NONLINEAR ROBUST CONTROL OF LARGE-SCALE SYSTEM WITH INPUT SATURATION

The article studies control algorithms of multiply connected system for dynamic plants with control saturation and nonlinear cross-connections. The authors of the article offer a decentralized control law based on the hyperstability criterion. They also use this law to constuct the MIMO servo system with input saturation. To illustrate the capability of the proposed decentralized robust control system the authors use an inverted pendulums connected by a spring.


Introduction
The controlling of the nonlinear large-scale systems has a very high theoretical and practical importance. Examples of unstable plants can be found in the various fields of knowledge. In mechanics, for example, it is multiply-connected inverted pendulum (which is multiply connected system itself), or the interconnected system of several inverted pendulums. In technology -operation of a gas turbine engine at low speeds, or control of statically unstable aircraft. The tasks of controlling such plants can also be complicated by the presence of input signals saturations, which can be caused by some design features of technical systems, or by the operating conditions of technical plants.
This article develops the results of ( [Eremin, 2016], [Eremin, 2017]) and expands the area of their application in solving the problem of controlling multiply connected dynamic plants with nonlinear cross-links when the dimensions of vectors of the input and output variables of the plant coincide.

Preliminaries
This section is devoted to the mathematical models of the interconnected control system with input saturation that we can present as input-output model or in the statespace forms. Also in this section the problem statement is given.
Let the input-output model of the plant have the following form: is vector of the external disturbances; F j (y j (t)) are some functions to describe the nonlinear cross-connections; σ(u i (t)) is the nonlinear function of saturation where σ 0i are known values.
The plant (1), (2) is supposed to satisfy the following assumptions: 1. Disturbances f i (t) and unknown functions F j (y j (t)) satisfy the inequalities where f * i and F * j are unknown numbers. 2. Transfer functions of the linear links have the form like where a i (p) = p ni + a 1i p (ni−1) + ... + a nii and b i (p) = b mi 0i + b (mi−1) 1i + ... + b mii are normalized polynomials with unknown coefficients; b i (p) are Hurwitz polynomials, b 0i > 0; a i (p) are polynomials with arbitrary roots distribution; deg b i (p) = m i ≥ 0, deg a i (p) = n i ≥ 1 are unknown degrees; (max n i ), (max m i ) are known limit values; ρ = (deg a i (p) − deg b i (p)) = (n i − m i ) ≥ 1 are relative orders of the W i (p); ρ i = max ρ i = (max n i − max m i ) are known values; 3. For the direct measurement only vector y(t) is available, i. e. the variables y 1 (t), ..., y k (t) (the local subsystems outputs) are available.
Since unknown values ρ i = const belong to a known interval ρ i ≥ ρ i ≥ 1 (Assumption 2), it is expedient (see [Eremin, 2018]) to pass the measurable signals y i (t) through the outputs filter-correctors (OFC) where y Fi (t) and y i (t) are respectively output and input signals; W Fi (p) are transfer functions of the OFC; T i and T * i are time constants. As a result of serial connection of the plant (1) and OFC (6) mathematical model (1) will be transformed like Considering that products W Fi (p)W i (p) have the relative degrees ∆ i = degã i − degb i ≥ 1, where degã i = n i + ρ i − 1, degb i = m i + ρ i − 1 and also taking into account the identities and choosing a small values for time constants T * i we can show (by analogy with [Eremin, 2018]) that with the respect to it is always permissible to replace model (7) with the following approximate model where the relative degrees ofŴ i (p) will satisfy the It should be noted that degrees of numerators (ρ i + m i ) and denominators (ρ i + m i − 1) of the transfer functionsŴ i (p) are priory unknown. Let us rewrite model (9) in the state-space form then can be represented as follows: and (ρ i + m i − 1) × 1 sizes with an appropriate coefficients.

Control Goals
Let the main control goal is to provide the desired dynamics of the outputs y i (t) that consists in the quality serving to given signals r i (t), i. e. in achieving following conditions at t → ∞: where ∆ 0i are required values. Wherein the desired dynamics for outputs of the main control loop y Fi and also outputs of the OFC are formed with the help of command filter-correctors (CFC) [Eremin, 2018]: wherer i (t) are auxiliary command signals.
Then for the plant with OFC (10) that operates in the conditions of functional and parametric uncertainties it is possible to formulate following additional control goal: it is necessary to synthesize an explicit form of the control law so that at measuring only signals y i (t), any initial conditions x i (0), any disturbances f i (t) (3) and nonlinear cross-connections F i (y j (t)) (4) at t → ∞ it will fulfill the following requirements where∆ 0i are maximum allowable errors in the tracking mode; y * i (t) are outputs of an implicit reference model (IRM) in the input-output form: It is well known that at χ * i 0 we can rewrite the expression (15) like Thus, if we provide the additional targets (14), then the main control targets (11) will be fulfilled by virtue of full equivalence of the transfer functions in equations (6) and (12).
Note that for model (10) instead of IRM (15) it is expedient to consider their equivalent analogues [Eremin, 2018] (ρ i +mi−1)i , that in the state-space can be represented as follows: T is a vector which elements are determined by values of polynomial coefficients calculated from

Main Results
In this section with the help of hyperstability criterion the synthesis of the control algorithms for the considered decentralized system is discussed.
Considering the deviation e i (t) = x T i (t) − x i (t) of the state vectors of the IRM (18) and plant with OFC (10), the model of this system can be described by the equations where v i (t) and µ i (t) are modified outputs and control signals respectively.
Since the frequency inequality (20) is fulfilled due to where s is complex variable; then it is necessary to determine the conditions that lead to inequality (21) fulfillment. Let us show the synthesis of the control law (13) as follows are switching functions; 0 < δ 0i < 1 are scaling factor; δ i (t) are outputs of the dynamic switches (23); it is possible to satisfy the inequality (21). For this purpose we define control signal as u i (t) = u 1i (t) + u 2i (t) + u 3i (t) and rewrite the left side of (21) taking into account (19) like If we write for the integral η 1i (0, t) taking into account constraints (2) and condition δ i (t) ≥ δ 0i following relation: , then we can equate the relation in square brackets to zero, and obtain the explicit form of u 1i : where h 1i = const > 0 are arbitrary numbers; and obtain for summand η 1i (0, t) following fair estimate η 01i = const, ∀t > 0.
Let us transform integral η 2i (0, t) in the following way: Then the component u 2i (t) will take the form: where h 21 = const > 0 are arbitrary numbers; and for the integral η 2i (0, t) we will have the estimate like: η 02i = const, ∀t > 0.
We can transform the integral η 3i (0, t) as follows: The component u 3i (t) is synthesized in the following form where h 3i = const > 0 are arbitrary numbers; then for η 03i (0, t) we can obtain following estimate η 03i = const, ∀t > 0.
If now, similarly to [Eremin, 2017], the integral on the right side of this inequality is rewritten as where t * is a moment in time starting from which in the considered system a condition |u i (t)| ≤ σ 0i is obvious, then it can be argued that the following inequalities will be true: due to the boundedness of integrable functions on a finite time interval; since at t > t * it will be fair σ i (u i (t)) = u i (t). Therefore, taking into account conditions (33) and (34), relation (32) will satisfy the estimate that confirms the fulfillment of inequality (21). Remark. In [Khalil, 2002] it was noted that one of the disadvantages of the strong feedback observers is emergence of peaks in transient processes which often lead to the system instability. In the considered case a similar situation may arise because of fast-acting OFC. Indeed the OFC are forcing links at the output of which at small values of time constants T * i and non-zero initial conditions as a rule significant peak emissions are formed.
Therefore, in order to weaken the influence of peaks on the formation of control signals, similar to [Khalil, 2002], we limit the output of the OFC using nonlinearities of the saturation type and rewrite the control law (22) -(24) as follows i = 1, 2, ..., k.

Simulation
In this section we apply the obtained decentralized nonlinear robust regulator to control two inverted pendulums connected by a hard spring (Fig. 1).

Figure 1. Inverted pendulums
Each pendulum is positioned by a control torque u i (t), i = 1, 2 applied by servomotor at its base. We assume that only angular displacements θ 1 (t) and θ 2 (t) are available to the direct measurements Nonlinear mathematical model which describes the motion of the such pendulums can be represented as follows [Karimi, and Menhaj, 2010]: y 1 (t) = x 11 (t), y 1 (0) = 0.5; y 2 (t) = x 21 (t), y 2 (0) = −0.5, where x 11 (t) = θ 1 (t) and x 21 (t) = θ 2 (t) are vertical angular displacements of the pendulums; m 1 = 2 kg and m 2 = 2.5 kg are pendulums end masses; j 1 = 0.5 kg·m 2 and j 2 = 0.625 kg·m 2 are moments of inertia; k = 100 N/m is the spring constant; l = 0.5 m is the natural length of the spring; r = 0.5 m is the pendulum height; α 1 = α 2 = 25 are the control input gains; g = 9.81 m/s 2 is the gravitation acceleration; b = 0.4 m is the distance between the pendulums binges, b < 1 indicates that the pendulums repel one another when both are in the upright position.
At first, let us attempt to regulate the angular position of each pendulum to zero. In this case we define command signals of the subsystems as r 1 (t) = r 2 (t) = r * = 0. In the course of simulation parameters of the robust regulator were chosen with following values: In Fig. 2 and Fig. 3 the pendulums positions and the signals r * are depicted.  At second, we set the desired trajectory of each pendulums as follows r 1 (t) = sin(6.28t), (42) r 2 (t) = 0.5 sin(9.42t), From the presented results, one can see that the proposed control algorithm (36) -(38), (41) ensures high quality of the system for various operating modes (stabilization of the pendulums position and the movement of the pendulums along given trajectories) without reconfiguring the parameters of the regulator (41). In both cases, due to the formation of control torques, a good enough performance of the control system is ensured (control errors in both local control loops are practically equals zero).

Conclusion
With the help of hyperstability criterion, the solution of the problem of decentralized control for class of multiply connected dynamic plant which contains nonlinear cross-links is considered. Using the high-speed dynamic correctors and implicit reference model the robust control law is synthesized. This robust regulator, as it is shown at the stage of simulation in relation to the control of two inverted pendulums, makes it possible to achieve a given control goal with a sufficiently high quality.
The obtained results may be useful to construct the control system for mechanical systems such as control systems for robotic manipulators. Also an approach considered in paper can be used for developing the control system for plants which contain state delay.