Decentralized control design for large-scale systems with strong interconnections using neural networks

We propose a decentralized neural network (NN) controller for a class of large-scale nonlinear systems with the strong interconnections. The NNs are used to approximate the unknown subsystems and interconnections. Due to the functional approximation capabilities of NNs, the additional precautions are not required to be made for avoiding the possible control singularity problems. Semiglobal asymptotic stability results are obtained and the tracking error converges to zero. Furthermore, the issue of transient performance of the subsystems is also addressed under an analytical framework.     

Stabilisation of discrete-time bilinear systems by constant control inputs

ABSTRACT

In this paper, the stabilisation problem of discrete-time bilinear systems by using constant control inputs is considered. It is shown that the spectrum of a matrix derived from the system matrices plays a key role in the stabilisation design. Sufficient conditions for the systems to be stabilizable by constant control inputs are presented in the cases when the derived matrix has no complex eigenvalue as well as in the cases when it has complex eigenvalues. Particularly, it is proved that, if the derived matrix has only nonzero real eigenvalues, then the systems can always be stabilised by constant control inputs. Finally, simulations are given to demonstrate the obtained results.     

Quadratic stabilization of bilinear control systems

In this paper, a stabilization problem of bilinear control systems is considered. Using the linear matrix inequality technique and quadratic Lyapunov functions, an approach is proposed to the construction of the so-called stabilizability ellipsoid such that the trajectories of the closed-loop system emanating from any point inside this ellipsoid asymptotically tend to the origin. The approach allows for an efficient construction of nonconvex approximations to stabilizability domains of bilinear systems.     

Constrained control of SISO bilinear systems

Relative degree and nonminimum phase difficulties limit the applicability of input-output feedback linearization; hence the need for approximations. Earlier work on predictive control of bilinear systems overcame these problems by means of interpolation between feedback linearization and state feedback, the former providing optimality and the latter guaranteeing feasibility and stability through the use of invariant/feasible polytopes. The current work also makes use of polytopes in preference to ellipsoids but achieves distinctly different objectives. First, it is shown that feedback linearization can be used over particular polytopes without needing to resort to either approximation or interpolation. Then, it is shown that invariant polytopes based on bilinear controllers can be much larger. These two approaches are combined in an algorithm that guarantees stability over much larger initial condition sets and gives much improved closed-loop performance.     

Optimal control of distributed bilinear systems

The optimal control problem for a bilinear distributed parameter system subject to a quadratic cost functional is solved. It is shown that the optimal control is given by a convergent power series in the state with tensor coefficients.     

Optimal control of weakly coupled bilinear systems

The optimization of the time-invariant bilinear weakly coupled system with a quadratic performance criterion is considered. A sequence of linear state and costate equations is constructed such that the open-loop solution of the optimization problem is obtained in terms of the reduced-order subsystems. This leads to a reduction in the size of the required computations and allows parallel processing of information. The near-optimal closed-loop control is obtained in the form of a linear feedback law, with the feedback gains calculated from two reduced-order independent time-varying linear-quadratic optimal control problems.     

Trajectory funnels of two-dimensional bilinear control systems

Two-dimensional linear nonstationary systems under the conditions of parametrical interval indeterminacy, when the only known about the coefficient matrix of these systems is that it satisfies some matrix range restriction, are considered. Qualitative methods of the investigation of two-dimensional dynamic systems, based on the concept of a phase portrait, are developed. Extreme matrix-valued functions determining the branches of the boundaries of trajectory funnels are synthesized by a differential geometric method. The geometric interpretation of the straight lines of switching is given. The properties of boundaries of trajectory funnels including the singular mode are investigated.     

Phase portraits of two-dimensional bilinear control systems

Phase portraits characteristic of two-dimensional bilinear control systems with different oscillatory properties are constructed.     

On minimum energy control of commutative bilinear systems

A minimum energy control problem is considered for commutative bilinear systems with and without terminal constraints. Optimal controls are shown to be constant vectors determined by the boundary conditions. Sufficient conditions are derived for uniqueness of the optimal control in the absence of a terminal constraint. A class of physical bilinear systems is discussed which possesses the commutative property.     

On asymptotically stabilizing feedback control of bilinear systems

Asymptotically stabilizing feedback controls for a class of bilinear systems are discussed; a simple modification which strengthens a result of Slemrod [1] is noted and illustrated by example.     

Deterministic and stochastic control of discrete-time bilinear systems

Optimal control of a class of time invariant single-input, discrete bilinear systems is investigated in this paper. Both deterministic and stochastic problems are considered.

In the deterministic problem, for the initial state in a certain set ∑₀, the solution is the same as the solution to the associated linear system. The optimal path may be a regular path or a singular path.

The stochastic control problem is considered with perfect state observation, and additive and multiplicative noise in the state equation. It is demonstrated that the presence of noise simplifies the analysis compared to that in the determinstic case.     

A class of µ-commutative bilinear control systems

A class of μ-commutative bilinear control systems is defined and the properties of coefficients of a μ-commutative system are studied. A μ-commutative scalar control system is studied and an explicit representation for its input-output mapping is derived. This mapping is used to study the properties of a system, particularly the geometry of its reachable set. Original Russian Text ? M.V. Topunov, 2006, published in Avtomatika i Telemekhanika, 2006, No. 6, pp. 113–125. This paper was recommended for publication by B.T. Polyak, a member of the Editorial Board     

Optimal control of bilinear systems: Time-varying effects of cancer drugs

A bilinear model is introduced to study the effects of anti-tumor drugs on the kinetics of the cell cycle. Optimal control theory is applied in the estimation of these time-varying effects based on a least squares fit of laboratory data. This methodology enables the authors to quantify the effects of a drug on each phase of the cell cycle.     

Decentralised sampled-data control for large-scale systems with nonlinear interconnections

This paper presents a decentralised sampled-data control technique for a class of large-scale systems, which are considered to consist of linear subsystems and nonlinear interconnections. The decentralised sampled-data controller design problem is established using a closed-loop subsystem. Based on the controller design problem, the stability condition is derived for a closed-loop large-scale system, and the maximum interconnection bound is guaranteed to satisfy the stability condition. Also, its sufficient condition is formulated in terms of linear matrix inequalities. Finally, the effectiveness of the proposed technique is verified by using an example of the multi-machine power system.     

时滞双线性系统的最优跟踪控制

研究带有时滞项的双线性系统的最优跟踪控制.依据求解规律转化成两点边值问题.通过采用迭代逐次逼近法,能够得到收敛与原系统最优控制律的序列.由于控制律含有外系统变量导致控制律物理不可实现,则采用状态重构解决这一问题.仿真结果表明该方法对解决这类问题有较好的鲁棒性及快速性.     

Frequency-interval control configuration selection for multivariable bilinear systems

Control configuration selection is the procedure of choosing the appropriate input and output pairs for the design of decoupled (SISO or block) controllers for multivariable systems. This step is an important prerequisite for a successful industrial control strategy. The focus of this paper is on the problem of control configuration selection for a class of nonlinear systems which is known as bilinear systems. First, new frequency-interval gramians are presented for bilinear systems. These gramians are devised in particular for many applications in which one is interested in analysis and control of a system within a frequency-interval. It is shown that these gramians are the solutions to the so-called frequency-interval generalized Lyapunov equations. These gramians are used in the interaction measure for control configuration selection of MIMO bilinear systems. Most of the results on control configuration selection, which have been proposed so far, can only support linear systems. The proposed gramian-based interaction measure supports bilinear processes, can show the input–output interactions for any frequency-interval of interest, and can be used to propose a richer sparse or block diagonal controller structure.     

Quadratic cost output feedback control for bilinear systems

This paper considers output feedback control problems for bilinear systems with a quadratic cost function and develops a robust control theory approach. A key point of the approach is to regard once the bilinear term as a term with unknown parameters and then use some robust control synthesis methods. This approach derives two types of nonlinear output feedback controller that guarantee an upper bound for the cost function. Efficiencies of the proposed approach are also demonstrated through a numerical example.     

Optimal linear nonanticipative control of partially observable bilinear systems

Via the concept of projection estimates a class of partially observable bilinear control problem is reduced to a completely observable one and the optimal linear nonanticipative control is sought via a deterministic Hamilton-Jacobi-Bellman theory.     

Decentralized output-feedback neural control for systems with unknown interconnections.

An adaptive backstepping neural-network control approach is extended to a class of large-scale nonlinear output-feedback systems with completely unknown and mismatched interconnections. The novel contribution is to remove the common assumptions on interconnections such as matching condition, bounded by upper bounding functions. Differentiation of the interconnected signals in backstepping design is avoided by replacing the interconnected signals in neural inputs with the reference signals. Furthermore, two kinds of unknown modeling errors are handled by the adaptive technique. All the closed-loop signals are guaranteed to be semiglobally uniformly ultimately bounded, and the tracking errors are proved to converge to a small residual set around the origin. The simulation results illustrate the effectiveness of the control approach proposed in this correspondence.     

Stability analysis of bilinear systems under aperiodic sampled-data control

This note considers the problem of local stability of bilinear systems with aperiodic sampled-data linear state feedback control. The sampling intervals are time-varying and upper bounded. It is shown that the feasibility of some linear matrix inequalities (LMIs), implies the local asymptotic stability of the sampled-data system in an ellipsoidal region containing the equilibrium. The method is based on the analysis of contractive invariant sets, and it is inspired by the dissipativity theory. The results are illustrated by means of numerical examples.