Universal W-tracking for a class of nonlinear systems

An adaptive output feedback strategy, incorporating gains of Nussbaum type, is described and shown to be a universal tracking controller for a class of nonlinearly-perturbed, minimum-phase, linear systems with reference signals of Sobolev class W^1,∞.     

Tracking control with prescribed transient behaviour for systems of known relative degree

Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered for multi-input, multi-output linear systems satisfying the following structural assumptions: (i) arbitrary—but known—relative degree, (ii) the “high-frequency gain” matrix is sign definite—but of unknown sign, (iii) exponentially stable zero dynamics. The first control objective is tracking, by the output y, with prescribed accuracy: given λ>0 (arbitrarily small), determine a feedback strategy which ensures that, for every reference signal r, the tracking error e=y-r is ultimately bounded by λ (that is, e(t)<λ for all t sufficiently large). The second objective is guaranteed output transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel (determined by a function ). Both objectives are achieved by a filter in conjunction with a feedback function of the filter states, the tracking error and a gain parameter. The latter is generated via a feedback function of the tracking error and the funnel parameter . Moreover, the feedback system is robust to nonlinear perturbations bounded by some continuous function of the output. The feedback structure essentially exploits an intrinsic high-gain property of the system/filter interconnection by ensuring that, if (t,e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact.     

Tracking control: Performance funnels and prescribed transient behaviour

Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of nonlinear systems, with output y, described by functional differential equations (a generalization of the class of linear minimum-phase systems of relative degree one with positive high-frequency gain). The primary control objective is tracking with prescribed accuracy: given λ>0 (arbitrarily small), determine a feedback strategy which ensures that for every admissible system and reference signal, the tracking error e=y-r is ultimately smaller than λ (that is, e(t)<λ for all t sufficiently large). The second objective is guaranteed transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel . Adopting the simple non-adaptive feedback control structure u(t)=-k(t)e(t), it is shown that the above objectives can be attained if the gain is generated by the nonlinear, memoryless feedback , where is any continuous function exhibiting two specific properties, the first of which ensures that if (t,e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact, and the second of which obviates the need for large gain values away from the funnel boundary.     

Inherent robustness of MRAC schemes

A standard model reference adaptive control (MRAC) scheme without modification of the adaptive law is inherently robust with respect to L^∞ ∩ L² disturbances in the sense that all closed-loop signals remain bounded and the tracking error belongs to L^∞ ∩ L². A MRAC scheme with a new adaptive law is inherently robust with respect to the disturbances in L^∞ ∩ L^1+α, 0 < α < ∞, with an L^1+β tracking error, for .     

Stabilization and H control of symmetric systems: an explicit solution

In this paper we address the H^∞ control analysis, the output feedback stabilization, and the output feedback H^∞ control synthesis problems for state-space symmetric systems. Using a particular solution of the Bounded Real Lemma for an open-loop symmetric system we obtain an explicit expression to compute the H^∞ norm of the system. For the output feedback stabilization problem we obtain an explicit parametrization of all asymptotically stabilizing control gains of state-space symmetric systems. For the H^∞ control synthesis problem we derive an explicit expression for the optimally achievable closed-loop H^∞ norm and the optimal control gains. Extension to robust and positive real control of such systems are also examined. These results are obtained from the linear matrix inequality formulations of the stabilization and the H^∞ control synthesis problems using simple matrix algebraic tools.     

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.     

Performance funnels and tracking control

Tracking of an absolutely continuous reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of non-linear, single-input, single-output, dynamical systems modelled by functional differential equations satisfying certain structural hypotheses (which, interpreted in the highly specialised case of linear systems, translate into assumptions of (i) relative degree one, (ii) positive high-frequency gain and (iii) stable zero dynamics). The control objective is evolution of the tracking error within a prespecified funnel, thereby guaranteeing prescribed transient performance and prescribed asymptotic tracking accuracy. This objective is achieved by a control which takes the form of linear error feedback with time-varying gain. The gain is generated by a non-linear feedback law in which the reciprocal of the distance of the tracking error to the funnel boundary plays a central role. In common with many established adaptive control methodologies, the overall feedback structure exploits an intrinsic high-gain property of the system, but differs from these methodologies in two fundamental respects: the funnel control gain is not dynamically generated and is not necessarily monotone. The main distinguishing feature of the present article vis à vis its various precursors is twofold: (a) non-linearities of a general nature can be tolerated in the input channel; (b) a more general formulation of prescribed transient behaviour is encompassed (including, for example, practical (M, μ)-stability wherein, for prescribed parameter values M > 1, μ > 0 and λ > 0, the tracking error e(·) is required to satisfy |e(t)| < max {Me ^?μt |e(0)|, λ} for all t ≥ 0).     

Decentralized robust disturbance attenuation for a class of large-scale nonlinear systems

We solve the problem of decentralized H_∞ almost disturbance decoupling for a class of large-scale nonlinear uncertain systems in the absence of matching conditions. The method combines ideas from decentralized adaptive control and centralized nonlinear H_∞ control. We relax earlier assumptions on the uncertain time-varying interconnections which are demanded to be only bounded by general nonlinear functions in this work.     

H disturbance attenuation for state delayed systems

In this note, the differential game and dissipation inequality are applied to the disturbance attenuation or H^∞-control for linear systems with delayed state. Firstly, a simple sufficient condition on the existence of a γ-suboptimal H^∞ state feedback controller is given, which is independent of delay, and an observer-based dynamic output feedback solution is presented in terms of Riccati inequalities (or Riccati equations). Secondly, a sufficient condition on the existence of a delay-dependent state feedback is presented and the criterion is presented by a matrix inequality which can be solved by numerical methods.     

On the weighted sensitivity minimization problem for delay systems

We consider the H^∞-optimal sensitivity problem for delay systems. In particular, we consider computation of μ:= inf {|W-φq|_∞ : q ε H^∞(j )} where W(s) is any function in RH^∞(j ), and φ in H^∞(j ) is any inner function. We derive a new explicit solution in the pure delay case where φ = e^−sh, h > 0.     

Nonlinear robust H∞ control via dynamic output feedback

The problem on robust H_∞ control for a class of nonlinear systems with parameter uncertainty is studied. Sufficient conditions for the existence of the dynamic output feedback controller are obtained. Under these conditions, the closed-loop systems have robust H_∞-performance. A numerical example is given to illustrate the design of a robust controller using the proposed approach.     

Adaptive suboptimal control of a linear system with bounded disturbance

An adaptive suboptimal control of a linear discrete system with unknown parameters is proposed. An additive disturbance v_t acting on the system is supposed to be uniformly bounded. The criterion is sup_vtI(y^∞₁, u^∞₁), where y_t is the output, u_t is the control. The adaptive control law gives almost the same guaranteed value of the criterion as the optimal linear feedback does for a system with known parameters.     

On standard control problems for systems with infinitely many unstable poles

In this paper, H^∞ control for a class of linear time invariant systems with infinitely many unstable poles is studied. An example of such a plant is a high gain system with delayed feedback. We formulate the problem via a generalized plant which consists of a rational transfer matrix and the inverse of a scalar (possibly irrational) inner function. It is shown that the problem can be decomposed into a finite-dimensional H^∞ control problem and an additional rank condition.     

A nonsmooth strict bounded real lemma

In the theory of linear H^∞ control, the strict bounded real lemma plays a critical role because it provides a connection between the stabilizing solutions to the H^∞ Riccati equations and the stability and disturbance attenuation of the closed-loop system. Nonlinear versions of the strict bounded real lemma are also important in nonlinear H^∞ control theory. In this paper, we investigate the extension of the linear strict bounded real lemma and its smooth nonlinear generalization to cases where the solutions of the associated nonlinear PDE are not necessarily differentiable.     

Bounded real lemma and robust control of 2-D singular Roesser models

This paper discusses the problem of robust H_∞ control for linear discrete time two-dimensional (2-D) singular Roesser models (2-D SRM) with time-invariant norm-bounded parameter uncertainties. The purpose is the design of static output feedback controllers such that the resulting closed-loop system is acceptable, jump modes free, stable and satisfies a prescribed H_∞ performance level for all admissible uncertainties. A version of bounded realness of 2-D SRM is established in terms of linear matrix inequalities. Based on this, a sufficient condition for the solvability of the robust H_∞ control problem is solved, and a desired output feedback controller can be constructed by solving a set of matrix inequalities. A numerical example is provided to demonstrate the applicability of the proposed approach.     

H∞ filtering for a class of uncertain nonlinear systems

This paper investigates the problem of H_∞ filtering for a class of uncertain continuous-time nonlinear systems with real time-varying parameter uncertainty and unknown initial state. We develop an infinite horizon H_∞ filtering methodology which provides both robust stability and a guaranteed H_∞ performance for the filtering error irrespective of the parameter uncertainty.     

Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems

The first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic Riccati equation for a class of infinite-dimensional systems of the form Σ(A,B,S^−1/2B^*,D), where A is dissipative and all the other operators are bounded. These systems are not exponentially stabilizable and so the standard theory is not applicable. The second part uses the Riccati equation results to give formulas for normalized coprime factorizations over H_∞ for positive real transfer functions of the form D+S^−1/2B^*(author−A)⁻¹,B.     

Transient performance analysis in robust nonlinear adaptive control

This paper deals with the problem of controlling unknown linear systems in the presence of strictly proper unmodelled dynamics and bounded disturbances. Adaptive controllers that ensure the closed-loop global (uniform) stability and asymptotic performances can be designed following either the backstepping approach or the certainty-equivalence method. The main shortcoming of the involved controllers is that they do not allow quantification of the closed-loop transient behaviour. In this paper, the transient issue is addressed for backstepping adaptive controllers. A L_∞ bound on the tracking error is explicitly given as a function of the design parameters. This shows that the error can be made arbitrarily small by sufficiently increasing the design gains.     

Singular H∞ Suboptimal Control for a Class of Nonlinear Cascade Systems

In the present paper singular state feedback _∞ suboptimal control for a class of nonlinear cascade systems is addressed. Under the assumption that a regular state feedback _∞ suboptimal control problem is solvable for a particular subsystem of the cascade system, an auxiliary nonlinear system is defined. It is shown that a state feedback solution to the singular _∞ suboptimal control problem for the auxiliary system also applies to the original problem. The advantage of the auxiliary problem to the original problem is that the auxiliary penalty variable has lower dimension than the original penalty variable. It is shown how this fact can simplify the problem considerably for the case when the auxiliary system can be strongly input-output decoupled. The theory is applied to a problem of a rigid spacecraft with actuator dynamics. Application to the special case when a subsystem of the nonlinear cascade system is passive is also considered.     

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.