A switching adaptive controller for feedback linearizable systems

One of the main open problems in the area of adaptive control of linear-in-the-parameters feedback linearizable systems is the computation of the feedback control law when the identification model becomes uncontrollable. The authors propose a switching adaptive control strategy that overcomes this problem. The proposed strategy is applied to nth-order feedback linearizable systems in canonical form. The closed-loop system is proved to be globally stable in the sense that all the closed-loop signals are bounded and the tracking error converges arbitrarily close to zero. No assumptions are made about the type of nonlinearities of the system, except that such nonlinearities are smooth. However, the proposed controller requires knowledge of the sign and lower bound of the input vector field     

A constrained optimal control of nonlinear systems

A new method for optimal control of nonlinear systems with input constraint is discussed in this paper. The system is optimized by minimizing a quadratic performance index. Considering this problem as a nonlinear programming problem, the necessary and sufficient conditions for optimal control are derived, which are later simplified to an integral equation. This integral equation becomes a necessary condition. The existence of a solution of this integral equation is studied, and a method of solving it is discussed. A numerical example is worked out at the end.     

Singularity for static-state feedback linearizable bilinear systems

This paper deals with the problem of singularities for a class of static-state feedback linearizable bilinear systems. For this class of nonlinear systems, the decoupling matrix is singular on an algebraic surface (which contains the origin), and this relates the static state feedback linearization to the difficult problem of the completeness of the trajectories of the closed-loop system and/or the boundedness of the feedback laws. In this work, we give a sufficient condition under which all the trajectories of the closed-loop system are complete and the used feedback law is bounded on each of these trajectories     

A second-order feedback method for optimal control computations

A second-order method for computing optimal control is described. The method computes control corrections from the state perturbations through feedback coefficients. On each successive trajectory a test is made for a conjugate point, and the algorithm automatically forms a control correction when such points occur. Unlike some earliermethods, terminal constraints of arbitrary dimensions are handled by the method. Numerical results for two cases of a non-linear example based on van der Pol's equation are included.     

Exponential tracking of feedback linearizable non-linear control systems with uncertainties

This paper studies the robust output tracking problem of feedback linearizable non-linear control systems with uncertainties. Utilizing the input-output feedback linearization technique and the Lyapunov method for non-linear state feedback synthesis, a robust globally exponential output tracking controller design methodology for a broad class of non-linear control systems with uncertainties is developed. The underlying theoretical approaches are the differential geometry approach and the composite Lyapunov approach. One utilizes the parametrized coordinate transformation to transform the original non-linear system with uncertainties into a singularly perturbed model with uncertainties and the composite Lyapunov approach is then applied for output tracking. To demonstrate the practical applicability, the paper has investigated a pendulum control system.     

Dynamic anti-windup scheme for feedback linearizable nonlinear control systems with saturating inputs

This paper proposes a dynamic compensation scheme for input-constrained feedback linearizable nonlinear systems to cope with the windup phenomenon. Given a dynamic feedback linearizing controller designed without considering its input constraint, an additional dynamic compensator is proposed to account for the constraint. This dynamic anti-windup is based on the minimization of a reasonable performance index. The proposed strategy is a nonlinear extended version of [Park, J.-K., & Choi, C.-H. (1995). Dynamic compensation method for multivariable control systems with saturating actuators. IEEE Transactions on Automatic Control, 40(9), 1635–1640] with simplified derivation of an optimization solution under relaxed assumptions. The parameter matrices and structure of the solution are explicitly decided by mathematical optimization for infinite horizon without tuning of design parameters unlike previous schemes. During input saturation, the role of the anti-windup scheme with the proposed dynamic feedback compensator is to maintain the controller states to be exactly the same as those without input saturation. The local asymptotic stability and the total stability of the resulting systems are proved. The usefulness of the proposed design method is illustrated by comparative simulations for a constrained control system.     

Remarks on nonlinear planar control systems which are linearizable by feedback

The possibility that certain nonlinear systems can be linearized by the nonlinear feedback group has recently attracted a great deal of attention in the literature. At least for mechanical systems, the apparent ubiquity of linearizable systems is often accounted for by the claim that scalar control systems defined on R² are generically linearizable. Since there are several possible interpretations of the term ‘generic’ and since, until now, no rigorous proof of genericity for any use of the term has been offered, I thought it worthwhile to investigate in what sense, if any, this ‘folklore theorem’ is valid. Contrary to the prevailing intuition, linearizable systems fail to be dense in any topology for which evaluation of vector fields at points is a continous operation. This includes, for example, both the weak and strong C^∞-topologies.What, then, are the global properties of the class of linearizable systems? In this note, it is also shown that this class has no interior points in any reasonable, complete topology making feedback operations continuous. In particular, linearizable systems in R² are far from being structurally stable in the weak C^∞-topology. In the other hand, one of the main results presented here is that globally linearizable systems are open in the strong, or Whitney, topology. Thus, globally linearizable planar systems are structurally stable. If the well-known local conditions actually implied global linearizability, this result would be trivial. Recent work of Boothby shows, however, that global linearization is far more subtle. For this reason, in the course of the proof it is necessary to modify the existing global linearization criteria. In particular, a ‘nonvanishing’ criterion for irjectivity of smooth maps on R″, sharpening the so-called ‘ratio condition’, is presented. This result appears to be of independent interest and leads to improvements of the global linearization condition in R″, due to Hunt, Su and Meyer, and of the global non-linear observability conditions in R″, due to Kou, Elliott and Tarn.     

Adaptive control of linearizable systems

The authors give some initial results on the adaptive control of minimum-phase nonlinear systems which are exactly input-output linearizable by state feedback. Parameter adaptation is used as a technique to make robust the exact cancellation of nonlinear terms, which is called for in the linearization technique. The application of the adaptive technique to control of robot manipulators is discussed. Only the continuous-time case is considered; extensions to the discrete-time and sampled-data cases are not obvious     

Low-sensitivity optimal feedback control for linear discrete-time systems

The problem of reducing sensitivity of discrete-time systems to parameter variations is considered. State and output feedback gains are obtained to minimize a quadratic criterion which includes sensitivity functions. Necessary conditions for optimality are derived and numerical examples are discussed.     

A constrained optimal control problem

In this note the problem of optimizing a standard quadratic cost functional subject to a hard constraint is solved (i.e., the poles of the resulting system must lie in a specified region of the left-halfs-plane). This can he viewed as a generalization of the problem of finding the optimal controller that yields a system with specified relative stability. The problem is phrased in terms of a design procedure using coprime fractional representations.     

Stable adaptive control of feedback linearizable time-varying non-linear systems with application to fault-tolerant engine control

Stable indirect and direct adaptive controllers are presented for a class of input–output feedback linearizable time-varying non-linear systems. The radial basis function neural networks are used as on-line approximators to learn the time-varying characteristics of system parameters. Stability results are given in the paper, and the performance of the indirect and direct adaptive schemes is demonstrated through a fault-tolerant engine control problem where the faults are naturally time-varying.     

The pseudospectral Legendre method for discretizing optimal controlproblems

This paper presents a computational technique for optimal control problems including state and control inequality constraints. The technique is based on spectral collocation methods used in the solution of differential equations. The system dynamics are collocated at Legendre-Gauss-Lobatto points. The derivative x˙(t) of the state x(t) is approximated by the analytic derivative of the corresponding interpolating polynomial. State and control inequality constraints are collocated at Legendre-Gauss-Lobatto nodes. The integral involved in the definition of the performance index is discretized based on the Gauss-Lobatto quadrature rule. The optimal control problem is thereby converted into a mathematical programming program. Thus existing, well-developed optimization algorithms may be used to solve the transformed problem. The method is easy to implement, capable of handling various types of constraints, and yields very accurate results. Illustrative examples are included to demonstrate the capability of the proposed method, and a comparison is made with existing methods in the literature     

A nonsmooth Newton’s method for control-state constrained optimal control problems

We investigate optimal control problems subject to mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer–Burmeister function the minimum principle is transformed into an equivalent nonlinear and nonsmooth equation in appropriate Banach spaces. This nonlinear and nonsmooth equation is solved by a nonsmooth Newton’s method. We will show the local quadratic convergence under certain regularity conditions and suggest a globalization strategy based on the minimization of the squared residual norm. A numerical example for the Rayleigh problem concludes the article.     

Geometric properties of linearizable control systems

In this paper the distributions associated with a non-linear system of the form $$\frac{{dx}}{{dt}} = f(x) + \sum\limits_{\alpha = 1}^m {u_\alpha (t)g_\alpha (x)} ,f(0) = 0andx \in U_0 \subset R^n$$ are studied in relation to nonlinear state feedback $$u(x,v) = \hat a(x) + \hat S(x)v$$ withâ, u, v ∈ R ^m and ? a nonsingularm×m matrix withâ, ? functions ofx. Bothf andg are vector fields onU ₀, generally assumed to be real analytic. Two nested families of distributions {G ^j } and {M ^j } associated with the system are examined with emphasis on generic points ofU ₀, where it is shown that the usual conditions for feedback linearizability contain some redundancy. A characterization of state linearizability in terms of invariant factors of the equivalent linear form are given, and a criterion in terms of the distributions for a type of partial linearization is found.     

Robust feedback model predictive control of constrained uncertain systems

We propose a novel procedure for the solution to the problem of robust model predictive control (RMPC) of linear discrete time systems involving bounded disturbances and model-uncertainties along with hard constraints on the input and state. The RMPC (outer) controller – responsible for steering the uncertain system state to a designed invariant (terminal) set – has a mixed structure consisting of a state-feedback component as well as a control-perturbation. Both components are explicitly considered as decision variables in the online optimization and the nonlinearities commonly associated with such a state-feedback parameterization are avoided by adopting a sequential approach in the formulation. The RMPC controller minimizes an upper bound on an H₂/H_∞-based cost function. Moreover, the proposed algorithm does not require any offline calculation of (feasible) feedback gains for the computation of the RMPC controller. The optimal Robust Positively invariant set and the inner controller – responsible for keeping the state within the invariant set – are both computed in one step as solutions to an LMI optimization problem. We also provide conditions which guarantee the Lyapunov stability of the closed-loop system. Numerical examples, taken from the literature, demonstrate the advantages of the proposed scheme.     

Dynamic programming for constrained optimal control of discrete-time linear hybrid systems

In this paper we study the solution to optimal control problems for constrained discrete-time linear hybrid systems based on quadratic or linear performance criteria. The aim of the paper is twofold. First, we give basic theoretical results on the structure of the optimal state-feedback solution and of the value function. Second, we describe how the state-feedback optimal control law can be constructed by combining multiparametric programming and dynamic programming.     

Output-feedback finite-time stabilization of disturbed feedback linearizable nonlinear systems

A novel methodology for designing multivariable High-Order Sliding-Mode (HOSM) controllers for disturbed feedback linearizable nonlinear systems is introduced. It provides for the finite-time stabilization of the origin of the state-space by using output feedback. Only the additional assumptions of algebraic strong observability and smooth enough matched disturbances are required. The control problem is solved in two consecutive steps: firstly, designing an observer based on the measured output and, secondly, designing of a full-state controller computed from a new virtual output with vector relative degree. The introduced notion of algebraic strong observability allows recovering the state of the system using the measured output and its derivatives. By estimating the required derivatives through the HOSM differentiator, a finite-time convergent observer is constructed.     

An interior penalty method for inequality constrained optimal control problems

This paper presents a penalty function approach to the solution of inequality constrained optimal control problems. The method begins with a point interior to the constraint set and approaches the optimum from within, by solving a sequence of problems with only terminal conditions as constraints. Thus, all intermediate solutions satisfy the inequality constraints. Conditions are given which guarantee that the un "constrained" problems have solutions interior to the constraint set and that in the limit these solutions converge to the constrained optimum. For linear systems with convex objective and concave inequalities, the unconstrained problems have the property that any local minimum is global. Further, under these conditions, upper and lower bounds in the optimum are easily available. Three test problems are solved and the results presented.     

A canonical structure for constrained optimal control problems

We consider a general class of optimal control problems with regional pole and controller structure constraints. Our goal is to show that for a fairly general class of regional pole and controller structure constraints, such constrained optimal control problems can be transformed to a new one with a canonical structure. A three-step transformation procedure is used to achieve our goal, which essentially amounts to repeated augmentations of plant dynamics and repeated reductions of the controller. The transformed problem is one of the standard optimal static output feedback with a decentralized and repeated structure.     

Stabilizing low complexity feedback control of constrained piecewise affine systems

Piecewise affine (PWA) systems are powerful models for describing both non-linear and hybrid systems. One of the key problems in controlling these systems is the inherent computational complexity of controller synthesis and analysis, especially if constraints on states and inputs are present. In addition, few results are available which address the issue of computing stabilizing controllers for PWA systems without placing constraints on the location of the origin.This paper first introduces a method to obtain stability guarantees for receding horizon control of discrete-time PWA systems. Based on this result, two algorithms which provide low complexity state feedback controllers are introduced. Specifically, we demonstrate how multi-parametric programming can be used to obtain minimum-time controllers, i.e., controllers which drive the state into a pre-specified target set in minimum time. In a second segment, we show how controllers of even lower complexity can be obtained by separately dealing with constraint satisfaction and stability properties. To this end, we introduce a method to compute PWA Lyapunov functions for discrete-time PWA systems via linear programming. Finally, we report results of an extensive case study which justify our claims of complexity reduction.