Consensus seeking in a network of discrete-time linear agents with communication noises

This paper studies the mean square consensus of discrete-time linear time-invariant multi-agent systems with communication noises. A distributed consensus protocol, which is composed of the agent's own state feedback and the relative states between the agent and its neighbours, is proposed. A time-varying consensus gain a[k] is applied to attenuate the effect of noises which inherits in the inaccurate measurement of relative states with neighbours. A polynomial, namely ‘parameter polynomial’, is constructed. And its coefficients are the parameters in the feedback gain vector of the proposed protocol. It turns out that the parameter polynomial plays an important role in guaranteeing the consensus of linear multi-agent systems. By the proposed protocol, necessary and sufficient conditions for mean square consensus are presented under different topology conditions: (1) if the communication topology graph has a spanning tree and every node in the graph has at least one parent node, then the mean square consensus can be achieved if and only if ∑^∞_{k = 0}a[k] = ∞, ∑^∞_{k = 0}a²[k] < ∞ and all roots of the parameter polynomial are in the unit circle; (2) if the communication topology graph has a spanning tree and there exits one node without any parent node (the leader–follower case), then the mean square consensus can be achieved if and only if ∑^∞_{k = 0}a[k] = ∞, lim_{k → ∞}a[k] = 0 and all roots of the parameter polynomial are in the unit circle; (3) if the communication topology graph does not have a spanning tree, then the mean square consensus can never be achieved. Finally, one simulation example on the multiple aircrafts system is provided to validate the theoretical analysis.     

Parameterization of nonlinear H∞ state-feedback controllers

State-space formulas are derived for a family of controllers solving the nonlinear H^∞ state-feedback control problem. These controllers are obtained by interconnecting the ‘central controller’ with an asymptotically stable, free system that satisfies one additional cascade condition. All proofs given are simple and clear, and also provide a deeper insight into the synthesis of the corresponding linear H^∞ controllers.     

Robust performance control of uncertain nonlinear systems: A self-tuning approach

The problem of output regulation with a guaranteed H^∞ performance, besides robust stability, for a class of feedback linearizable nonlinear systems via self-tuning controllers is investigated. The H^∞ performance consists of the desired disturbance attenuation and internal finite L²-gain stability. We show that, under the perturbations of matched parametric uncertainties, the sufficient condition for the existence of a self-tuning controller reduces to the solvability of a single Hamilton-Jacobi-Isaacs inequality (or algebraic Riccati equation) which indicates that the design can be performed as if the system uncertainties were absent. Under certain situations, the sufficient condition is also necessary. Once a solution of this nonlinear differential inequality (or algebraic equation) is available, a desired self-tuning controller with gradient-type parameter estimator can easily be constructed. The present work falls into the category of singular nonlinear H^∞ control since the desired H^∞ performance does not require any penalty on control input variables. The results also provide an immediate application to H^∞ self-tuning model reference control in linear systems under full state measurement.     

On input-output linearization of discrete-time nonlinear systems

We consider a nonlinear discrete-time system of the form Σ: x(t+1)=f(x(t), u(t)), y(t) =h(x(t)), where x ε R^N, u ε R^m, y ε R^q and f and h are analytic. Necessary and sufficient conditions for local input-output linearizability are given. We show that these conditions are also sufficient for a formal solution to the global input-output linearization problem. Finally, we show that zeros at infinity of ε can be obtained by the structure algorithm for locally input-output linearizable systems.     

Nonlinearizable single-input control systems do not admit stationary symmetries

The main result of the paper states that almost any analytic single-input control system, which is truly nonlinear, that is not feedback linearizable, with controllable linearization at an equilibrium point, does not admit any symmetry preserving that point. By almost any system, we mean that we exclude a small class of odd systems, that admit just one nontrivial symmetry conjugated to minus identity. The obtained results are based on a recent classification of nonlinear single-input systems under formal feedback. We also describe symmetries of feedback linearizable systems.     

Constrained controllability of perturbed discrete-time systems

In this paper we study the global null controllability with bounded controls of perturbed linear discrete-time systems in Rⁿ . More explicitly, a globally null-controllable autonomous discrete-time system is perturbed by ‘suitably small’ terms V(k) and B(k) to obtain a system of the form x(k + 1) = [A + V(k)]x(k) + [B + B(k)]u(k) and sufficient conditions are given to ensure the global null controllability of the resulting perturbed system.     

Global bilinearization of non-linear systems and the existence of Volterra series

A global Volterra series is developed for an analytic system defined on an analytic manifold M. Local bilinearizations are pieced together using the theory of fibre bundles, giving rise to ‘twisted’ bilinear systems denned globally on M.     

Robust H2/H∞ global linearization filter design for nonlinear stochastic time-varying delay systems

One can design a robust H_∞ filter for a general nonlinear stochastic system with external disturbance by solving a second-order nonlinear stochastic partial Hamilton-Jacobi inequality (HJI), which is difficult to be solved. In this paper, the robust mixed H₂/H_∞ globally linearized filter design problem is investigated for a general nonlinear stochastic time-varying delay system with external disturbance, where the state is governed by a stochastic Itô-type equation. Based on a globally linearized model, a stochastic bounded real lemma is established by the Lyapunov–Krasovskii functional theory, and the robust H_∞ globally linearized filter is designed by solving the simultaneous linear matrix inequalities instead of solving an HJI. For a given attenuation level, the H₂ globally linearized filtering problem with the worst case disturbance in the H_∞ filter case is known as the mixed H₂/H_∞ globally linearized filtering problem, which can be formulated as a linear programming problem with simultaneous LMI constraints. Therefore, this method is applicable for state estimation in nonlinear stochastic time-varying delay systems with unknown exogenous disturbance when state variables are unavailable. A simulation example is provided to illustrate the effectiveness of the proposed method.     

A NOTE ON ROBUST STABILIZATION OF FEEDBACK LINEARIZABLE SYSTEMS

The problem of robust stabilization of nonlinear systems with feedback linearizable nominal part and norm-bounded nonlinear uncertainties is investigated. Necessary and sufficient conditions are obtained for robust stabilization of such systems. A design procedure is developed which combines feedback linearization technique and quadratic stabilization via linear feedback to achieve robust global asymptotic stability. © 1997 by John Wiley & Sons, Ltd.     

Outer factor ‘absorption’ for H∞ control problems

We present a constructive method for ‘absorbing’ an irrational outer factor of a plant into the ‘Q-parameter’ in the H ^∞ optimal weighted sensitivity problem for single-input/single-output distributed parameter systems, when the plant has finitely many irrational zeros on the imaginary axis. This problem could not be solved using previous results. We also extend our new results to the mixed sensitivity problem.     

Robust output feedback stabilization via a small gain theorem

In this paper, we give sufficient conditions for designing robust globally stabilizing controllers for a class of uncertain systems, consisting of ‘nominal’ nonlinear minimum phase systems perturbed by uncertainties which may affect the equilibrium point of the nominal system (‘biased’ systems). The constructive proof combines a systematic step-by-step procedure, based on H_∞ arguments, with a small gain theorem, recently proved for nonliner systems. At each step, one finds two Lyapunov functions, one for a state-feedback problem and the other one for an output injection problem. Combining these two functions, one derives at each step a Lyapunov function candidate for solving an ouptut feedback stabilization problem. This approach allows one to put into a unified framework many existing results on robust output feedback stabilization. © 1998 John Wiley & Sons, Ltd.     

Some generalizations on the ultimate boundedness control of uncertain systems

Some generalizations on the ultimate boundedness control of uncertain systems are made in this paper. First, the ‘cone-bounded assumption’ on the plant uncertainties of a class of nominally linear uncertain systems is weakened to a more general ‘K _∞-bounded assumption’. Second, we show that ‘half’ of the uncertain systems with the ‘cone-bounded assumption’ can be practically stabilized via a linear control. Third, for a class of nominally nonlinear uncertain systems with the ‘matching assumption’, a similar smooth nonlinear state feedback control can be constructed to guarantee the uniform ultimate boundedness of the closed-loop trajectories.     

A scaled feedback stabilization of power integrator triangular systems

Using unbounded time-varying scaling of the states we design C¹ feedback laws for power integrator triangular systems which globally asymptotically stabilize (GAS) the origin despite the uncontrollability of the linearization. With bounded scaling the feedback laws achieve global practical stability (GPS). For a trade-off between GAS/GPS of the origin and unboundedness/boundedness of the scaling we construct a dynamic version of these feedback laws.     

Globally exponential stability and stabilization of interconnected Markovian jump system with mode-dependent delays

This paper focuses on the problems of globally exponential stability and stabilization with H_∞ performance for a class of interconnected Markovian jump system with mode-dependent delays in interconnection. By constructing a Lyapunov-Krasovskii functional, delay-range-dependent globally mean-square exponential stability conditions are established in terms of linear matrix inequalities. Based on the obtained conditions, state feedback control utilizing global state information and state feedback control utilizing global state information of decentralised observers are developed to render the closed-loop interconnected Markovian jump time-delay system globally exponential stable with H_∞ performance. Numerical simulation of a power system, composed of three coupled machines, is used to illustrate the effectiveness of the obtained results.     

Adaptive control of nonlinearly parameterized systems: the smooth feedback case

Studies global adaptive control of nonlinearly parameterized systems with uncontrollable linearization. Using a parameter separation technique and the tool of adding a power integrator, we develop a feedback domination design approach for the explicit construction of a smooth adaptive controller that solves the problem of global state regulation. In contrast to the existing results in the literature, a key feature of our adaptive regulator is its minimum-order property, namely, no matter how big the number of unknown parameters is, the order of the dynamic compensator is identical to one, and is therefore minimal. As a consequence, global state regulation of feedback linearizable systems with nonlinear parameterization is achieved by one-dimensional adaptive controllers, without imposing any extra (e.g., convex/concave) conditions on the unknown parameters.     

非线性系统的自适应实用输出跟踪控制

研究了一类具有不可控不稳定线性化和线性化参数的非线性系统的全局自适应实用输 出跟踪控制问题,这类系统既不具有反馈线性化性质,对输入也没有仿射特性.因为这类系统的 线性化系统具有不可控模式且该模式在右半平面内有根,所以全局渐近输出跟踪是不可能的(甚 至是局部的).应用修正的自适应增加幂积分方法,设计了一种光滑状态反馈控制器,保证闭环系 统所有信号全局一致最终有界,且使跟踪误差进入零的小邻域内.仿真结果表明该控制器是可行 的并且是有效的.     

Comparison of H∞ and state-space induced quotient topologies on stable minimal systems

It is shown that on the set of w-input, p-output asymptotically stable, minimal, nth order systems that the H_∞ ^p·m and the R^N quoiient induced topologies coincide. Implications of this in frequency domain system identification are discussed     

ℋ︁∞‐filtering for singularly perturbed nonlinear systems

In this paper, we consider the ??_∞‐filtering problem for singularly perturbed (two time‐scale) nonlinear systems. Two types of filters are discussed, namely, (i) decomposition and (ii) aggregate, and sufficient conditions for the solvability of the problem in terms of Hamilton–Jacobi–Isaac's equations (HJIEs) are presented. Reduced‐order filters are also derived in each case, and the results are specialized to linear systems, in which case the HJIEs reduce to a system of linear‐matrix‐inequalities (LMIs). Based on the linearization of the nonlinear models, upper bounds ε^* of the singular parameter ε that guarantee the asymptotic stability of the nonlinear filters can also be obtained. The mixed ??₂/??_∞‐filtering problem is also discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Measuring the sizes of concavities

A concavity in a region R is a simply-connected region S whose boundary is composed of a boundary B to R and the chord K of B. Three measures of the size of S are studied: the area of S; the maximum distance of B from K; and the maximum distance of K from B. Relationships among these measures are established. It is shown that only the last measure guarantees (when its value is high) that S contains a large ‘block’ of the complement of R, so that S is ‘detectable’ if R is coarsely digitized. Similarly, if the last measure is large relative to the length of K, S is ‘locally detectable’ if R is digitized to a suitable degree of coarseness. Local detectability of S at a coarse level of digitization corresponds, intuitively, to S being visually conspicuous.