A stabilization algorithm for a class of uncertain linear systems

This paper presents an algorithm for the stabilization of a class of uncertain linear systems. The uncertain systems under consideration are described by state equations which depend on time-varying unknown-but-bounded uncertain parameters. The construction of the stabilizing controller involves solving a certain algebraic Riccati equation. Furthermore, the solution to this Riccati equation defines a quadratic Lyapunov function which is used to establish the stability of the closed-loop system. This leads to a notion of ‘quadratic stabilizability’. It is shown that the stabilization procedure will succeed if and only if the given uncertain linear system is quadratically stabilizable.The paper also deals with a notion of ‘overbounding’ for uncertain linear systems. This procedure enables the stabilization algorithm to be applied to a larger class of uncertain linear systems. Also included in the paper are results which indicate the degree of conservativeness introduced by this overbounding process.     

LMI optimization approach to robust H∞ observer design and static output feedback stabilization for discrete‐time nonlinear uncertain systems

A new approach for the design of robust H_∞ observers for a class of Lipschitz nonlinear systems with time‐varying uncertainties is proposed based on linear matrix inequalities (LMIs). The admissible Lipschitz constant of the system and the disturbance attenuation level are maximized simultaneously through convex multiobjective optimization. The resulting H_∞ observer guarantees asymptotic stability of the estimation error dynamics and is robust against nonlinear additive uncertainty and time‐varying parametric uncertainties. Explicit norm‐wise and element‐wise bounds on the tolerable nonlinear uncertainty are derived. Also, a new method for the robust output feedback stabilization with H_∞ performance for a class of uncertain nonlinear systems is proposed. Our solution is based on a noniterative LMI optimization and is less restrictive than the existing solutions. The bounds on the nonlinear uncertainty and multiobjective optimization obtained for the observer are also applicable to the proposed static output feedback stabilizing controller. Copyright © 2008 John Wiley & Sons, Ltd.     

Limitations in remote stabilization over unreliable channels without acknowledgements

We consider the remote control of linear systems using unreliable communication channels where transmitted messages may be randomly lost and aim at clarifying the limitations arising there. In particular, we study the case without acknowledgement messages from the actuator to the remote controller regarding the arrival of the control input. It is shown that for a class of plants, stabilization is possible if and only if the loss rates satisfy critical bounds which have been known in the literature when acknowledgements are available. We present a constructive proof employing a nonlinear controller that makes control input estimation.     

On system gains for linear and nonlinear systems

System gains, and bounds for system gains, are determined for stable linear and nonlinear systems in different signal setups which include ℓ_p signal setups and certain persistent signal generalizations of the ℓ₂ and ℓ₁ signal setups. These results show that robust H_∞ and ℓ₁ control generalize to very versatile persistent signal settings. Relationships between different system gains are also derived. Finally, an application of nonlinear system gain bounds is given by establishing induced ℓ_∞ modelling error bounds for a class of (generalized) piecewise linear systems approximated with simpler linear time-invariant models.     

Error bounds for polynomial tensor product interpolation

We provide estimates for the maximum error of polynomial tensor product interpolation on regular grids in ${\mathbb{R}^d}$ . The set of partial derivatives required to form these bounds depends on the clustering of interpolation nodes. Also bounds on the partial derivatives of the error are derived.     

An interpolated controller for stabilization of a plant withvariable operating conditions

This paper considers stabilization of a linear time-invariant single-input/single-output plant with variable operating conditions. It is assumed that the plant is described by a linear interpolation of proper stable coprime factorizations of the transfer functions of two representative models which are defined at two representative operating points. A linear interpolation of stabilizing controllers for the representative models is applied to the interpolated plant. The necessary and sufficient condition for the interpolated plant to be stabilizable by the interpolated controller is presented using the Nevanlinna-Pick theory. It is also shown that simultaneous stabilizability of the two representative models is necessary for such stabilization. An example is given, which illustrates that the class of stabilizable plants via interpolated controllers is larger than that of plants stabilizable by fixed controllers     

Univariate Strip Interpolations by Nonlinear Parametric Splines

 In this paper, we develop algorithms for treating several constrained interpolation problems using the parametric cubic splines introduced in reference [11]. Explicit bounds for the parameters occurring in the spline class are given such that within these bounds the considered problems are always solvable. Received March 9, 2000     

Delay-dependent robust stability and control for jump linear systems with delays

This paper considers robust stochastic stability, stabilization and H_∞ control problems for a class of jump linear systems with time delays. By using some zero equations, neither model transformation nor bounding for cross terms is required to obtain the delay-dependent results, which are given in terms of linear matrix inequalities (LMIs). Maximum sizes of time delays are also studied for system stability. Furthermore, solvability conditions and corresponding H_∞ control laws are given which provide robust stabilization with a prescribed H_∞ disturbance attenuation level. Numerical examples show that the proposed methods are much less conservative than existing results.     

Robust stabilisation of nonlinear plants via left coprime factorizations

In this paper we take steps towards the development of a robust stabilization theory for nonlinear plants. An approach using the left coprime factorizations of the plant and controller under certain differential boundedness assumptions is used. We first focus attention on a characterization of the class of all stabilizing nonlinear controllers K_Q for a nonlinear plant G, parameterized in terms of an arbitrary stable (nonlinear) operator Q. Also, we consider the dual class of all plants G_S stabilized by a given nonlinear controller K and parameterized in terms of an arbitrary stable (nonlinear) operator S. We show that a necessary and sufficient condition for K_Q to stabilize G_S with Q, S not necessarily stable, is that S stabilizes Q. This robust stabilization result is of interest for the solution of problems in the areas of nonlinear adaptive control and simultaneous stabilization. It specializes to known results for linear operators.     

On continuity properties of the two-block ℓ1 optimal design

This paper investigates continuity properties of the two-block ℓ₁ optimization resulting from the optimal design of BIBO stability robustness for discrete time systems in the presence of coprime factor perturbations. In general, the two-block ℓ₁ optimal design might lack continuity properties, since it might have no finite dimensional solution. However, this paper shows that for a suitable given truncated order, the two-block ℓ₁ suboptimal design is continuous as a map from the plant to the suboptimal closed loop solution, if the plant has no zeros on the unit circle and has a unique suboptimal solution. Furthermore, if the set of plants is compactness, we show that there exists a uniform bound on the degree of suboptimal solution for all plants in the set such that their deviation of suboptimal value from the optimal one have the same bound, and the two-block ℓ₁ suboptimal design is uniformly continuous. In particular, in the case in which the plant has nonunique solution, a procedure is also developed on how to choose a solution that preserves continuity. The results enable the stability of robust ℓ₁ adaptive scheme to hold in the presence of coprime factor perturbations.     

On 0′-computable reals

Given a 0′-computable real x we are interested in the relative complexity of the sets A_z = {i: z_i < x} for possible computable sequences of rationals z = {z_s} with lim_sz_s = x, with respect to a strong reducibility ≤ = _r. It turns out that the r-degree structure of these sets (excluding the trivial, i.e. the finite and co-finite ones) is a substructure of the ≤ = _r-degrees inside the Turing degree of x and it can be non-trivial. In fact, we construct a real x = lim_sz_s = lim_sw_s such that A_zwttA_w. Also, it can be trivial even for non-computable reals x: there is a non-computable c.e. real x such that for all sequences z with lim_sz_s = x the sets A_z (if co-infinite) have the same m-degree. Assigning such a degree structure to each 0′-computable real we propose the study of the complexity of a single real by means of the complexity of the correspondent degree structure. The variety of these degree structures from real to real indicates that a fine classification of the 0′-computable reals may be possible in this way. Finally, we study the immunity properties of A_z: we prove that it cannot be (co-)hyperhyperimmune but it is always bi-hyperimmune or (co-)hypersimple if x is non-computable.     

Coprime factors reduction methods for linear parameter varying and uncertain systems

We present a generalization of the coprime factors model reduction method of Meyer and propose a balanced truncation reduction algorithm for a class of systems containing linear parameter varying and uncertain system models. A complete derivation of coprime factorizations for this class of systems is also given. The reduction method proposed is thus applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured ℓ₂-induced stability constraint required in the standard nonfactored case. Reduction error bounds in the ℓ₂-induced norm of the factorized mapping are given.     

A linear, robust and convergent interpolatory algorithm for quantifying model uncertainties

This paper presents a linear, interpolatory, convergent uncertainty estimation scheme which is robust in the face of disturbance. The idea is that by simply changing from Lagrange interpolation to Hermite interpolation an effective scheme is derived for estimation of uncertainty. It is shown that the proposed scheme has the advantage that in the face of corrupted data, the error in the derived uncertainty bound is independent of the degree of the approximating polynomial, i.e. if ε denotes the maximum magnitude of the disturbance, then the maximum possible error in the uncertainty bound is ε.     

State-feedback implementation of cascade compensators

For general dynamical system Σ and cascade compensator conditions are developed for the existence of a state-feedback law F which when applied to Σ causes the resulting closed-loop system Σ_F to exhibit the same input-output behavior as the cascade connection of Σ with . For a controllable, observable linear system Σ with transfer matrix T_Σ, it is shown that an invertible linear cascade compensator with transfer matrix can be implemented with state feedback if and only if the McMillan degree of T_Σ equal the McMillan degree of .     

Robust control of a class of uncertain nonlinear systems

This paper considers the robust control of a class of nonlinear systems with real time-varying parameter uncertainty. Interest is focused on the design of linear dynamic output feedback control and two problems are addressed. The first one is the robust stabilization and the other is the problem of robust performance in an H_∞ sense. A technique is proposed for designing stabilizing controllers for both problems by converting them into ‘scaled’ H_∞ control problems which do not involve parameter uncertainty.     

Estimating the distance to uncontrollability: A fast method and a slow one

For a linear control system (A,B), the distance to uncontrollability is characterized by

, where σ_n([A − λI,B]) is the smallest singular value of the augmented matrix [A − λI,B]. Two methods are developed to estimate the distance to uncontrollability, giving both a lower bound and an upper bound. One method is fast, requiring only one spectral decomposition of A and computations of three smallest singular values and being used for well-conditioned A. The other is slow, requiring computations of a large number of the smallest singular values, but it produces bounds as tight as possible and also a region containing global minimizers. Newton's method can be used to compute the global minimizers if one really wants.     

Static output feedback stabilisation with H∞ performance for a class of plants

In this paper, the problem of static output feedback control of a linear system is considered. The existence of a static output feedback control law is given in terms of the solvability of two coupled Lyapunov inequalities which result in a non-linear optimisation problem. However, using state-coordinate and congruence transformations and by imposing a block-diagonal structure on the Lyapunov matrix, we will see that the determination of a static output feedback gain reduces, for a specific class of plants, to finding the solution of a system of linear matrix inequalities. The class of plants considered is those which are minimum phase with a full row rank Markov parameter. The method is extended to incorporate H_∞ performance objectives. This results in a sub-optimal static H_∞ control law found by non-iterative means. The simplicity of the method is demonstrated by a numerical example.     

Universal stabilization of a class of nonlinear systems with homogeneous vector fields

Under relative-degree-one and minimum-phase assumptions, it is well known that the class of finite-dimensional, linear, single-input (u), single-output (y) systems (A,b,c) is universally stabilized by the feedback strategy u = Λ(λ)y, λ = y², where Λ is a function of Nussbaum type (the terminology “universal stabilization” being used in the sense of rendering /s(0/s) a global attractor for each member of the underlying class whilst assuring boundedness of the function λ(·)). A natural generalization of this result to a class ^k of nonlinear control systems (a,b,c), with positively homogeneous (of degree k 1) drift vector field a, is described. Specifically, under the relative-degree-one (cb ≠ 0) and minimum-phase hypotheses (the latter being interpreted as that of asymptotic stability of the equilibrium of the “zero dynamics”), it is shown that the strategy u = Λ(λ)/vby/vb^k−1y, assures ^k-universal stabilization. More generally, the strategy u = Λ(λ)exp(/vby/vb)y, assures -universal stabilization, where = _{k 1} ^k.