\le 0.5$, and the admissible values of $\frac{1}{T_p\left(\delta \right)}$ are within the range

$$(1+{\delta }_2)\frac{1}{T_p},$$

where $|{\delta }_2|\le 0.25$.

The goal is to design the linear static state feedback

$$u=Kx$$

which minimizes the $H_{\infty }$-norm of the transfer function of the considered system. Based on the above results, we can establish the following LMI-based statement.

Proposition 1.

Let $\widehat{P}$, $\widehat{Y}$ and $\widehat{\gamma }$ be the solution of the optimization problem

$$min\gamma$$

subject to the constraints

$$\left(\begin{array}{ccccc}\mathrm{\Omega }& \overline{w}D& P{C}^T& P{H}_1^T& P{H}_2^T\\ \overline{w}{D}^T& -\gamma & 0& 0& 0\\ CP& 0& -1& 0& 0\\ H_{1}P& 0& 0& -{\epsilon }_1& 0\\ H_{2}P& 0& 0& 0& -{\epsilon }_2\end{array}\right)⪯0,$$

$$\left(\begin{array}{cc}P& Y^T\\ Y& {\overline{u}}^2\end{array}\right)⪰0,$$

$$CP{C}^T\le \overline{z},P\succ 0,$$

where

$$\mathrm{\Omega }=AP+P{A}^T+B_{1}Y+Y^T{B}_1^T+{\epsilon }_1{F}_1{F}_1^T+{\epsilon }_2{F}_2{F}_2^T,$$

and the optimization is performed in the matrix variables $P=P^T\in {\mathbb{R}}^{3\times 3}$, $Y\in {\mathbb{R}}^{1\times 3}$, and the scalar variables γ, ${\epsilon }_1$, ${\epsilon }_2>0$.

Then, the output Z of the closed-loop uncertain system embraced with $H_{\infty }$-optimal controller

$$\widehat{K}=\widehat{Y}{\widehat{P}}^{-1},$$

remains in the ball

$$\parallel \mathit{Z}\parallel \le \overline{Z}$$

for bounded control input $\parallel u\left(t\right)\parallel \le \overline{u}$ and for all unknown-but-bounded disturbances $\parallel w\parallel \le \overline{w}$.

Namely, for the plant with parameters

$$\overline{z}=0.02\mathrm{m},\overline{u}=1\mathrm{V},\overline{w}=1.5461\times {10}^3\mathrm{A},$$

the gain matrix

$$\widehat{K}=\left(\begin{array}{ccc}-0.0005& -0.0009& -130.0341\end{array}\right)$$

is obtained.

During the numerical simulation the reference step signal (15) with

$$r=0.03\mathrm{m},T_r=0.1\mathrm{s},$$

and disturbance (16) with

$$w=-1500\mathrm{A},T_w=0.1\mathrm{s},$$

have been applied to the plant model. The simulation results are shown in Figure 12.

Figure 13 depicts the transient processes at the extreme points of the polyhedron.

8. Stabilizing Output Feedback: Linear Dynamical Controller

In this section, the full-order linear dynamical controller is constructed for the plant (4) subjected to the unknown-but-bounded exogenous disturbances such that $\left|w\right|\le \overline{w}$ (Figure 14). The state of the considered system is unavailable for observation and the information about the plant is provided by its scalar observable output (14).

The corresponding result has the following form, see [35] for the details.

Proposition 2.

Let $\widehat{P}$, $\widehat{Q}$, $\widehat{\alpha }$ be the solution of the optimization problem

$$minCP{C}^T$$

subject to the constraints

$$\left(\begin{array}{cc}AP+P{A}^T+\alpha P-{\mu }_1{B}_1{B}_1^T& \overline{w}D\\ \overline{w}{D}^T& -\alpha \end{array}\right)⪯0,$$

$$\left(\begin{array}{cc}QA+A^{T}Q+\alpha Q-{\mu }_2{C}^{T}C& \overline{w}QD\\ \overline{w}{D}^{T}Q& -\alpha \end{array}\right)⪯0,$$

$$\left(\begin{array}{cc}P& I\\ I& Q\end{array}\right)⪰0,$$

with respect to the matrix variables $P=P^T$, $Q=Q^T$, the scalar variables ${\mu }_1,{\mu }_2$, and the scalar parameter α.

Then, the parameters

$$\Delta =\left(\begin{array}{cc}{A}_r& B_r\\ C_r& D_r\end{array}\right)$$

of the full-order linear output dynamic controller

$$\begin{array}{cc}\hfill {\dot{x}}_r& =A_r{x}_r+B_{r}y,x_r\left(0\right)=0,\hfill \\ \hfill u& =C_r{x}_r+D_{r}y,\hfill \end{array}$$

defined as the solution of the linear matrix inequality

$$\tilde{A}\tilde{P}+\tilde{P}{\tilde{A}}^T+\widehat{\alpha }\tilde{P}+\frac{1}{\widehat{\alpha }}{\overline{w}}^2\tilde{D}{\tilde{D}}^T+M\Delta N+{(M\Delta N)}^T⪯0,$$

where

$$\tilde{P}=\left(\begin{array}{cc}\widehat{P}& V\\ V& V\end{array}\right),V=\widehat{P}-{\widehat{Q}}^{-1},$$

$$\tilde{A}=\left(\begin{array}{cc}A& 0\\ 0& 0\end{array}\right),\tilde{D}=\left(\begin{array}{c}D\\ 0\end{array}\right),$$

$$\tilde{M}=\left(\begin{array}{cc}0& B_1\\ I& 0\end{array}\right),\tilde{N}=\left(\begin{array}{cc}0& I\\ C& 0\end{array}\right).$$

For the considered plant with $\overline{w}=1.5461\times {10}^3$ A, the following matrices of the linear output dynamical controller are obtained:

$$A_r=\left(\begin{array}{ccc}-0.1795& -0.3922& -36802.1944\\ 0.0238& -0.0021& -0.0001\\ -0.0006& -0.0031& -35275.2534\end{array}\right),$$

$$B_r=\left(\begin{array}{c}-3.7739\\ 0.0000\\ 3.3031\end{array}\right)\times {10}^8,$$

$$C_r=\left(\begin{array}{ccc}-0.0025& -0.0065& -607.2362\end{array}\right),$$

$$D_r=-6.2269\times {10}^2.$$

Figure 15 depicts the respective transient processes.

9. Robust Stability Radius

This section is devoted to obtaining the boundaries of the stability regions of the control systems for the plasma vertical position in the tokamak with synthesized controllers in the space of control system parameters.

9.1. The Three-Loop Control System

The Hurwitz criterion [23] is used to calculate the stability boundaries of a three-loop control system. The characteristic equation of the system (Figure 3) is as follows:

$$\begin{array}{c}\hfill a_0{s}^3+a_1{s}^2+a_{2}s+a_3=0,\end{array}$$

(37)

where

$$\begin{array}{cc}\hfill a_0& =T_a{T}_c{T}_p,\hfill \\ \hfill a_1& =-T_a{T}_c+T_a{T}_p+T_c{T}_p+K_a{T}_c{T}_p{k}_1,\hfill \\ \hfill a_2& =T_p-T_a-T_c-K_a{T}_c{k}_1+K_a{T}_p{k}_1+K_a{T}_p{K}_c{k}_2,\hfill \\ \hfill a_3& =-K_a{k}_1-K_a{K}_c{k}_2+K_a{K}_p{K}_c{k}_3-1.\hfill \end{array}$$

According to the Hurwitz criterion, the necessary and sufficient condition for the stability of the system represents the fulfillment of the following conditions:

$$\begin{array}{cc}\hfill a_0& >0,\hfill \\ \hfill a_1& >0,\hfill \\ \hfill a_2{a}_1-a_0{a}_3& >0,\hfill \\ \hfill a_3& >0.\hfill \end{array}$$

The boundaries of the stability area of the system are defined as the boundaries of the area defined by the system of inequalities:

$$\begin{array}{cc}\hfill T_p& >\frac{T_a{T}_c}{T_a+T_c+K_a{T}_c{K}_1},T_p>\frac{T_a+T_c+K_a{T}_c{K}_1}{1+K_a{K}_1+K_a{K}_c{K}_2},\hfill \\ \\ \hfill K_p& >\frac{1+K_a{K}_1+K_a{K}_c{K}_2}{K_a{K}_c{K}_3},K_p<\frac{1}{b_4}\left(T_p{b}_1+b_2+\frac{b_3}{T_p}\right),\hfill \end{array}$$

(38)

where

$$\begin{array}{cc}\hfill b_1& =(T_a+T_c+K_a{T}_c{K}_1)(1+K_a{K}_1+K_a{K}_c{K}_2),\hfill \\ \hfill b_2& =-{(T_a+T_c+K_a{T}_c{K}_1)}^2,\hfill \\ \hfill b_3& =T_a{T}_c(-K_a{K}_c{K}_2-1-K_a{K}_1),\hfill \\ \hfill b_4& =T_a{T}_c{K}_a{K}_c{K}_3.\hfill \end{array}$$

9.2. The One-Loop Control System

The transfer function of the output controller with matrices $A_r$, $B_r$, $C_r$ has the following form:

$$\frac{-d{s}^3-e{s}^2-fs-m}{s^3+a{s}^2+bs+c},$$

(39)

with the following numerical values of the controller parameters: $d=622.7$, $e=4.202\times {10}^{11}$, $f=1.363\times {10}^{14}$, $m=2.729\times {10}^{15}$, $a=3.528\times {10}^8$, $b=6.387\times {10}^{11}$, and $c=3.402\times {10}^{14}$.

For the considered one-loop system (Figure 14), the characteristic equation is as follows:

$$\begin{array}{c}{T}_a{T}_c{T}_p{s}^6+[T_a{T}_c{T}_{p}a+T_a{T}_p+T_p{T}_c-T_a{T}_c]s^5+[T_a{T}_c{T}_{p}b+a(T_a{T}_p+T_p{T}_c-T_a{T}_c)+\hfill \\ T_p-T_a-T_c]s^4+[T_a{T}_c{T}_{p}c+b(T_a{T}_p+T_p{T}_c-T_a{T}_c)+a(T_p-T_a-T_c)-1+\\ K_a{K}_c{K}_{p}d]s^3+[e{K}_a{K}_c{K}_p+c(T_a{T}_p+T_p{T}_c-T_a{T}_c)+b(T_p-T_a-T_c)-a]s^2+\\ \hfill [f{K}_a{K}_c{K}_p+c(T_p-T_a-T_c)-b]s-c+m{K}_a{K}_c{K}_p=0.\end{array}$$

(40)

Each point on the plane of the equation of plasma parameters $K_p$ and $T_p$ corresponds to certain coefficients of the characteristic equation. In this case, there are n roots of (40), where n is the order of the system, which also have fixed values. The entire parameter plane can be divided into $n+1$ regions, and each of them consists of points characterized by the fact that k roots are in the left half-plane, and $(n-k)$ are in the right half, where k takes values from 0 to n. When $k=n$, the region is the stability region. This method is called D-partitioning [36].

Any point located on the boundary of such regions corresponds to the location of the roots, in which there is a root $p=j\omega$ on the imaginary axis. After substituting the expression $p=j\omega$ ($\omega$ is a real number of a circular frequency) into the characteristic equation, separating the real and imaginary parts, equating them to 0, the following system of equations is obtained:

$$\begin{array}{cc}\hfill T_p{Q}_1\left(\omega \right)+K_p{P}_1\left(\omega \right)+R_1\left(\omega \right)& =0,\hfill \\ \hfill T_p{Q}_2\left(\omega \right)+K_p{P}_2\left(\omega \right)+R_2\left(\omega \right)& =0,\hfill \end{array}$$

(41)

where

$$\begin{array}{cc}\hfill Q_1=& -{\omega }^6{T}_a{T}_c+{\omega }^4(T_a{T}_{c}b+T_{a}a+T_{c}a+1)-{\omega }^2(T_{c}c+T_{a}c+b),\hfill \\ \hfill P_1=& -{\omega }^{2}e{K}_a{K}_c+m{K}_a{K}_c,\hfill \\ \hfill R_1=& -{\omega }^4(T_a{T}_{c}a+T_a+T_c)+{\omega }^2(T_a{T}_{c}c+(T_a+T_c)b+a)-c,\hfill \\ \hfill Q_2=& {\omega }^5(T_a{T}_{c}a+T_a+T_c)-{\omega }^3(T_a{T}_{c}c+b(T_a+T_c)+a)+\omega c,\hfill \\ \hfill P_2=& -{\omega }^3{K}_a{K}_{c}d+\omega f{K}_a{K}_c,\hfill \\ \hfill R_2=& -{\omega }^5{T}_a{T}_c+{\omega }^3(T_a{T}_{c}b+a{T}_a+a{T}_c+1)-\omega ((T_a+T_c)c+b).\hfill \end{array}$$

The Kramer method is used to solve the resulting system (41) with respect to the plasma parameters (see Appendix A):

$$\begin{array}{cc}\hfill T_p& =\frac{\left|\begin{array}{cc}-R_1\left(\omega \right)& P_1\left(\omega \right)\\ -R_2\left(\omega \right)& P_2\left(\omega \right)\end{array}\right|}{\left|\begin{array}{cc}{Q}_1\left(\omega \right)& P_1\left(\omega \right)\\ Q_2\left(\omega \right)& P_2\left(\omega \right)\end{array}\right|}=\frac{{\Delta }_1}{\Delta },\hfill \\ \hfill K_p& =\frac{\left|\begin{array}{cc}{Q}_1\left(\omega \right)& -R_1\left(\omega \right)\\ Q_2\left(\omega \right)& -R_2\left(\omega \right)\end{array}\right|}{\left|\begin{array}{cc}{Q}_1\left(\omega \right)& P_1\left(\omega \right)\\ Q_2\left(\omega \right)& P_2\left(\omega \right)\end{array}\right|}=\frac{{\Delta }_2}{\Delta }.\hfill \end{array}$$

(42)

Formulas (42) define the equation of the stability boundary on the plane of the parameters of the plasma model ($T_p,K_p$) in parametric form.

If the determinant $\Delta$ is 0 for some value of $\omega$, then the Equations (41) are no longer linearly independent and degenerate into single equation. With such an exceptional value of $\omega$, there is not a point on the plane of parameters ($T_p,K_p$), but straight lines, which divide the plane of parameters into the regions. In our case $\omega =0$, straight lines on the plane of the plasma parameters are obtained:

$$\begin{array}{cc}\hfill T_p& =\frac{-cf+m\left((T_a+T_c)c+b\right)}{mc},\hfill \\ \hfill K_p& =\frac{c}{m{K}_a{K}_c}.\hfill \end{array}$$

(43)

9.3. The Robust Stability Radii for Three-Loop and One-Loop Control Systems

Definition 2.

The radius of robust stability is the radius of a circle whose center is located at a nominal point on the plane of parameters $K_p$ and $T_p$ of the control system in the stability region and which touches the boundary of the stability region. As a consequence, this radius is the minimum distance from the nominal point to the boundary of the system stability region [37].

Formulas (38), (42)–(43) specify the stability boundaries of closed-loop control systems for the vertical position of the plasma in the tokamak. After normalizing the coordinate axes to the coordinates of the nominal point, it is possible to determine the radii of robust stability for systems with different controllers. The approach for determining the radii of robust stability is shown in Figure 16, where the numbering corresponds to an increase in the radius of robust stability of control systems (

Table 1. Radius of robust stability and peak power for the closed-loop control systems with different controllers.

	Controller Type	Section	Radius	Peak Power (W)
				At the Disturbance	After Drop of the Disturbance	At the Reference Step Signal	After Drop of the Reference Step Signal
1.	Triple pole controller	6	0.2238	$1.33\times {10}^6$	$1.35\times {10}^6$	$1.62\times {10}^6$	$4.28\times {10}^6$
2.	$H_2$ controller	5	0.4168	$1.18\times {10}^6$	$6.35\times {10}^5$	$4.95\times {10}^5$	$3.05\times {10}^6$
3.	Mixed robust $H_2$ with $D_{\alpha ,r,\vartheta }$ pole region controller	7.1	0.6515	$1.65\times {10}^6$	$1.88\times {10}^6$	$5.67\times {10}^7$	$7.74\times {10}^7$
4.	$D_{\alpha ,\beta }$ pole region controller	4.1	0.6609	$1.63\times {10}^6$	$1.74\times {10}^6$	$4.08\times {10}^7$	$5.95\times {10}^7$
5.	Mixed $H_2$ with $D_{\alpha ,r,\vartheta }$ pole region controller	6	0.6630	$1.16\times {10}^6$	$1.11\times {10}^6$	$4.92\times {10}^6$	$10.14\times {10}^6$
6.	Output controller	8	0.6848	$9.22\times {10}^5$	$1.24\times {10}^6$	$9.48\times {10}^6$	$1.85\times {10}^7$
7.	$D_{\alpha ,r,\vartheta }$ pole region controller	4.2	0.7234	$1.15\times {10}^6$	$1.11\times {10}^6$	$4.77\times {10}^6$	$9.88\times {10}^6$
8.	$H_{\infty }$ robust controller	7.2	0.8349	$1.95\times {10}^6$	$3.1\times {10}^6$	$1.57\times {10}^8$	$2.02\times {10}^8$

).

10. Comparative Analysis of Closed-Loop Vertical Position Plasma Control Systems

Control systems for dynamic plants must provide guaranteed reliability and performance in the presence of disturbances and noise, variable operating conditions, nonlinearities, actuator limitations, etc. [38].

These requirements for control systems can be taken into account during the design for a sufficiently comprehensive controlled plant and the competent selection or development of a method for synthesizing the control system. However, there is a gap between theoretical control methods and their applications [39].

Definition 3.

Analysis is the determination of certain characteristics of an already known system when signals pass through it, i.e., when the control system processes information.

Definition 4.

Synthesis is the development of new control systems with predetermined properties.

Mathematical models of plants are widely used both in the development of control systems for dynamic plants, and in their analysis of operation. In some cases, there is insufficient knowledge to define clear requirements to the plant model under control and their exact mathematical formulation. Therefore, when solving the problem of controller synthesis for a given model, it is necessary to take into account the incomplete correspondence of the plant model and the real physical plant. This issue becomes even more important when one does not have precise knowledge about the laws governing the behavior of the system.

Therefore, it is important to investigate the influence of approximations and/or assumptions used during the design of system models. In doing so, the practical implementation of the control system synthesis problem can be subdivided into steps that typically include [40]:

• Mathematical modeling and analysis of physical phenomena, and selection of sensors and actuators;
• Designing a control system that provides a given behavior, satisfies the imposed constraints, and minimizes the resources consumed;
• Verification of control efficiency using simulation studies on plant models (including the real-time simulation on digital platforms specifically “digital twins”) [5];
• Practical implementation in a real experiment or production process.

It should be noted that when formulating the problem of controller synthesis for a feedback system, the tuning task includes a certain combination of requirements, among which the key ones are ensuring stability of the closed-loop system dynamics (limited disturbances lead to limited errors) and ensuring the desired behavior of the system (acceptable attenuation of perturbations, fast response to changes in the operating point, etc.). Various modeling and analysis techniques are used to reveal the essential dynamics of the system and to explore the possible behavior in the presence of uncertainties, noise, and component failure.

Meanwhile, the central task is the synthesis and implementation of control algorithms. Engineers face the critical challenge of reducing costs, including power losses, while maintaining or improving quality and ensuring safe operation. As systems become more complex, it becomes equally important to ensure the reliability of implemented systems. Consequently, hardware and software reliability are issues that must be addressed. Different control algorithms may be considered during the design process, in this regard, an assessment should be made of how a particular controller achieves certain design goals, and then an acceptable control system with appropriate settings must be selected and implemented in the experimental, and technical practices [41]. A compromise must be made between the different characteristics of the synthesized systems, in particular between the stability margin, the performance (speed of response and accuracy), and the power of the actuators. The problem becomes even more acute when the engineer is faced with the choice of sensors and actuators, the establishment of communication channels, saving computing resources of the device, etc. This suggests that when synthesizing plasma control systems in tokamaks, one should consider not only the robustness and control quality criteria, but also practical criteria that must be satisfied by control systems when applied in practice [42,43,44,45,46].

The control systems synthesized using the modern LMI theory for the vertical plasma position in the D-shaped tokamak were compared with each other by practical criteria, which allowed us to reveal the practical suitability or unsuitability of the controllers for implementation in an actual plant, not the model of the plant. Comparison of control systems synthesized for plasma in the tokamak was carried out according to three criteria (Table 1):

• The stability margin defined by the robust stability radius (Figure 16 and Figure 17);
• The criterion of the required power of the actuator in the presence of an external disturbance (Figure 18a);
• The criterion of the required power of the actuator in the presence of the reference signal (Figure 18b).

These criteria should be taken into account when synthesizing control systems for operating tokamaks, since the systems with a large stability margin can remain functional when the controllers synthesized on plasma models are connected in a real physical experiment on the tokamak (the model parameters may differ from the experimental plasma parameters). It should also be noted that plasma parameters can change during the discharge, and the robust controller can ensure that the stability and performance of the closed-loop control system is maintained when the plasma parameters change. The criterion of the required power of the actuator is significant in the synthesis of control systems, since this criterion is decisive in the selection and development of the actuator.

Analysis of the data obtained for 8 different controllers used in the feedback of the closed-loop control system for the unstable vertical position of the plasma showed that:

• The largest stability radius of 0.8349 was obtained for the system with the $H_{\infty }$ robust controller, the smallest 0.2238 is for the system with triple pole controller (Figure 17);
• The highest power of the actuator in the presence of the reference step signal is required for the system with the $H_{\infty }$ robust controller $2.02\times {10}^8$ W, the lowest for the system with the $H_2$ controller $3.05\times {10}^6$ W (Figure 18a);
• The highest power of the actuator in the presence of external disturbances is required for the system with $H_{\infty }$ robust controller and is equal to $3.1\times {10}^6$ W, the lowest one for the system with the pole arrangement in the area $D_{\alpha ,r,\vartheta }$ is $1.15\times {10}^6$ W (Figure 18b).

It could be objected that the systems with the highest peak power values have larger imaginary parts of the poles (Figure 19,

Table 2. Poles of the closed-loop control systems with different controllers.

	Controller Type	Poles
1.	Triple pole controller	{$-300;-300;-300$}
2.	$H_2$ controller	{$-37476737;-238;-48$}
3.	Mixed robust $H_2$ with $D_{\alpha ,r,\vartheta }$ pole region controller	{$-300\pm 670i;-288$}
4.	$D_{\alpha ,\beta }$ pole region controller	{$-294\pm 595i;-278$}
5.	Mixed $H_2$ with $D_{\alpha ,r,\vartheta }$ pole region controller	{$-273\pm 151i;-289$}
6.	Output controller	{$-352752540;-952;-647;$ $-303;-163;-21$}
7.	$D_{\alpha ,r,\vartheta }$ pole region controller	{$-280\pm 138i;-287$}
8.	$H_{\infty }$ robust controller	{$-1388\pm 1200i;-580$}

). In particular, for the $D_{\alpha ,\beta }$ controller with placing of the poles of the closed loop system in the strip region, the imaginary part of the poles is four times larger than for the controller with the poles of the system placed in the area limited by the sector, for example, for the controller with placing the poles in the $D_{\alpha ,r,\vartheta }$ area. Placing the poles in the $D_{\alpha ,r,\vartheta }$ - area allows one to limit the range of imaginary part of the system poles. The required power of the actuator in the presence of external disturbance for $D_{\alpha ,r,\vartheta }$ controller is 1.5 times less than for $D_{\alpha ,\beta }$ controller (Figure 18a), and the required power of the actuator in the presence of the reference signal for $D_{\alpha ,r,\vartheta }$ controller is an order of magnitude less than for $D_{\alpha ,\beta }$ controller (Figure 18b). For this reason, when synthesizing control systems, it is necessary to introduce an LMI condition that allows limiting the imaginary part of the poles, which, as a result, will reduce the peak power required for the actuator.

The controller with poles located in the area $D_{\alpha ,r,\vartheta }$ is the best controller according to three criteria simultaneously. It is the controller number 7 in Table 1 and histograms in Figure 15 and Figure 16. Thus, the analysis performed allowed to identify the research directions in solving the actual problem of controlling the unstable vertical plasma position in vertically elongated tokamaks on the example of the plasma model in the T-15MD tokamak, and also allowed to synthesize an acceptable for practice robust controller, which satisfies the given criteria.

In the article, we discussed and estimated the acceptable disturbance. The constraints were then used when the modeling of the closed-loop systems was carried out. Additionally, in Section 7.1, that focuses on the mixed $H_2$ robust control system design with desired pole region providing the stability of the plasma vertical position in the tokamak, the possible deviation of plasma parameters from nominal values by 20% was used. This value was obtained from the experimental practice. No other constraints were considered in this article because of the comparison of the control systems without any restrictions to find out their possibilities.

11. Modeling of the Control System on the Real-Time Digital Control Testbed

A unique real-time digital control testbed was created for simulation of digital plasma control systems in tokamaks (Figure 20) [5,47], it was done within the framework of the scientific and technical cooperation agreement between the Faculty of Physics of the Lomonosov Moscow State University and V. A. Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences (ICS RAS) in the field of real-time modeling and control of plasma physical processes in tokamaks. The core of the testbed is two Speedgoat Performance real-time target machines from the Swiss company Speedgoat. The first real-time target machine is a “Plant”, it works in conjunction with the second real-time target machine “Controller” in the feedback.

Figure 21 shows the structural scheme of the real-time digital control system with poles located in the area $D_{\alpha ,r,\vartheta }$ for the vertical plasma position in tokamak T-15MD, taking into account its simulation on two real-time target machines. The real-time target machine “Controller” contains a digital controller. The vertical position of the plasma from the digital model of the plant is fed to the input of the digital controller through the DAC, and the control signal is sent from the output of the digital controller through the ADC to the model of the actuator (multiphase thyristor rectifier). The second real-time target machine contains a digital model of the plant, which includes a discrete model of the multiphase thyristor rectifier, a discrete model of the HFC, and a discrete plasma model in the T-15MD tokamak, to which the external perturbation w is applied during the simulation process. Signals between real-time target machines are analog, the use of ADCs and DACs at the input and output of the digital models allow simulating the operation of the digital controller in the feedback with the real plant, for example, during a real experiment on the T-15MD tokamak.

The simulation of the vertical plasma control systems is executed on the real-time testbed, so it is required to discretize the model with the sample time $T_s$ = 100 μs. The model discretization is realized by the Zero Order Hold method [48].

The unstable discrete linear plasma model as a difference equation obtained from the original differential Equation (1) is

$$Z(T_{s}k+T_s)=A^{d}Z\left(T_{s}k\right)+B^d(I\left(T_{s}k\right)+w\left(T_{s}k\right)),$$

where $A^d=exp\left(A{T}_s\right),A=T_p^{-1},B^d=A^{-1}(A^d-1)B,B=K_p{T}_p^{-1}$, the control coil discrete model is

$${\displaystyle I\left(z\right)=\frac{R^{-1}\left(1-exp\left(-\frac{T_{s}R}{L}\right)\right)}{z-exp\left(-\frac{T_{s}R}{L}\right)}U\left(z\right),}$$

and the multiphase thyristor rectifier discrete linear model is

$${\displaystyle U\left(z\right)=\frac{K_a\left(1-exp\left(-\frac{T_s}{T_a}\right)\right)}{z-exp\left(-\frac{T_s}{T_a}\right)}V\left(z\right)}$$

obtained by application of the Z-transform [48] to (2) and (3).

Figure 22 shows real-time signals, similar to the simulation signals in MATLAB/Simulink, in the form of a screenshot from the oscilloscopes. The signals have been decoded to be analog. These figures show the performance of the control system and provide information about the processes in the system in real time. This is important for understanding these processes when introducing the system into the practice of a physical experiment.

The feedback controller in that case is only a matrix-row derived using the LMI approach. The digital testbed used in this work can be applied to any control system developed in MATLAB/Simulink regardless of its complexity. The methodology to transfer these systems into C-code and then into the real-time testbed is the same.

12. Discussion

The results obtained showed that the basic plasma vertical position control system depends very much on the feedback controller. So, it is critical to design and optimize the controller, which gives the required stability margins and performance as well as a chance to minimize the control power. The main originality and novelty of the work consists in the optimization of robust control systems of the unstable plant with uncertain disturbance, namely, a minor disruption type disturbance on the set of LMI synthesis approaches. In this study, the plasma model in the T-15MD tokamak is used in the simplest form as the first order unstable unit where the control coil influences the plasma vertical position by its horizontal field. As this takes place, the plant uncertainties are located in the additive disturbance. The reason for such model simplicity is that when plasma is closed by the feedback and moves around the equilibrium position, the plasma position displacements are relatively small in comparison with minor and major plasma radii. It gives a chance to linearize plasma model equations with the usage of only the first linear term and to keep only the small linear deviations of the plasma horizon or vertical positions around equilibrium. The original performance criteria such as control power and robust stability radius are applied to the designed control systems. These two criteria allowed us to compare plasma unstable vertical position control systems with a multiphase thyristor rectifier as an actuator designed by LMI. The controller with poles located in the area $D_{\alpha ,r,\vartheta }$ is the best controller according to two criteria simultaneously. The control power requirements for the multiphase thyristor rectifier as an actuator in closed loop control systems are done for different types of controllers. The proven LMI theorems of the paper are directed to design the robust output and state feedback controllers for systems subjected to uncertainties and exogenous disturbances and study the systems with the aim of elimination of the gap between theory and practice.

In any vertically elongated tokamak, where the actuator connected to the HFC may be approximated by a linear dynamic model, for plasma unstable vertical position stabilization the LMI technique may be used and the experience gained in the given article. The ASDEX Upgrade machine (Germany) may serve as an example of such a tokamak, where the vertical plasma position is stabilized by a multiphase thyristor rectifier that is approximated by an inertial stable unit of the first order [19]. The same approach would be appropriate for ITER (France) [42]. Another illustration is the Russian spherical Globus-M2 tokamak [49], which uses a current inverter in self-oscillation mode [18] as the actuator for plasma vertical stabilization. This particular regime is organized to allow one to approximate the current inverter by a static gain.

The LMI technique may be used to solve other plasma control problems in tokamaks, such as designing and analyzing the plasma current and shape control systems. This relates to the USDEX Upgrade (Germany), ITER (France), Globus-M2 (Russian Federation), DIII-D (USA), JET (Great Britain), TCV (Switzerland), EAST (China), KSTAR (South Korea), and other contemporary tokamaks with vertically elongated plasmas.

Other approaches can be used to design a vertical plasma control system, not just the LMI methodology for robust control. The most attractive of them after LIMs are artificial neural networks when they automatically approximate the plants under control with unknown structure and parameters and adjust feedback controller parameters on line [50]. Fussy logics is another way to be used for plasma control in tokamaks as a plant with uncertainties [51]. Sliding modes in control systems have some robust features, but such sliding may appear only along some switching surfaces and around reference points these sliding modes transfer to auto-self oscillations [4]. Such sliding modes may be useful only inside algorithms of identification and control [52].

13. Conclusions

The calculation of the normalized stability boundaries and robust stability radii of the closed-loop control systems with a set of LMI-controllers for the unstable plasma vertical position of the T-15MD tokamak have shown that the radii are between 0.2238 and 0.8349. The simulation of the control systems designed have demonstrated that the peak control power when minor disruption occurs is in the range of 1.15 to 3.1 MW. The controller with pole placement in the $D_{\alpha ,r,\vartheta }$ region is the optimal controller according to two criteria at the same time: 1.15 MW is the peak power in the presence of external disturbance and 0.7234 is the robust stability radius. The controller with poles located in the area $D_{\alpha ,r,\vartheta }$ is the best controller according to the relevant criteria.

An important result is that the control power under the influence of the step reference signal on the control systems varies in the range of 3.03 to 202 MW. This means that this criterion is very much dependent on the vertical plasma position feedback controller. Therefore, it is important to include the control power in the controller design methodology. When synthesizing control systems, an LMI condition must be added that allows limiting the imaginary part of the poles, which will subsequently reduce the peak power required for the actuator.

One of the main results of the work is the simulation of the developed digital control system with the poles located in the area $D_{\alpha ,r,\vartheta }$ for the unstable vertical plasma position of the T-15MD tokamak on the real-time digital control testbed. The system provides stabilization of the vertical plasma position on the unstable model with the perturbation. The tuned digital controller in the feedback of the control system can be applied directly on the T-15MD tokamak.