Abstract
A new approach to the analytical design of linear and nonlinear hierarchical control loops and multifunctional automatic control systems of real (accelerated) time, based on the combined use of dynamic-programming technologies and the quasilinearization method, is presented. For continuous dynamical systems, the fundamentals of the theory of nonlinear synthesis are presented in a formulation that allows the formation of optimal, approximately optimal, and suboptimal control strategies with respect to a vector function of optimal control that is previously unknown, but determined at small optimization lengths.
Similar content being viewed by others
Notes
Formally, the conditions of Theorem 1 coincide with L. S. Pontryagin. The connection of these conditions with sufficient optimality conditions was established in the works of V. F. Krotov [10]. However, they also correspond to the traditional DP scheme (in the form of the minimum principle [5]) and determine the solution of not one, but a family of optimal-control problems.
In optimal control, these conditions were obtained and proved by V. A. Baturin [9].
REFERENCES
Handbook of Automatic Control Theory, Ed. by A. A. Krasovskii (Nauka, Moscow, 1987) [in Russian].
V. N. Bukov, Adaptive Predictive Flight Control Systems (Nauka, Moscow, 1987) [in Russian].
R. Gabasov and F. M. Kirillova, “Principles of optimal control,” Dokl. NAN Belarusi. 48, 15–18 (2004).
R. Bellman and R. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems (Am. Elsevier, New York, 1965).
M. M. Athans and P. L. Falb, Optimal Control: An Introduction to the Theory and Its Applications, Dover Books on Engineering (Dover, New York, 2006).
M. M. Khrustalev, “On sufficient conditions for an absolute minimum,” Dokl. Akad. Nauk SSSR 174, 1026–1029 (1967).
A. I. Moskalenko, Optimal Control of Economic Dynamics Models (Nauka, Novosibirsk, 1999) [in Russian].
D. H. Jacobson, “Differential dynamic programming methods for solving bang-bang control problems,” IEEE Trans. Autom. Control 13, 661–675 (1968).
V. A. Baturin and D. E. Urbanovich, Approximate Optimal Control Methods Based on the Extension Principle (Nauka, Novosibirsk, 1997) [in Russian].
V. F. Krotov and V. I. Gurman, Methods and Problems of Optimal Control (Nauka, Moscow, 1973) [in Russian].
V. N. Sizykh and A. V. Daneev, “Quasilinearization and sufficient conditions of optimality in the problem of improvements and containment,” Izv. Samar. Nauch. Tsentra RAN 18, 1250–1260 (2016).
V. N. Sizykh, A. V. Daneev, and J. G. Dambaev, “Methodology of approximately optimal synthesis of fuzzy controllers for circuit of improvement and localization,” Far East J. Math. Sci. 101, 487–506 (2017).
V. N. Sizykh, “Iterative method of suboptimal control synthesis on base relaxing phase space extension,” Dokl. Akad. Nauk 371, 571–574 (2000).
V. G. Boltyansky, “Separability of convex cones – a general method for solving extremal problems,” in Optimal Control (Znanie, Moscow, 1978) [in Russian].
V. F. Dil’ and V. N. Sizykh, “Methodology of aircraft control principles synthesis on the basis of trajectory prognostication and method of reverse dynamics problems,” Sovrem. Tekhnol., Sist. Anal., Model., No. 4 (48), 134–138 (2015).
G. N. Konstantinov, Rationing of Influences on Dynamic Systems (Irkut. Univ., Irkutsk, 1983) [in Russian].
F. L. Chernousko and V. P. Banichuk, Variational Problems of Mechanics and Control (Nauka, Moscow, 1973) [in Russian].
N. N. Moiseev, Elements of the Theory of Optimal Systems (Nauka, Moscow, 1974) [in Russian].
L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis (Fizmatgiz, Moscow, 1962) [in Russian].
N. A. Bobylev, S. V. Emel’yanov, and S. K. Korovin, Geometric Methods in Variational Problems (Magistr, Moscow, 1998) [in Russian].
V. N. Bukov and V. N. Sizykh, “Approximate synthesis of optimal control in a degenerate problem of analytical design,” Avtom. Telemekh., No. 12, 16–32 (1999).
V. F. Dil’ and V. N. Sizykh, “Optimal aircraft control synthesis based on the equations of non-linear dynamics,” Nauch. Vestn. MGTU GA 20 (3), 139-148 (2017).
A. V. Daneev, V. F. Dil’, and V. N. Sizykh, “Optimization of processes of management of the spatial movement of the aircraft on the basis of the equations of nonlinear dynamics,” Izv. Samar. Nauch. Tsentra RAN 19 (1), 195–200 (2017).
V. N. Bukov and V. N. Sizykh, “The method and algorithms for solving singular-degenerate problems of the analytical construction of regulators,” J. Comput. Syst. Sci. Int. 40, 740 (2001).
V. N. Sizykh, “Nonlinear controller design: An iterative relaxation method,” Autom. Remote Control 66, 892 (2005).
A. V. Daneev and V. N. Sizykh, Methodology for Designing Algorithmic Support for Integrated Control Systems for Aircraft Vehicles Based on Equations of Nonlinear Dynamics (Nauka, Moscow, 2021) [in Russian].
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare that they have no conflicts interest.
APPENDIX
APPENDIX
The procedure for proving Theorem 3 reduces to the following. We introduce the extended state vector \(y = (x,\delta \,u)\) and transform equations (2.15) and (2.16) to the form
where \(f(t,y) = (f(t,{{x}_{0}},{{u}_{0}}) + \partial \,f{\text{/}}\partial \,u \cdot \delta \,u,\,0)\) is a vector function obtained by quasilinearization according to the differential DP scheme; \({{{{\Gamma}}}_{1}} = {{\left[ {O\,\,E} \right]}^{{\text{T}}}}\) is a rectangular matrix with a “new” control vector ϑ; E1 is the identity matrix of dimension m × m respectively; \(y({{t}_{0}}) = (x({{t}_{0}}),\,0)\). Then we form the GWF for the extended space of states and controls X × U × T in the form (3.2) (\({{S}_{{\text{z}}}}(y({{t}_{{\text{f}}}})) = {{S}_{{\text{z}}}}(x({{t}_{{\text{f}}}})))\).
We write out sufficient optimality conditions:
from which we determine the local-optimal “new” controls ϑ:
If in the last expressions we introduce the notation \({{p}_{{\delta \,u}}} = \partial \,{{\varphi }^{{\text{T}}}}{\text{/}}\partial \,\delta u\), then condition (3.5) of the theorem is satisfied. Formula (4.3) can be obtained differently through the stationary condition: \(\partial H{\text{/}}\partial \,\delta u = 0\), where H(t, y, φy) = \(\partial \,\varphi (t,y){\text{/}}\partial \,y(f(t,y) + \) \({{{{\Gamma}}}_{1}}\vartheta ) + {{Q}_{p}}(t,y) + \) 0.5\(\vartheta _{0}^{T}\)r–1ϑ0 is the Hamiltonian of system (4.1), and \(\partial \varphi (t,y){\text{/}}\partial y\) is a row vector of dimension \(1 \times (n + m)\).
We formulate formula (A.3) into expression (A.2), as a result of which the sufficient optimality conditions will be written in the form of the equation
where the function φ(t, y) has the meaning of the Lyapunov function in stability theory.
Formula (A.4) determines the “free” motion of system (4.1). The total derivative calculated for “free” motion is calculated using the expression
Equation (A.4), taking into account expression (A.5), takes the form
which implies condition (3.6) of Theorem 3.
Using the method of characteristics, we determine the solution of the partial differential equation (A.4) in the form of a canonically conjugate system [1]
where \(H\left( {t,y,p} \right) = \partial \varphi \left( {t,y} \right){\text{/}}\partial t + \partial \varphi \left( {t,y} \right){\text{/}}\partial y \cdot f\left( {t,y} \right)\) is the Hamiltonian of the “free” motion of the system (A.1), \(p = \partial {{\varphi }^{{\text{T}}}}\left( {t,y} \right){\text{/}}\partial y\)is a column vector of partial derivatives of the size n × m.
Revealing vectors y, p through subvectors x, δu and px, \({{p}_{{\delta \,u}}}\) as a result of the decomposition of relations (A.7), we obtain formulas corresponding to conditions (2.15), and (3.3)–(3.4) of Theorem 3.
Through the procedure of searching for a weak local minimum \({{u}_{{{\text{opt}}}}}(t,\tau )\mathop \to \limits_{\tau \to t} {{u}_{{{\text{opt}}}}}(t,t) = {{u}_{{\text{0}}}}(t)\) we find the optimal one in the sense of achieving a local minimum of the functional (1.4) of the process (1.3). Thus, all the conditions of Theorem 3 turn out to be satisfied. The theorem has been proven.
Proof of Theorem 4 for the differential DP scheme (Theorem 3). The equations of the canonically conjugate system (1.3), (5.1) and formula (5.4) are obtained from (2.15), (3.3) and (3.6) with u = u0(t). The stationary conditions (5.2) and condition (3) of Theorem 2 can be deduced by following the scheme of proof from contradiction (Bliss scheme). For this, we consider the control variations determined by the gradient procedure (3.1), which, at short optimization lengths, \(\Delta t\) represented by the relation \(u(t)\, = {{u}_{0}}(t) + \,\vartheta \,\Delta t.\)
It can be seen from the last expression that the weak local minimum (x0, u0) is formed through fulfillment at each stationary point of the condition u = u0(t): exactly, by ensuring that the limit elements of minimizing sequences are equal to zero in terms of u: \(\vartheta = {{\vartheta }_{0}} = 0\); and approximately, by reducing the optimization lengths: Δt → 0.
We assume the opposite, i.e., there is a control \(\tilde {u}\)(t) = u0(t) + Δu(t) for which the minimum of the local criterion is less than the minimum of the local functional: \(\tilde {I}\)(t)< I(t). Then \(d(\tilde {u}(t) - {{u}_{0}}(t)){\text{/}}dt\, = \,d\Delta u(t){\text{/}}dt,\) which contradicts the conditions of local optimality of the control in (5.10): \(d\,\delta \,u{\text{/}}d\,t = 0\). In this way, \(\Delta u(t) = 0\) and \(\vartheta = {{\vartheta }_{0}} = 0\).
Further, it follows from formula (3.5) of Theorem 3 that, in the absence of left zero divisors, the subvector \({{p}_{{\delta \,u}}}\) of the extended skew-state vector is equal to \({{p}_{{\delta \,u}}} = 0.\)
The derivative of this subvector will also be equal to zero: \({{\dot {p}}_{{\delta \,u}}} = 0,\) whence, by virtue of the relation \({{\dot {p}}_{{\delta \,u}}} = \partial \,{{H}^{{\text{T}}}}{\text{/}}\partial \,\delta \,u\) Equation (5.2) of condition 2) of Theorem 4 turns out to be valid. The theorem has been proven.
The proof of estimate (5.5) is as follows. Let \(u_{0}^{j}\) be the initial approximation of the local-optimal control vector u0, and the general recurrence relation is written from (3.1):
where ϑ(\(u_{0}^{j}\)) – rpδu(\(u_{0}^{j}\)).
To verify the validity of the estimate in (A.8) for i components of vectors \(u_{0}^{{j + 1}}\), u0, ϑ(\(u_{0}^{j}\)), we write
where \(\xi _{i}^{1}\)(u) = ui + ϑi(u)Δt, ϑi(u0) = –ri\(p_{{\delta u}}^{i}\)(u0) = 0, (see Theorem 4).
Expression (A.9) is represented by a Taylor series, in which we take into account the first three terms, including the residual \(u_{{0i}}^{{j + 1}} - {{u}_{{0i}}} = (u_{{0i}}^{j} - {{u}_{{0i}}})\dot {\xi }_{i}^{1}\left( {{{u}_{0}}} \right) + 0.5{{(u_{{0i}}^{j} - {{u}_{{0i}}})}^{2}}\ddot {\xi }_{i}^{1}\left( \theta \right).\) Here, by the mean-value theorem (Lagrange’s theorem [4]), the remainder term is equal to
where θ is some value of the independent variable, intermediate between \(u_{{0i}}^{{j + 1}}\), u0i; u0i ≤ θ ≤ \(u_{{0i}}^{j}\).
Because the \(\dot {\xi }_{i}^{1}\)(u) = \({{\dot {u}}_{i}}\) + \({{\dot {\vartheta }}_{i}}\)(u)Δt, then at the stationary points ui = u0i: \(\dot {\xi }_{i}^{1}\)(u) = 0, since u0i is a parameter that does not vary over optimization lengths, but the condition \(\dot {p}_{{\delta u}}^{i}\)(u0) = 0 follows from Theorem 4. Therefore, the estimate
where \({{k}_{{1i}}} = \mathop {\max }\limits_{{{u}_{0}} \leqslant \theta \leqslant u_{0}^{'}} (\ddot {\xi }_{i}^{1}\left( \theta \right){\text{/}}2)\).
Formula (A.10), written in vector form, is the first desired estimate in (5.5). The statement has been proven.
Rights and permissions
About this article
Cite this article
Daneev, A.V., Sizykh, V.N. Approximate-Optimal Synthesis of Operational Control Systems for Dynamic Objects on the Basis of Quasilinearization and Sufficient Optimality Conditions. J. Comput. Syst. Sci. Int. 61, 918–934 (2022). https://doi.org/10.1134/S1064230722060065
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064230722060065