Structural properties of minimax controllers for a class of differential games arising in nonlinear H control

This paper introduces, in precise mathematical terms, two properties (named, certainty equivalence and generalized certainty equivalence) that nonlinear minimax controller problems might possess. The certainty equivalence is a generalization of the one introduced earlier by Ba ar and Bernhard (1991) and Bernhard (1990), which applies to problems where the ‘worst-case disturbance’ may not be unique (but the worst-case state trajectory is). The generalized certainty equivalance, on the other hand, extends this to accomodate nonunique worst-case state trajectories, and leads to the constrollers that guarantee a bounded upper value for the underlying game. The paper also shows that for a large class of games (and under certain conditions), certainty-equivalent (as well as generalized certainty-equivalent) controllers admit (infinite-dimensional) estimator (Kalman-filter) structures, where the estimator gain depends on the state of the estimator.     

Singular nonlinear H∞ optimal control problem

The theory of nonlinear H_∞ of optimal control for affine nonlinear systems is extended to the more general context of singular H_∞ optimal control of nonlinear systems using ideas from the linear H_∞ theory. Our approach yields under certain assumptions a necessary and sufficient condition for solvability of the state feedback singular H_∞ control problem. The resulting state feedback is then used to construct a dynamic compensator solving the nonlinear output feedback H_∞ control problem by applying the certainty equivalence principle.     

Robustness of the tuning functions adaptive backstepping design forlinear systems

We study robustness of the adaptive backstepping design with tuning functions for linear systems. Under assumptions on unmodeled dynamics and disturbances equal to those for certainty equivalence schemes, we address an adaptive scheme not based on the certainty equivalence principle. In the process of redesign for robustness we employ only leakage in the estimator; we do not employ normalization, neither static nor dynamic. A fundamental difference between the tuning functions design and the certainty equivalence designs is that the controller in the former is inherently nonlinear, while in the latter it is nonlinear only in the parameter estimate. As a result, achievable robustness results for the tuning functions scheme are not global but regional, with a region of attraction inversely proportional to the “size” of the unmodeled dynamics. The tracking error is proportional to the size of the uncertainties     

A note on reduced order stabilizing output feedback controllers

In this paper we show that if a certain class of nonlinear systems is globally asymptotically stabilizable through an n-dimensional output feedback controller then it can be always stabilized through an (n − p)-dimensional output feedback controller, where p is the number of outputs and n is the dimension of the state space. This result gives an alternative construction of reduced order controllers for linear systems, and recovers in a more general framework the concept of dirty derivative, used in the framework of rigid and elastic joint robots, and gives an alternative procedure for designing reduced-order controllers for nonlinear systems considered in the existing literature.     

Limitations of nonlinear discrete periodic control for disturbance attenuation and robust stabilization

This paper presents a performance analysis of nonlinear periodically time-varying discrete controllers acting upon a linear time-invariant discrete plant. Time-invariant controllers are distinguished from strictly periodically time-varying controllers. For a given nonlinear periodic controller, a time-invariant controller is constructed. Necessary and sufficient conditions are given under which the time-invariant controller gives strictly better control performance than the time-invariant controller from which it was obtained, for the attenuation of l_p exogenous disturbances and the robust stabilization of l_p unstructured perturbations, for all p[1,∞].     

The parallel projection operators of a nonlinear feedback system

This paper defines and studies a pair of nonlinear parallel projection operators associated with a nonlinear feedback system. These operators have been seen to play an important role in the robustness and design of linear systems especially in the theory of the gap metric, the use of weighted gaps in control system design and Glover-McFarlane loop-shaping. We show that the input-output L₂-stability of a feedback system amounts to a ‘coordinatization’ of the input and output spaces, which is also equivalent to the existence of a pair of nonlinear parallel projection operators onto the graph of the plant and the inverse graph of the controller respectively. These projections are shown to have equal norms whenever one of the feedback elements is linear. A bound on this norm is given in the case of passive systems with unity negative feedback.     

Nonlinear H ∞ control of a Quadrotor (UAV), using high order sliding mode disturbance estimator

A nonlinear H _∞ output feedback controller is proposed and coupled to a high-order sliding mode estimator to regulate an UAV in the presence of the unmatched perturbations. The plant to be controlled is a Quadrotor helicopter described by nonlinear dynamics with plant uncertainties due to the variations of inertia moments and payload operation. A robust state estimation is considered under model uncertainties as well as external/measurement disturbances. Performance issues of the controller are illustrated in a simulation study made for an UAV prototype.     

On nonlinear H∞ control under sampled measurements

This paper deals with the nonlinear H_∞ control problem with sampled measurement feedback. This problem has already been studied in Suzuki et al. (1995), where, using a certainty equivalence principle, a control solution involving a state estimator with a linear injection gain is proposed. Using the same general framework, we propose a more refined estimator with a nonlinear injection gain. This gain is shown to be connected to a periodic solution of a Hamilton-Jacobi inequality with jumps     

Feedback equivalence for general control systems

We discuss the solution to the problem of local equivalence of control systems with n states and p controls in a neighbourhood of a generic point, under the Lie pseudo-group of local time independent feedback transformations. We have shown earlier that this problem is identical with the problem of simple equivalence of the time optimal variational problem. Here we indicate the way in which this identification may be used to obtain closed loop time critical controls for general systems. We show that the classification of general nonlinear systems depends on the classification of all n−p dimensional affine subspaces of the space of symmetric forms in p variables and that the case of control linear systems depends on the classification of all n−p dimensional affine subspaces of the space of skew forms in p variables. We show that in the latter case the G-structure is the prolongation of one determined by a state space transformation group. We give a complete list of normal forms for control linear systems in the case p=n−1.     

Certainty equivalence in nonlinear output regulation with unmeasured error

In this article, we consider a nonlinear output regulation problem in which the controlled output and the measured output are not the same. It is assumed that the controlled plant has a single control input, and that it can be transformed into Gauthier–Kupka's observability canonical form. Then, it is shown that a design based on certainty equivalence is effective for determining a controller that solves the given problem.     

Adaptive control of a class of discrete-time affine nonlinear systems

The adaptive control problem is addressed in the paper for a class of discrete-time affine nonlinear input/output stochastic models with linear unknown parameters. The controller is a certainty equivalence weighted one-step-ahead control and is constructed by using the weighted-least-squares and random regularization methods. Global stability of the closed-loop systems is established, which shows that arbitrarily large growth rate is allowed for the multiplicative nonlinear part of the systems.     

Adaptive control of a class of discrete-time affine nonlinear systems

The adaptive control problem is addressed in the paper for a class of discrete-time affine nonlinear input/output stochastic models with linear unknown parameters. The controller is a certainty equivalence weighted one-step-ahead control and is constructed by using the weighted-least-squares and random regularization methods. Global stability of the closed-loop systems is established, which shows that arbitrarily large growth rate is allowed for the multiplicative nonlinear part of the systems.     

H∞ controller synthesis for pendulum-like systems

This paper focus on a stabilization problem for a class of nonlinear systems with periodic nonlinearities, called pendulum-like systems. A notion of Lagrange stabilizability is introduced, which extends the concept of Lagrange stability to the case of controller synthesis. Based on this concept, we address the problem of designing a linear dynamic output controller which stabilizes (in the Lagrange sense) a pendulum-like system within the framework of the H_∞ control theory. Lagrange stabilizability conditions for uncertainty-free systems and systems with norm-bounded uncertainty in the linear part are derived, respectively. When these conditions are satisfied, the desired stabilization output feedback controller can be constructed via feasible solutions of a certain set of linear matrix inequalities (LMIs).     

Use of the design freedom of time-optimal control

A method for synthesizing a state-feedback time-optimal controller for linear n-state, r-input discrete-time systems is developed. The nonuniqueness of the solution is organized in such a way as to select the controller with the minimum value of the Frobenius norm of the feedback gain matrix K. It is shown that among the rn design parameters in the parametric form of K only r(n − rp_r) need to be considered in the search technique, where p_r is the order of the smallest Jordan block in the Jordan form of the closed-loop system.     

On the stability of the information state system

The purpose of this paper is to present some preliminary results on the stability of the information state system. The information state system underlies the (infinite dimensional) dynamics of an H_∞ controller for a nonlinear system. Thus it is important to understand its stability and the structure of its equilibrium points. We analyse the important case corresponding to the mixed sensitivity problem. We prove the existence of an equilibrium information state, convergence under very general conditions to such an equilibrium state p_e and uniqueness of this state (up to an irrelevant constant). In this case the equilibrium p_e is usually singular in the sense that it takes on the value − ∞ except on a low dimensional subset of its domain.This meshes with the article [9] which analysed the effect of using p_e to initialize the information state controller and gave explicit formulas which in many cases produce a dramatic reduction in the amount of computation required to implement the controller. What this article suggests is that indeed p_e is the only equilibrium initialization possible.     

Adaptive backstepping control of wheeled inverted pendulum with velocity estimator

In this paper, we introduce a backstepping control design of a wheeled inverted pendulum. Based on a second-order motion equation of the body angle, an adaptive integral backstepping controller is designed to stabilize the body angle. It is shown that the σ-modification rule in the adaptive update law guarantees the boundedness of the errors in estimating the time-varying signal that is an output of a linear system with every bounded input signal. Then, the stabilizing controller for the wheel angle is constructed by a PD-type positive feedback. The derived controller requires the full-state measurements. In the output feedback case, the K filter or the observer backstepping is needed. However, the structure of the controller becomes complicated. We propose a non-model-based differentiator based on the adaptive update law. Since the non-model-based differentiator does not require any knowledge of the dynamic structure of the signal, we can use it as a velocity estimator for unknown nonlinear systems. Therefore, we replaced the velocity measurement with the estimates by the non-model-based differentiator. Finally, simulation results for the proposed controller are presented.     

Fliess operators on Lp spaces: convergence and continuity

Fliess operators as input–output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory. In the classical analysis of these operators, certain growth conditions on the coefficients in their series representations insure uniform and absolute convergence, provided every input is uniformly bounded by some fixed upperbound. In some emerging applications, however, it is more natural to consider other classes of inputs. In this paper, L_p function spaces are considered. In particular, it is shown that the classic growth conditions also provide sufficient conditions for convergence and continuity when the admissible inputs are from a ball in L_p[t₀,t₀+T], where T is bounded and p1. In addition, stronger global growth conditions are given that apply even for the case where T is unbounded. When the coefficients of a Fliess operator have a state space representation, it is shown that the state space model will always locally realize the corresponding input–output map on L_p[t₀,t₀+T] for sufficiently small T>0. If certain well-posedness conditions are satisfied then the state space model will globally realized the input–output mapping for unbounded T when the coefficients satisfy the global growth condition.     

Robust static output feedback control for linear discrete-time systems with time-varying uncertainties

This paper is concerned with the problem of designing robust static output feedback controllers for linear discrete-time systems with time-varying polytopic uncertainties. Sufficient conditions for robust static output feedback stabilizing controller designs are given in terms of solutions to a set of linear matrix inequalities, and the results are extended to H₂ and H_∞ static output feedback controller designs. Numerical examples are given to illustrate the effectiveness of the proposed design methods.