A separation principle for distributed dissipative bilinear systems

The problem of input-output stabilization is generalized from finite dimensional bilinear systems to a class of distributed bilinear systems. The solution is based on the nonlinear feedback stabilization results given previously by the authors (1999), with the observers presented by the same authors (1997) in another work.     

A separation principle for bilinear systems with dissipative drift

Input-output stabilization of a class of bilinear systems using a nonlinear stabilizing feedback control law together with an observer, is studied. In some cases, it can be proved that the result is a globally asymptotically stable system     

A separation principle for a class of non-UCO systems

This paper introduces a new approach to output feedback stabilization of single-input-single-output systems which, unlike other techniques found in the literature, does not use quasilinear high-gain observers and control input saturation to achieve separation between the state feedback and observer designs. Rather, we show that by using nonlinear high-gain observers working in state coordinates, together with a dynamic projection algorithm, the same kind of separation principle is achieved for a larger class of systems which are not uniformly completely observable. By working in state coordinates, this approach avoids using knowledge of the inverse of the observability mapping to estimate the state of the plant, which is otherwise needed when using high-gain observers to estimate the output time derivatives.     

Local input-output decoupling of discrete-time non-linear systems

A local treatment of the block input-output decoupling problem is given for discrete-time non-linear control systems. The major tools employed are the invariant and locally-controlled invariant distributions that have recently been extended to the discrete-time domain. A sufficient condition for the solvability of the problem with local stability about an equilibrium point is also given.     

Local asymptotic approximation of non-linear control systems!

Let g₁, …, g_m be smooth vector fields on R^d. Using a canonical linear rescaling of time and space coordinates, the control system is transformed into a system, on the same state space, which is arbitrarily close to a nilpotent one. This approximation takes place in the C^∞ topology, and is particularly useful for the study of local properties of (*) which depend on the high-order variational structure of the vector fields g_i.     

Local controllability of non-linear systems: An example

An example of a locally controllable nonlinear system on R³ is given. The system is of the type with f, g analytic vector fields and u bounded. The fields f and g are such that the dimension of the vector space spanned at x₀ by the Lie brackets which contain g an odd number of times is 2.     

Local and global optimal control for non-linear systems using two-level methods

The recent co-state coordination algorithm for the two-level optimization of nonlinear discrete dynamical systems with separable quadratic cost functions is specialized to ensure the satisfaction of both necessary and local sufficiency conditions. For an important subclass of the problems, global sufficiency can be guaranteed. This is demonstrated on a simple example where previous methods have yielded worse local optima.     

A separation principle for the stabilization of a class ofnonlinear systems

It is shown that the performance of a globally bounded partial state feedback control of a certain class of nonlinear systems can be recovered by a sufficiently fast high-gain observer. The performance recovery includes recovery of asymptotic stability of the origin, the region of attraction, and trajectories     

Feedback estimation systems and the separation principle of stochastic control

Estimation systems with a feedback link have a structure similar to that encountered in the study of stochastic feedback control systems. A feedback estimation algorithm of a specific form is derived in this paper. This algorithm is shown to be superior to one based upon the separation principle of stochastic control and inferior to one employing a different feedback signal structure. The observed differences in performance of these algorithms give insight into a basic limitation on control policies derived from formal application of the separation principle.     

Bounded smooth state feedback and a global separation principle for non-affine nonlinear systems

This paper develops sufficient conditions for a general nonlinear control system to be locally (resp. globally) asymptotically stabilizable via smooth state feedback. In particular, it is shown that as in the case of affine systems, this is possible if the unforced dynamic system of ∑₁ is Lyapunov stable and appropriate controllability-like rank conditions are satisfied. Our results incorporate a series of well-known stabilization theorems proposed in the literature for affine control systems and extend them to nonaffine nonlinear control systems.     

Stochastic control of discrete systems: a separation principle forWiener and polynomial systems

A new separation principle is established for systems represented in discrete frequency-domain Wiener or polynomial forms. The LQG or H ₂ optimal controller can be realized using an observer based structure estimating noise free output variables that are fed back through a dynamic gain control block. Surprisingly, there are also two separation principle theorems, depending upon the order in which the ideal output optimal control and the optimal observer problems are solved     

Set-point changing for non-linear systems

In many control systems, we sometimes face a situation such that we have to change the set-point of the system in the course of operation. Theoretically ‘set-point changing’ means letting the system be at another equilibrium point. If the system is linear, this operation presents no problem because the properties of linear systems are always global. However, in non-linear systems, this type of global operation has many unsolved problems. We only know experimentally that, if the system is locally stable at every equilibrium point, then we can slowly change the set-point without exciting unstable motion. This paper proposes a theoretically guaranteed method of changing the set-point of non-linear systems and showing a sufficient condition needed for this operation.     

High-gain observers for non-linear systems

State estimation for non-linear dynamic systems is discussed. A high-gain injection from the output variables is used to attenuate to any desired degree, the effect of the non-linear terms on the estimation errors, which can be made arbitrarily small, from a particular observable form. Some Liapunov arguments are used to study the stability properties of the resulting error dynamics. It is shown that any completely observable multi-input multi-output non-linear system, not necessarily linear in the input variables, can be transformed, by means of a change of coordinates depending on the input variables, in such an observable form. In particular, the solvability of partial differential equations is not needed for the design of this transformation.     

Observers for Lipschitz non-linear systems

In an interesting paper R. Rajamani and Y. M. Cho have proposed a systematic methodology to design observers. They have introduced a new problem: relation between distance to unobservability and convergence of 'Luenberger-like' observers. A result for the convergence of the observer has also been given. They have presented a quantity denoted by i, claimed as the distance to unobservability. We show that this number is not actually the distance from the system to the set of unobservable systems. Moreover, the result for the observer is incorrect. We provide a counterexample for the result of convergence. In this paper results for convergence are obtained, with additional strengthened hypothesis, to correct Rajamani and Cho's result. The results are used to design an observer for a, now classical, single-link flexible robot joint. The behaviour of the observer to noisy output is quite satisfactory.     

Finite horizon quadratic optimal control and a separation principle for Markovian jump linear systems

In this note, we consider the finite-horizon quadratic optimal control problem of discrete-time Markovian jump linear systems driven by a wide sense white noise sequence. We assume that the output variable and the jump parameters are available to the controller. It is desired to design a dynamic Markovian jump controller such that the closed-loop system minimizes the quadratic functional cost of the system over a finite horizon period of time. As in the case with no jumps, we show that an optimal controller can be obtained from two coupled Riccati difference equations, one associated to the optimal control problem when the state variable is available, and the other one associated to the optimal filtering problem. This is a principle of separation for the finite horizon quadratic optimal control problem for discrete-time Markovian jump linear systems. When there is only one mode of operation our results coincide with the traditional separation principle for the linear quadratic Gaussian control of discrete-time linear systems.     

A separation principle for non-UCO systems: the jet engine stall and surge example

The problem of controlling surge and stall in jet engine compressors is of fundamental importance in preventing damage and lengthening the life of these components. In this theoretical study, we illustrate the application of a novel output feedback control technique to the Moore-Greitzer mathematical model (1986) for these two instabilities assuming that the plenum pressure rise is measurable. This problem is particularly challenging since the system is not uniformly completely observable and, hence, none of the output feedback control techniques found in the literature can be applied to recover the performance of a full state feedback controller.     

Practical exponential stability of perturbed triangular systems and a separation principle

In this paper, we present a practical stability result for perturbed dynamic systems depending on a parameter and we study the practical exponential stability of perturbed triangular systems. These results are applied to show that a separation principle for nonlinear uncertain systems can be achieved and which considers practical global uniform exponential stability. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

State feedback for non-linear control systems

A theory of static state feedback for non-linear discrete-time systems is developed. The theory applies to non-linear systems possessing a recursive representation of the form x _k+1 =?(x _k , u_k ), where ? is a continuous function, and it deals with the construction of continuous state feedback functions that internally stabilize a given system. The theory yields an explicit method for the computation of stabilizing feedback functions, and several examples of the computation of such functions are provided.     

High-gain filters for non-linear stochastic systems

The use of high-gains in the filtering problem for non-linear stochastic systems is studied. It is shown that a high-gain filter has the same structure as a Luenberger observer. The use of high-gains is studied in the cancellation of non-linearities in order to simplify the filter design for non-linear systems. Singular perturbation techniques are used to show how the error dynamics reaches stable equilibrium in a very fast transient time, ensuring that the slow dynamics of the filter are just those of the non-linear stochastic system.     

Robust stabilization for non-linear discrete-time systems

This paper deals with the problem of robust stabilization for linear discrete-time systems under non-linear perturbations. Without limitation to single-input systems and restriction on the structures of both the positive definite matrix and the corresponding variable, a new sufficient condition is presented which guarantees the stability of the closed-loop system and at the same time maximizes the bound on the non-linearity. In addition, our main result is extended to discrete time-delay systems. All the conditions are given within the framework of linear matrix inequalities. The effectiveness of the methods is illustrated with three examples.