Approximate Synthesis of Н_∞ – Controllers in Nonlinear Dynamic Systems over a Semi-Infinite Time Period

Introduction. Methods of modern control theory play an important role in the development of complex aerospace systems, providing their efficient operation. To achieve high productivity, stability and efficiency of such systems, it is necessary to develop algorithms for synthesizing controllers capable of operating under conditions of uncertainty in the description of external influences. The modern foundation for their development include the state space method, frequency analysis, and the approach based on linear matrix inequalities [1]. To solve problems of finding optimal control, sufficient optimality conditions in the form of the Bellman equation and the relations following from it in special cases are usually applied. Linear matrix inequalities can be used to search for Н_∞ – controllers. They determine the existence of a regulator that satisfies certain performance criteria and provides the stability of the system to external influences. These criteria are usually associated with a norm, which is a measure of the sensitivity of the system to external disturbances. These criteria are usually related to Н_∞ – norm, which is a measure of the sensitivity of the system to external disturbances. The problem is to find a regulator that minimizes this norm, while providing the stability of the system and satisfying the quality criterion of control. The solution method is based on finding the extremum of a convex objective function, where the conditions are presented in the form of linear matrix inequalities [2]. By using this method, it is possible to reduce the solution of complex systems of linear and nonlinear algebraic matrix equations of a certain type to the solution of convex optimization problems. However, the solution of linear matrix inequalities can be difficult when considering complex technical problems.

An alternative method based on stochastic minimax is presented in the anisotropic theory of stochastic robust control described in [3]. The main idea in applying this method is that robustness in stochastic control is reached by explicitly including different noise distribution scenarios in a single performance indicator to be optimized. Statistical uncertainty is expressed through entropy, and the robust quality indicator is selected in such a way as to make it possible to quantify the system's ability to suppress the worst external impact. The application of such an approach to solving complex systems of interrelated equations requires the development and use of specialized algorithms.

It should be noted that methods of Н_∞ – optimization are used to solve numerous different applied problems, such as aircraft [4], helicopter [5], quadcopter [6] and multi-agent systems [7] control, robot stabilization [8], rocket engine design [9], where, when compared to other controllers, these methods show good results and lower error values under limited disturbances. It is also worth mentioning their use in filtering problems [10], state vector estimation [11] and neural network design [12]. Thus, the development and advancement of Н_∞ – optimization methods are highly topical issues for research. Previously, the authors considered the problems of synthesizing Н_∞ – controller [13] and Н_∞ – observer [14] for linear dynamic systems, for the solution of which sufficient optimality conditions based on the expansion principle were used. Their application made it possible to justify the synthesis procedures and, as a result, to form step-by-step algorithms for solving problems.

Despite significant achievements in this area, a number of problems related to the analysis and synthesis of controllers for nonlinear systems remain unsolved. In this regard, the paper considers the problem of synthesis of controllers for nonlinear dynamic systems, linear in control and disturbance, on a semi-infinite time interval. The research objective is to formulate and prove sufficient conditions for the existence of control. This will not only create a basis for new research and development, but also fill existing gaps in the field of knowledge. Specifically, the work provides for the use of sufficient conditions as a theoretical justification for the formulation of approximate control search algorithms for the class of dynamic systems under consideration. To test the efficiency of the proposed algorithm, two model examples will be solved.

Materials and Methods. Let there be a mathematical model of the control object:

(1)

and the model of the measuring system:

(2)

where x ∈ Rⁿ —state vector, u ∈ R^q —control vector, w ∈ R^p — external influences vector, y ∈ R^m — output vector, t ∈ T = [0, ∞) — current time, 0 — zero matrix-column of dimensions (n × 1). Assume that the continuously differentiable vector function f(x) of dimensions (n × 1), as well as the matrix functions B₁(x) of dimensions (n × p), B₂(x) of dimensions (n × q), C(x) of dimensions (m × n) are given. The model of the object is described by an equation that is nonlinear in state, but linear with respect to control and external influences.

It is implied that:

1. w(.) ∈ L₂[0,∞), u(.) ∈ L₂[0,∞);
2. m≤n, rg С(x) = m;
3. the origin of coordinates x≡ 0 is the equilibrium point, e.g., f(0) = 0;
4. B₁(x) ≠O, B₂(x) ≠O;

where O — zero matrix of corresponding dimensions.

An indicator describing the current behavior of the control object model (1) with measuring system (2) is defined:

(3)

where for all x ∈ Rⁿ Q(x) > 0 — symmetric positive definite square matrix of order q, and S(x) > 0 — symmetric non-negative definite square matrix of order m. Functional (3) is quadratic in control, but non-quadratic in state.

Note that we are considering models of an object and a measuring system, the matrices in which depend on the state vector.

It is required to ensure the correctness of the condition:

(4)

where ∀ x ∈ Rⁿ P(x) > 0 — symmetric square matrix of order p, γ > 0 — some number. As an additional condition, the necessity of fulfilling the property of asymptotic stability of the closed system “object-controller” is considered. Note that it is important to find the smallest value of parameter γ* that provides the preservation of the required properties of the closed system. This is possible only if the conditions of minimizing the numerator and maximizing the denominator of the expression are simultaneously fulfilled (4).

We rewrite condition (4) in the form:

(5)

This means that it is required to provide that inequality (5) is satisfied while minimizing control costs under conditions of maximum counteraction of external influences (disturbances).

Sufficient conditions for the existence of Н_∞ – controllers

Assume that function V(x) ∈ C¹(Rⁿ) is known. Let us define the function:

(6)

where

.

Theorem. If there exists function V(x) ∈ C¹(Rⁿ), satisfying conditions V(0) = 0 and

(7)

where

(8)

and function V(x) is determined by solving a partial differential equation:

(9)

then condition (4) is satisfied.

Proof. Suppose that the conditions of the theorem are satisfied. We find

,

applying the required conditions for an unconditional extremum, since no restrictions are imposed on the variables u, w:

Solving the matrix equations, we obtain:

where u*(x), w*(x) — control structures of the object model and external influence (disturbance).

Since

is true, then the sufficient conditions for the minimum in control are satisfied. Also true are the sufficient conditions that guarantee the achievement of the maximum external influence w,
since

.

In this case

From this we get

(10)

i.e., the conditions for the presence of a saddle point are satisfied.

Assume that function V(x) ∈ C¹(Rⁿ) satisfies conditions V(0) = 0 and R(x, u*(x), w*(x)) = 0.

Then the relation that is fulfilled along the trajectories of system (1) is valid, namely:

For u = u*(x), let us rewrite the left side of the inequality (10), i.e.,

,

in the form:

.

By integrating the left and right sides of the resulting inequality over the time interval from 0 to t₁, we obtain:

.

Since it is required to provide the fulfillment of the condition of asymptotic stability of the closed system, then x(t₁) → 0 when t₁ → +∞, therefore V(x(t₁)) → V(0) = 0. Since x(0) = 0 then V(x(0)) → V(0) = 0. From this we can conclude that when t₁ → +∞ the inequality is valid:

indicating that condition (4) is satisfied, which was to be proved.

As a remark, we emphasize that when conditions P(x) = E,

,

i.e., the energy of external influences is limited, an inequality of the form

is valid.

Approximate synthesis of Н_∞ – state-based controllers using the SDRE method

Due to the nonlinearity of equation (9) and difficulties in obtaining its solution, a method based on the algebraic Riccati equation with coefficients dependent on the state vector is used for further analysis [15].

As a result of applying the factorization operation, we obtain a nonlinear system transformed to a structure similar to a linear one, with matrices dependent on the state vector.

It is known [16] that if f(0) = 0 and f(x) ∈ C¹(Rⁿ), then there exists matrix function A(x), such that:

(11)

Notes

1. The factorization procedure for n= 1 in unique ∀x≠ 0, i.e., A(x) = f(x) / x = a(x).

2. For n> 1, the factorization procedure yields a non-unique result [16]. For example, for n= 2, there are at least two options: f(x) = A₁(x)x and f(x) = A₂(x)x, i.e., for a system of the form:

we obtain

3. If there are two parametrization options, i.e., f(x) = A₁(x)x= A₂(x)x, then there is an infinite family of options of the form [16]: A(x, α) = αA₁(x) + (1 – α)A₂(x) ∀α. The selection of parameter α allows for flexibility in designing the control system. The solution to the Riccati equation and the corresponding control become functions of this parameter.

As a result of factorization, the mathematical model of system (1) takes the form:

(12)

On the trajectories of system (12), functional (5) is specified.

It is assumed that system (12) is controllable and observable, i.e., ∀x ∈ Rⁿ the conditions are fulfilled point-to-point [16]:

,

.

Equation (9) takes the form:

(13)

Assume that

(14)

where K₂(x) > 0 unknown matrix function (for fixed x ∈ Rⁿ, matrix K₂(x) is a symmetric positive definite numerical matrix). Thus, an assumption is made not about the type of function V(x), but only about the structure of its partial derivative.

Then, the control structures of the object and disturbance take the form:

(15)

Equation (13) takes the form:

or

Applying the transpose operation, we obtain:

Summing up the last two expressions, we arrive at the equality:

or finally:

(16)

Note that due to the dependence of all matrices on the state vector, the equality of the zero matrix in square brackets does not follow from (16).

By analogy with the case of a linear stationary system, it is proposed to solve the algebraic Riccati equation, all matrices in which are functions of the system's state vector (State Dependent Riccati Equation, SDRE). In this case, a positive definite solution to the Riccati equation is sought, generating a control law that guarantees that the system will be asymptotically stable in the vicinity of the equilibrium position. To check this property, the stability criterion by the roots of the characteristic equation, checked point-to-point, or the Routh–Hurwitz criterion, is used.

In the problem under consideration, it is proposed to solve the equation:

, (17)

i.e., to look for matrix K₂(x) > 0, that satisfies the Riccati equation, whose coefficients depend on x. Equation (17) is solved repeatedly for fixed x ∈ Rⁿ. The coordinates of the state vector are determined in the process of integrating the differential equation (12) together with the controls of the object and the disturbances:

(18)

In this case, the solution to the Riccati equation must be such that the criterion

∀x ∈ Rⁿ, is satisfied, where σ^– matrix spectrum, C^– — open left half-plane of the complex plane. Note that the stability criterion of a closed system can be replaced by checking the point-to-point fulfillment of the Routh-Hurwitz criterion.

Algorithm for approximate synthesis of Н_∞ – controllers of state

Step 1. Set parameter γ > 0.

Step 2. Find the solution to the equation:

with controls

one of the numerical methods of integration with constant step h (explicit Euler method, Euler-Cauchy method, Adams-Bashforth, Milne, Hamming methods of various orders).

In this case, for each of the discrete moments of time t_i = ih, i = 0, 1, 2, …, solve the Riccati equation:

for x = x(t_i ). As a result, find matrix K₂(x) and use it to form control laws.

Step 3. Find the minimum γ*. To do this, it is required to consistently decrease γ until the stability property of solutions of the differential equation

,

describing the dynamics of the system with the obtained controls, remains valid.

Research Results. To test the efficiency of the proposed approximate algorithm for synthesizing Н_∞ – state-based controllers, two model examples were solved.

Model example No. 1. A one-dimensional case is considered, when equations (1), (2) and functional (5) have the form:

Solution. The control structures follow from (18):

and equation (17) has the form:

Let us write the resulting quadratic equation in canonical form:

The solution is as follows:

Since K₂ > 0, then

Let us take a closer look at a particular case:

where A(x) = (1 – x²), B₁(x) = 1, B₂(x) = 1, C(x) = 2, Q(x) = 1, S(x) = 1, P(x) = 1.

Then (omitting the dependence on х) we get:

Roots of the quadratic equation:

Note that when considering the case γ ≠ 1, it is not yet possible to exclude the extra roots, since A(x) and (1 – γ⁻²) can change the sign.

As a result, we obtain the control structures:

In this case, the equation of the closed system has the form:

To provide asymptotic stability, we take the minus sign, and in the expression for K₂ − plus. For this example, we get:

• if γ≠1, then
  
  

• if functions A(x), C(x) are not equal to zero simultaneously, then discriminant A²(x) + C² (x)( 1– γ⁻²).

Then,

For the example being solved

,

i.e., for each current x, there is its own value .

For γ = γ_min, x = 0, x(0) = 0 is satisfied (the condition of asymptotic stability is not satisfied, but x(t) ≡ 0 is valid).

For γ = γ_min we have

.

If γ = 1, then

To fulfill condition K₂ > 0, condition A(x) = 1 – x² < 0 that defines set |x| < 1 of possible functioning of the system must be fulfilled.

Modeling. For modeling under different initial conditions, finite time interval T = [0, 20] was selected, since all transient processes in a closed system are practically completed.

According to Figures 1–3, the value of the state vector asymptotically tends to zero for different initial conditions, which indicates the stability of the system and the correct selection of parameters, under which the system retains the property of stability subject to any given limited disturbances.

Model example No. 2. One of the options of the two-dimensional case is considered, when equations (1), (2) and functional (5) have the form:

,

i.e., C(x) = E₂, S(x) = E₂, Q(x) = 1, P(x) = 1.

Solution. The control structures of the object and the disturbance follow from (18):

and equation (17) has the form:

From here,

where

Solution to the first equation has the form:

Solution to the third equation:

Solution to the second equation:

In the obtained solutions, positive signs were selected taking into account condition K₂ > 0.

Modeling. To model the system, it is required to select the minimum possible value γ, so that it satisfies (4) and at the same time guarantees the asymptotic stability of the closed system. In example 1, the value of parameter γ was found analytically, but a problem arose with determining the value of the optimal parameter γ*, therefore value γ, used in the modeling, was selected experimentally. When γ* = 1.5, the system remained stable according to expression (19). For modeling, the time interval T = [0, 10] was selected, since transient processes in the closed system decayed fairly quickly. Modeling was performed for various initial conditions.

According to Figures 4–5, it can be concluded that the coordinates of the state vector asymptotically tend to zero. This result is observed for each of the initial conditions considered. This indicates that the system is stable, and its parameters are selected correctly, which allows maintaining the property of stability under any initial conditions and under the worst disturbances.

The initial conditions significantly affect the trajectories of the change in the coordinates of the state vector, but from Figures 4–5, it is clear that the proposed approach allows us not only to compensate for external disturbances, but also to stabilize the trajectory of motion.

Discussion of the Results. As a result of the study, sufficient Н_∞ – control conditions were formulated and substantiated, an approximate solution method was developed. The method proposed within the framework of the problem was tested on two model examples. The simulation results allow us to conclude that the use of the developed controller synthesis method guarantees the required quality of transient processes and provides asymptotic stability of closed systems.

Conclusion. The results and methods proposed in this paper can be applied to solve control problems of varying complexity — from designing simple autopilots to developing complex automatic navigation systems for manned and unmanned aerial vehicles. This emphasizes the prospects of using the proposed approach and makes it an attractive option for further research.