Controllability Analysis and Optimization of Hydrocannon Nozzle Shape Based on Direct Extreme Approach

Introduction. Despite the significant amount of research on the control theory, there is still no unified approach to the analysis of controllability for systems with distributed parameters. Existing studies are usually limited to considering controllability as the possibility of transferring a system from an initial state to a given final state [1]. However, this approach turns out to be insufficient for optimization problems of spatially distributed systems described by partial differential equations. Controllability by the final state does not guarantee controllability by the conditions specified in the objective functional, which makes the analysis of such systems complex and non-obvious. In this paper, the authors use the concept of controllability proposed by V.K. Tolstykh [2] and apply it to find the optimal shape of the hydrocannon nozzle.

Hydraulic cannons designed to generate pulsed jets of high-pressure liquid are widely used, e.g., in the mining industry to destroy rocks [3]. The efficiency of such devices largely depends on the shape of the nozzle [4], which makes the task of its optimization urgent. Despite significant interest in this problem, existing papers, such as the works of Zuikova Z.G. [5], Zubov V.I. [6], and Atanova G.A. [7], are primarily theoretical. These authors formulated the required conditions for optimality, but the numerical results were not supported by evidence of their optimality. Moreover, as it is shown in this research, the previously obtained “optimal” nozzle shapes are not such. Thus, despite multi-year research, the problem of designing an optimal nozzle shape remains unsolved.

The solution to this complex problem is possible only through using a direct extreme approach with original adaptive gradient methods described in the work of V.K. Tolstykh [8]. The objective of this work was to apply the new approach proposed in [2] to the analysis of the controllability of a system with distributed parameters for the problem of optimal design of the hydrocannon nozzle shape. Thus, the article is aimed at developing the controllability theory for systems with distributed parameters and demonstrating its practical applicability using the example of optimizing the shape of a hydrocannon nozzle.

Materials and Methods. The essence of the direct extreme approach is the direct maximization of some objective functional using gradient methods:

, (1)

Here, the control (in our case — function of the nozzle shape along the length x) u(x) ∈ U(S), S = (x_a, x_b) — domain of control definition, U — admissible set of controls, v(x, t) ∈ V() — state of the nonstationary process of ultra-jet formation on the closed space-time set {x, t} = . Operator includes a specific type of differential equations of water flow in a hydrocannon and acts on v. Objective function I(v, u) is defined on the set ω, and its value clearly depends on the parameters v and u.

The direct approach does not use any intermediate conditions (e.g., required optimality conditions), but solves the problem immediately:

(2)

where u_* — optimal control.

The original problem with the equations of the distributed system (v, u)v = 0 is characterized by a direct mapping:

At the same time, optimization problem (1) is inverse. Such problems are usually ill-posed in the classical sense [9]. The solution to direct and inverse problems differs significantly. The latter require regularization of the solution to narrow the set of possible solutions U to the compact of correctness ⊆ U, which leads to conditional correctness according to Tikhonov. In the direct extreme approach, regularization is performed by gradient methods.

According to the definition of controllability [2], the distributed system in problem (1) will be controllable by u(x) ∈ U(S) with respect to objective functional J, when the inverse problem of mapping the elements of space V(ω) of model states into element u_* is correct according to Tikhonov, provided that max J:

Next, when solving problem (2), we will perform a controllability analysis and obtain controllability conditions that can allow us to correctly formulate and solve the problem of optimizing the shape of the hydrocannon nozzle using gradient methods [10].

Paper [8] describes in detail the formulation of the problem under consideration for subsonic, axisymmetric flows of compressible fluid in smoothly changing channels. Let us recall it in the form required for further research. The isentropic motion of water in the nozzle is described by a quasi-one-dimensional, quasi-linear hyperbolic system of equations [11]:

(3)

The state of the system is characterized by vector function v = {ρ, w} ∈ V(), where ρ — density of water, w — speed of water. Operator its matrix and vector φ = ρwuΘ(x – x_a), Θ — Dirac theta function, x_a — beginning of the nozzle at the end of the barrel of hydrocannon, — square of the speed of sound in water, B and n — constants in the equation of state of water in theta form.

The control is described by the formula:

(4)

Here, σ — nozzle cross-sectional area, , x ∈ [x_a, x_b], σ_a = σ(x_a). In the barrel of the hydrocannon, when x ≤ x_a, there is no control u(x), and free term φ = 0.

Boundary and initial conditions of problem (3):

Here, m_p — piston mass, ρ₀ — density of water at atmospheric pressure, w₀ — speed of water and piston before it starts flowing into the nozzle. The boundaries of process Г for domain Ω, are shown in Figure 1. The type of Ω is important for the controllability analysis.

We specify Ω. The origin of coordinates is aligned with the nozzle entrance x_a, and t₀ — time of the beginning of water flow into the nozzle. On the one hand, the water is limited by the piston moving in the barrel of the hydrocannon along the trajectory Г_p, and on the other hand — by the free surface of the inflow Г_b0 from t₀ to t₁ and the outflow Г_b1 from t₁ to t₂. The line of the initial state of system (3) is Г₀, coordinates x_p0 and x_p2 — initial and final positions of the piston. The specified Г-lines for Ω form the closure of . The domain of definition of the control i.e., for x > x_a, S is the projection of a part of domain Ω onto x axis.

The optimization problem (optimal design of the nozzle shape) is formulated as follows: it is necessary to find the control u(x) that delivers the maximum to the functional:

(5)

The objective functional is set at ω = Г_b1, i.e., at the nozzle exit of the hydrocannon, and determines the average force of the jet on a possible obstacle [12].

The gradient algorithm for maximizing functional (5) has the form:

(6)

where k — iteration number; b^k — step multiplier that controls the rise to max J in the direction of gradient ∇J^k.

The gradient is the functional Frechet derivative , which can be found from the first variation of the objective functional . Here, the angle brackets denote the scalar product, in this case, in the conjugate control space U^*(S). The superscript * denotes conjugacy.

It should be noted that sometimes the gradient of a functional on a Hilbert space is confused with the Frechet derivative of this functional [13]. Derivative may be insensitive to the control u(x) on the entire set S or on parts of S of nonzero measure. Therefore, in the general case, in (6) will not indicate a reliable direction of correction u^k for the directed search for the optimal solution The gradient from the Frechet derivative can be obtained only when the controllability conditions are realized.

Approaches to solving problems of hydrocannon nozzle optimization with the aim of maximizing the average jet impulse force [5], the functional depending on the flow parameters [6], and maximizing the outflow velocity [7] were mentioned earlier. In these papers, after varying δJ, a formal expression for derivative was obtained. It depends on the solution of the linear conjugate hyperbolic problem:

(7)

The superscript T means transposition, — deriva F tive of the free term F with respect to v. Boundary and initial (terminal on Г₂) conditions:

Frechet derivative is sometimes called the residual or gradient. It is more convenient to represent it in operator form:

(8)

Here, the conjugate inhomogeneous operator ^*— conjugate homogeneous operator, the dot denotes the location of argument f of the operators, κ — weight coefficient for equalizing the computational noise of the numerical solution of the original and conjugate problems [8]. Expression (8) contains the value of the functional in the form of the number J. This is the value of derivative .

The heterogeneity of operator is a consequence of the dependence of the objective function I(w, u) on the control u. Such a dependence is a rare feature of optimization problems and can significantly complicate the calculation of the gradient.

The value of homogeneous operator ^* in derivative (8) has the form:

where integration is performed from the lower nonlinear boundary Г_b0 (Fig. 1) when water flows into the nozzle.

The meaning and objective of the conjugate problem is to map derivative (sensitivity of J to w) from domain ω to domain S, where the gradient and control are defined. Such a mapping with the help of f is done using the intermediate set which, according to controllability, specifies the correct domain V^*(Ω) of the definition of the homogeneous operator ^*, so that in expression (8) from , we obtain gradient ∇J.

That is, the conjugate problem implements the mapping:

Next, using operator ^* : V^*(Ω) → U^*(S), we can obtain the gradient from Frechet derivative :

Domain Ω is determined from the controllability analysis.

Research Results. The requirements for controllability conditions within the framework of the direct extreme approach are formulated in the following theorem (for the proof, see [2]).

Theorem. The mathematical model (v, u)v = 0 in problem (3) is controllable by u(x) on S with respect to functional J if:

1) there exists domain V^*(Ω), of a correct conjugate state, which is the domain of definition of operator ^* with its values in the gradient domain U^*(S);

2) operator ^* — non-degenerate;

3) algorithm (6) for u⁰∈ uses satisfactory regularization parameters b^k.

We start with the first and most difficult requirement of the theorem. First, it is necessary to verify the classical correctness of the original and conjugate problems. Original (3) and conjugate (7) systems are of the hyperbolic type. The eigenvalues of the matrices A and A^T are the same. Therefore, in both systems, the characteristics ξ_1,2 will be the same — as the trajectories of the propagation of disturbances in the plane (x, t) along the characteristic directions . The conjugate waves generated by derivative at the nozzle exit Г_b1 = ω, will move with the same characteristics as the original ones, but in the opposite direction. The initial condition for the conjugate problem is specified on the terminal line Г₂.

All characteristics in domain Ω emerge from the boundary sections with known solutions given by the boundary conditions. In the case of shock-free wave flows (it is precisely such flows that are considered in this research), the characteristics of the same family do not intersect, and at the intersection of two characteristics of a different family ξ₁ and ξ₂ at any point , a solution to the hyperbolic system of two equations [14] can be found in the form of two-dimensional vector functions v and f.

To find the domain of operator ^*, it is necessary to conduct an analysis and identify the existence of the following sequence of mappings, starting from the control u ∈ U(S) and ending with gradient ∇J ∈ U^*(S):

(9)

The problem under discussion can be described more simply. First, objective functional J(u), specified on ω, must be sensitive to the control defined on S (sensitivity is characterized by derivative ). Second, from the set of conjugate solutions on the entire , it is necessary to select such a subset Ω, where the conjugate solutions will uniquely depend on the values of the objective functional in the form . There may not be such a dependence on the entire domain . Thirdly, the set Ω must provide operator ^* with the ability to map the conjugate states into U^*(S). This mapping is represented by the last branch, where operator ^* from the obtained domain of definition can produce a mapping into the domain of values U^*(S). where there is gradient .

From the correctness of the original direct problem, it follows that any functions u(x) ∈ U(S) will uniquely affect the value of the derivative of the objective function on ω = Г_b1 through the characteristics ξ₁, if at least one of them has passed through the entire nozzle. The loss of such effect is possible in the presence of dissipation in the system, but this is not the case with isentropic flows. That is, there is a left branch of mappings in (9):

Let us proceed to identifying the set Ω, required for domain V^*(Ω) of the definition of operator ^*. Term in the boundary condition of conjugate problem (7) causes perturbations of the conjugate solution f. They propagate in the form of waves along the characteristics of the first family ξ₁ in the reverse time direction from the nozzle exit toward the piston (Fig. 2).

These disturbances propagate along the entire nozzle and transfer information about the objective functional from the points to the points on S = (x_a, x_b). On the piston, the waves described by the characteristics of the first family ξ₁, are reflected and, changing the direction of their propagation, continue to transfer information received from ξ₁ about disturbances , adding new information about the piston motion. This process of wave reflections from the piston and from the inner part of the nozzle continues until the moment t₀.

Starting from the moment t₃ and below, two conjugate waves ξ₁ and ξ₂, generated by different values of and with unnecessary information (noise), at a minimum, will arrive at the same point of some sections of the set S from the piston. And below the characteristic ξ₁, which came out of the nozzle below t₁, unnecessary information from the nozzle will be added. This information is not needed, since it does not contain information about the optimization goal from .

Figure 2 shows an example of a possible set Ω (the entire area shaded with different densities under the upper characteristic ξ₁ from x_a to x_b). In this case, Ω corresponds to the Frechet derivative . In domain Ω, under characteristic ξ₂ (reflection ξ₁, which came out at t₂) and under ξ₁ (which came out at t₁), a light area of ambiguous influence of the values of function on the conjugate state f is formed. Obviously, that it is pointless to solve the conjugate problem and calculate the gradient in such a domain Ω.

It is reasonable to limit ourselves to considering (Fig. 2) the conjugate state of f on the part of Ω, enclosed in the rectangle:

In this case, rectangle Ω₁ should be considered too large if the piston is relatively close to the beginning of the nozzle, affecting the conjugate state.

In this rectangle Ω₁, the set Ω (shaded with different density in Figure 2 from t₁ to t₂) will correspond to the redundant set ω. That is, in the objective functional J, the interval (t₁, t₂) will be redundant. At the same time, in the considered domain Ω, there may be unacceptable interference on the left (low density of shading in Ω₁) for calculating gradient ∇J.

Redundancy ω is eliminated by further reducing Ω₁ to t₃ = t₁, i.e., when t₃ corresponds to the start of the outflow (Fig. 3 a).

In this case, the entire domain of a sufficient solution to the conjugate problem is narrowed to an even smaller rectangle:

Here, the piston will not “interfere” with the display .

If the technical conditions for designing a hydrocannon allow for an even greater reduction in the flow time t₂ – t₁, then the solution rectangle of the conjugate problem can be reduced even further, to rectangle Ω₃ with a corresponding triangular domain Ω (Fig. 3 b):

Here there is a minimally sufficient set ω_min to form the domain of definition V^*(Ω) of operator ^*.

It is in the obtained domains Ω, located inside Ω₂ and Ω₃ (Fig. 3), that there is a domain of definition V^*(Ω) of operator ^* with a unique mapping of the derivatives by means of f into the domain of values of gradient U^*(S).

Conjugate problem (7) and its solution in the rectangles Ω_2,3 become significantly simpler:

(10)

Here, Г_a = x_a × (t₁, t₂). Now there is no impact of the piston from line Г_p, and there is no flow into the nozzle at the boundary Г_b0.

Formula (8) for calculating the gradient of the objective functional also takes a simplified form (there is no nonlinear integration boundary Г_b0):

(11)

The resulting set Ω ⊂ Ω_2,3 will correctly define domain V^*(Ω) of the definition of operator ^*(Ω) with the range of values in U^*(S). The corresponding expression for the time required to form such Ω depends on the characteristics of the first family ξ₁ and has the form:

(12)

That is, firstly, the upper characteristic ξ₁ must pass through the entire nozzle from x_b to x_a. Secondly, the start of the outflow t₁ should not be less than the moment t₃ of the start of the entry of waves reflected from the piston into the nozzle.

This expression is the controllability condition in the problem under consideration. In this case, the remaining branches of the mappings are fulfilled (9):

Now we discuss requirement 2) in the theorem on the nondegeneracy of operator ^*. Let us start with operator , which defines gradient (8). If the objective function I did not depend explicitly on the control u, then and the set of conjugate states in the kernel f_ker = {f : ^*f = 0 on S} would be zero for an unbounded optimal control u_*. In our case, for , the values of the elements of the kernel f_ker will not be zero, i.e., the optimal control will correspond to non-zero conjugate states. The subscript means the absence of a zero kernel. Obviously, if ^* was nondegenerate, then operator will also be nondegenerate. The nonhomogeneity of operator in our problem leads only to a shift in the zero kernel of the homogeneous operator ^*.

We estimate the possible degeneracy of the homogeneous operator ^*. It is obvious that for any values of ρ and w, the result of integration in can become zero on S only when f₁ = 0 on Ω. This means that operator ^* is nondegenerate, and therefore, is also nondegenerate.

The last requirement of the theorem remains. Regularization in the direct extreme approach is provided by:

The initial approximation u⁰(x) was set in the form of a pipe — as a continuation of the cannon barrel. At the first iterations of nozzle narrowing, functional J(u) grew with its convexity (growth of the norm ∇J). Then, the convexity changed to concavity (decrease of the norm ∇J), at the end of which there were very small regions of maximum (minimum norm ∇J with concavity of the functional) with a subsequent convex minimum. Nozzle 1 was obtained in the local maximum. The transition through these local extrema was further accompanied by unlimited convex growth of J(u). Only adding a constraint on the control made it possible to stop the uncontrolled expansion of the nozzle on the boundary with reasonable shape 2.

Recall that attempts were previously made to obtain a satisfactory solution using the classical calculus of variations. To do this, the authors [5] and [7] used relaxation methods to find root u_* from the required optimality condition (the Frechet derivative on the incorrect Ω from Figure 2). However, this approach did not give the desired results. Moreover, it requires an additional restriction on the nozzle exit area to prevent its collapse. Such collapse also confirms the incorrectness of using for the directed search for u_*(x) without isolating the controllability domain Ω inside Ω_2,3. In other words, instead of the Frechet derivative, it is required to obtain a gradient with the justification.

Discussion and Conclusion. The research results show that the application of the controllability analysis proposed in [2] made it possible to identify the key controllability conditions (12) required for the correct formulation and solution of the problem of optimizing the shape of the hydrocannon nozzle.

According to the controllability conditions, the optimization problem must be set and solved in a small rectangular domain Ω₂ or even Ω₃, and not in a large and complex domain Ω. This is due to the fact that the nozzle shape optimization problem with a statement in Ω does not reduce the Frechet derivative to gradient ∇J, which makes it impossible to search for the optimal solution. It was this circumstance that caused failures of the previous studies, where the optimality of solutions was not proven.

Our recommendations are as follows: first, perform a controllability analysis until the correct controllability domain Ω is identified, and then, for the resulting Ω, identify the solution region of the conjugate problem (in our case, it is Ω₂ or Ω₃) and find the variation . Next, you can continue the controllability analysis and obtain gradient ∇J from derivative . It is with the help of Ω inside Ω_2,3, that you can find the value of gradient ∇J(u; x), which is distributed along the entire nozzle and uniquely corresponds to the objective functional of the problem J(u). And then, you can purposefully search for the optimal nozzle shape.

The major advantage of the proposed approach is the use of a direct extreme method, which allows for direct maximizing the objective functional using gradient algorithms. This provides not only for the clarity of the controllability analysis, but also the possibility of obtaining numerically confirmed optimal solutions.

The theoretical value of the research is in the development of controllability analysis methods for distributed parameter systems, which opens up new prospects for solving similar problems in other areas. Further research can be aimed at expanding the method for more complex fluid flow models, as well as optimizing other devices operating on the basis of pulsed jets.