Uniting local and global controllers for uncertain nonlinear systems: beyond global inverse optimality

This paper provides a solution to a new problem of global robust control for uncertain nonlinear systems. A new recursive design of stabilizing feedback control is proposed in which inverse optimality is achieved globally through the selection of generalized state-dependent scaling. The inverse optimal control law can always be designed such that its linearization is identical to linear optimal control, i.e. optimal control, for the linearized system with respect to a prescribed quadratic cost functional. Like other backstepping methods, this design is always successful for systems in strict-feedback form. The significance of the result stems from the fact that our controllers achieve desired level of ‘global’ robustness which is prescribed a priori. By uniting locally optimal robust control and global robust control with global inverse optimality, one can obtain global control laws with reasonable robustness without solving Hamilton–Jacobi equations directly.     

-like control for nonlinear stochastic systems

In this paper we develop a H_∞-type theory, from the dissipation point of view, for a large class of time-continuous stochastic nonlinear systems. In particular, we introduce the notion of stochastic dissipative systems analogously to the familiar notion of dissipation associated with deterministic systems and utilize it as a basis for the development of our theory. Having discussed certain properties of stochastic dissipative systems, we consider time-varying nonlinear systems for which we establish a connection between what is called the L₂-gain property and the solution to a certain Hamilton–Jacobi inequality (HJI), that may be viewed as a bounded real lemma for stochastic nonlinear systems. The time-invariant case with infinite horizon is also considered, where for this case we synthesize a worst case-based stabilizing controller. Stability in this case is taken to be in the mean-square sense. In the stationary case, the problem of robust state feedback control is considered in the case of norm-bounded uncertainties. A solution is then derived in terms of linear matrix inequalities.     

Nonlinear model-state feedback control for nonminimum-phase processes

A general nonlinear controller design methodology for continuous-time nonminimum-phase systems is presented, which utilizes synthetic outputs that are statically equivalent to the original process outputs and make the system minimum-phase. A systematic procedure is proposed for the construction of statically equivalent outputs with prescribed transmission zeros. The calculated outputs are used to construct a model-state feedback controller. The proposed method is applied to a nonminimum-phase chemical reactor control problem where a series/parallel reaction is taking place.     

Viscosity solutions and optimal control problems with integral constraints

We discuss optimal control problems with integral state-control constraints. We rewrite the problem in an equivalent form as an optimal control problem with state constraints for an extended system, and prove that the value function, although possibly discontinuous, is the unique viscosity solution of the constrained boundary value problem for the corresponding Hamilton–Jacobi equation. The state constraint is the epigraph of the minimal solution of a second Hamilton–Jacobi equation. Our framework applies, for instance, to systems with design uncertainties.     

Global -gain design for a class of nonlinear systems

It is known that the so-called control problem of a nonlinear system is locally solvable if the corresponding problem for the linearized system can be solved by linear feedback. In this paper we prove that this condition suffices to solve also a global control problem, for a fairly large class of nonlinear systems, if one is free to choose a state-dependent weight of the control input. Using a two-way (backward and forward) recursive induction argument, we simultaneously construct, starting from a solution of the Riccati algebraic equation, a global solution of the Hamilton–Jacobi–Isaacs partial differential equation arising in the nonlinear control, as well as a state feedback control law that achieves global disturbance attenuation with internal stability for the nonlinear systems.     

On designing filters for uncertain sampled-data nonlinear systems

This paper is concerned with the problem of nonlinear filtering for sampled-data systems with nonlinear time-varying parameter uncertainty. The aim is to design a digital filter such that the ratio between the energy of the estimation errors and the energy of the exogenous inputs is minimised or guaranteed to be less or equal to a prescribed value for all admissible uncertainties. A nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty is provided. Based on this nonlinear bounded real lemma, the robust filtering problem is solved in terms of both continuous and discrete Hamilton–Jacobi equations.     

A new approach to the zero-dynamics assignment problem for nonlinear discrete-time systems using functional equations

The present research work aims at the development of a systematic method to arbitrarily assign the zero dynamics of a nonlinear discrete-time real analytic system by constructing the requisite synthetic output maps. The problem under consideration is motivated by the need to adequately address the control problem of nonminimum-phase nonlinear discrete-time systems, since the latter represent a rather broad class of systems due to the well-known effect of sampling on the stability of zero-dynamics. In the proposed approach, the above control objective can be attained through: (i) a systematic computation of synthetic output maps that induce minimum-phase behavior while being statically equivalent to the original output maps (both vanish on the equilibrium manifold) and (ii) the subsequent integration into the methodological framework of currently available nonminimum-phase compensation schemes for nonlinear discrete-time systems that rely on output redefinition. The mathematical formulation of the zero-dynamics assignment problem is realized via a system of nonlinear functional equations, and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of functional equations can be proven to be locally analytic, and this enables the development of a solution method that is easily programmable with the aid of a symbolic software package. The synthetic output maps that induce the prescribed zero dynamics for the original nonlinear discrete-time system can be explicitly computed on the basis of the solution to the aforementioned system of functional equations.     

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 the design of a reduced-order H ∞ controller for nonlinear sampled-data systems

In this paper, the problem of designing reduced-order H ^∞ controllers is studied for nonlinear continuous-time systems with sampled measurements. Using the concepts of dissipativity and differential game, sufficient conditions are derived for the existence of such reduced-order H ^∞ controllers. These conditions are expressed in terms of the solutions of two Hamilton–Jacobi inequalities, comprising a standard Hamilton–Jacobi inequality and a differential Hamilton–Jacobi inequality with jumps. These Hamilton–Jacobi inequalities are exactly those used in the construction of full-order H ^∞ controllers. When these conditions hold, state-space formulae are also given for such reduced-order controllers. An illustrative example is also included.     

Stability of current-mode control for DC–DC power converters

DC–DC power converters are switched devices whose averaged dynamics are described by a bilinear second-order system with saturated input. In some cases (e.g., boost and buck–boost converters), the input output dynamics can be of nonminimum-phase nature. Current-mode control is the standard strategy for output voltage regulation in high dynamic performance industrial DC–DC power converters. It is basically composed by a saturated linear state feedback (inductor current and output voltage) plus an output voltage integral feedback to remove steady-state offset. Despite its widespread usage, there is a lack of rigorous results to back up its stabilization capability and to systematize its design. In this paper, we prove that current-mode control yields semiglobal stability with asymptotic regulation of the output voltage.     

Suboptimal control for nonlinear systems: a successive approximation approach

This paper presents a successive approximation approach (SAA) designing optimal controllers for a class of nonlinear systems with a quadratic performance index. By using the SAA, the nonlinear optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. By using the finite-step iteration of the nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation examples are employed to test the validity of the SAA.     

Suboptimal mean controllers for bounded and dynamic stochastic distributions

Based on the recently developed algorithms for the modelling and control of bounded dynamic stochastic systems (H. Wang, J. Zhang, Bounded stochastic distributions control for pseudo ARMAX stochastic systems, IEEE Transactions on Automatic control, 486–490), this paper presents the design of a subotpimal nonlinear mean controller for bounded dynamic stochastic systems with guaranteed stability. The B-spline functional expansion based square root model is used to represent the output probability density function of the system. This is then followed by the design of a mean controller of the output distribution of the system using nonlinear output tracking concept. A nonlinear quadratic optimization is performed using the well known Hamilton–Jacobi–Bellman equation. This leads to a controller which consists of a static unit, a state feedback part and an equivalent output feedback loop. In order to achieve high precision for the output tracking, the output feedback gain is determined by a learning process, where the Lyapunov stability analysis is performed to show the asymptotic stability of the closed loop system under some conditions. A simulation example is included to demonstrate the use of the algorithm and encouraging results have been obtained.     

Exact slow–fast decomposition of the nonlinear singularly perturbed optimal control problem

We study the infinite horizon nonlinear quadratic optimal control problem for a singularly perturbed system, which is nonlinear in both, the slow and the fast variables. It is known that the optimal controller for such problem can be designed by finding a special invariant manifold of the corresponding Hamiltonian system. We obtain exact slow–fast decomposition of the Hamiltonian system and of the special invariant manifold into the slow and the fast ones. On the basis of this decomposition we construct high-order asymptotic approximations of the optimal state-feedback and optimal trajectory.     

A new Lyapunov design approach for nonlinear systems based on Zubov's method

The present research work proposes a new nonlinear controller synthesis approach that is based on the methodological principles of Lyapunov design. In particular, it relies on a short-horizon model-based prediction and optimization of the rate of “energy dissipation” of the system, as it is realized through the time derivative of an appropriately selected Lyapunov function. The latter is computed by solving Zubov's partial differential equation based on the system's drift vector field. A nonlinear state feedback control law with two adjustable parameters is derived as the solution of an optimization problem that is formulated on the basis of the aforementioned Lyapunov function and closed-loop performance characteristics. A set of system-theoretic properties of the proposed control law are examined as well. Finally, the proposed Lyapunov design method is evaluated in a chemical reactor example which exhibits nonminimum-phase behaviour.     

Robust global stabilization of a class of uncertain feedforward nonlinear systems

The problem of global asymptotically stabilizing a certain class of uncertain feedforward nonlinear systems is considered. The control law is obtained by nesting saturation functions whose amplitude can be rendered arbitrarily small. With respect to previous works on the subject the design procedure is able to deal with uncertain (possibly time-varying) parameters ranging within the prescribed compact sets which can affect also the linear approximation of the system. The small gain theorem for nonlinear systems which are input to state stable “with restrictions” is shown to be a key tool for designing a state feedback saturated control law.     

Remarks regarding the gap between continuous, Lipschitz, and differentiable storage functions for dissipation inequalities appearing in H∞ control

This paper deals with the regularity of solutions of the Hamilton–Jacobi inequality which arises in H_∞ control. It shows by explicit counterexamples that there are gaps between existence of continuous and locally Lipschitz (positive definite and proper) solutions, and between Lipschitz and continuously differentiable ones. On the other hand, it is shown that it is always possible to smooth-out solutions, provided that an infinitesimal increase in gain is allowed.     

Finite-element Discretization of Static Hamilton-Jacobi Equations based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton–Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss–Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.Dedicated to Peter Deuflhard on the occasion of his 60th birthday     

Input–output decoupling of nonlinear systems by static measurement feedback

This paper gives necessary and sufficient conditions for solvability of the strong input–output decoupling problem by static measurement feedback for nonlinear control systems.     

Singular PDEs and the assignment of zero dynamics in nonlinear systems

The present research work aims at the development of a systematic method to arbitrarily assign the zero dynamics of a nonlinear system by constructing the requisite synthetic output maps. The minimum-phase synthetic output maps constructed can be made statically equivalent to the original output maps, and therefore, they could be directly used for nonminimum-phase compensation purposes. Specifically, the mathematical formulation of the problem is realized via a system of first-order nonlinear singular PDEs and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of singular PDEs can be proven to be locally analytic and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package. The minimum-phase synthetic output maps that induce the prescribed zero dynamics for the original nonlinear system can be computed on the basis of the solution of the aforementioned system of singular PDEs. Moreover, static equivalence to the original output map can be readily established by a simple algebraic construction.