Unstable Zero Dynamics Research Articles

Output Stabilization for Nonminimum-Phase Systems Using Observer Backstepping

The article considers stabilization of the output programmed trajectory in nonminimum-phase dynamical systems of a special kind with unstable linear zero dynamics. An algorithm is proposed for the construction of a stabilizing control by observer backstepping. The backstepping method is generalized to nonminimum-phase nonlinear dynamical systems.

Path-following or reference tracking?: An answer relaxing the limits to performance

In path-following the control objective is to force the output to follow a geometric path without a timing law assigned to it. We highlight a fundamental difference between the path-following and the standard reference-tracking by demonstrating that performance limitation due to unstable zero-dynamics can be removed in the path-following problem.

Adaptive flight control design for nonlinear missile

The focus of this work is to investigate the possible benefits of modern nonlinear control design methods for missile autopilot design. The basic requirement for an autopilot is firstly fast responses because of the short amount of time involved in the end-game. A slow response could easily cause a miss if the target has the capacity to perform high-g evasive manoeuvers. Secondly, minimum error is an obvious requirement if the missile is to achieve a kill. Finally, robustness to model uncertainties is important in order for the missile to achieve its objective in the physical environment. In the first part of this paper input–output approximate linearisation of a nonlinear missile has been studied. A method for controlling the nonlinear system that is input–output linearisable is examined that retains the order of the system in the linearisation process, hence producing a linearised system with no internal or zero dynamics. Desired tracking performance for lateral acceleration of the missile is achieved by using a nonlinear control law that has been derived by selecting the lateral velocity as the linearisation output. Simulation results are shown to exercise the final design and show that the linearisation and controller design are satisfactory. Then an adaptive nonlinear controller is designed that guarantees tracking performance when the uncertain parameters vary within a stability bound. An autopilot combining an indirect adaptive controller with approximate feedback linearisation is proposed in order to achieve asymptotic tracking. Adaptation is introduced to enhance closed-loop robustness, while approximate feedback linearisation is used to overcome the problem of unstable zero dynamics. Computer simulations show that this approach offers a possible autopilot design for nonlinear missiles with uncertain parameters.

Generalized normal form and stabilization of non-linear systems

This paper considers the normal form of non-linear control systems. First we propose a generalized relative degree (relative degree vector) for non-linear single (respectively, multiple) input control system, which is called the point relative degree (respectively, point relative degree vector). For the systems without output, the concepts of essential relative degree (respectively, essential relative degree vector) and the essential point relative degree (respectively, essential point relative degree vector) are defined. Unlike the classical definition which requires regularity, the point relative degree (vector) is always well defined. Using these new concepts the generalized normal form is obtained. Its relationship with the Jacobian linearization is investigated. Using it, a straightforward computation algorithm is provided to achieve the generalized normal form. Based on the generalized normal form we prove that with an additional condition, if the zero-dynamics is stable the overall system is stabilizable by using pseudo-linear state feedback control. For the systems under generalized normal form with unstable zero dynamics, the centre manifold approach is applied. It is shown that the stabilization technique via a designed centre manifold is still applicable to this kind of general non-linear control system.

An interpolation strategy for discrete-time bilinear MPC problems

Input-output (I-O) feedback linearization suffers from a number of restrictions which have limited its use in model-based predictive control. Some of these restrictions do not apply to the case of bilinear systems, but problems with input constraints and unstable zero dynamics persist. This paper overcomes these difficulties by means of an interpolation strategy. Involved in this interpolation is a feasible and stabilizing trajectory, which is computed through the use of invariant feasible sets, and a more aggressive trajectory, which can be chosen to be either the unconstrained optimal trajectory or any alternative one.

ADAPTIVE FLIGHT CONTROL DESIGN FOR NONLINEAR MISSILE

An autopilot combining an indirect adaptive controller with approximate feedback linearisation is proposed in order to achieve asymptotic tracking. Adaptation is introduced to enhance closed-loop robustness, while approximate feedback linearisation is used to overcome the problem of unstable zero dynamics. Computer simulations show that this approach offers a possible autopilot design for non-linear missiles with uncertain parameters.

Controllability evaluation for nonminimum phase‐processes with multiplicity

Two approaches to determine the process inherent limitation on switchability imposed by unstable zero dynamics have been shown. The first solves the problem analytically and results in an expression for nonminimum phase (NMP) factor, which dictates the system inherent limitation. The method is restricted to two classes of rather small systems and also cannot be applied to models that have both NMP and minimum phase (MP) regions. The second which overcomes this restriction uses a methode numerically optimizing the dynamic model in order to find the integral square error (ISE) optimal closed-loop response. Due to the fact that the process is NMP, some precautions were taken when the problem was formulated in order to avoid finding the unstable inverse. The method was also used to study the system's inherent limitations for steps over the maximum in a process with input multiplicities. The nonlinearities do have an effect, but the extent will, of course, vary from problem to problem. The problems are still small, and there is some way to go before we have a method for problems of realistic size.

Control of Flexible-Link Multiple Manipulators

In this paper, we consider the object trajectory tracking control for flexible-link cooperating manipulators. In particular, we develop a stable inversion control law which is commonly known as the inverse dynamics control or the computed torque method for rigid manipulators. Difficulties in applying this method to the control of flexible link manipulators are due to the fact that the inverse dynamics system is generally unstable because of the inherently unstable zero dynamics. As such, bounded actuator torques cannot be guaranteed. For multiple manipulators handling a common object, there are more actuators than the degrees of freedom of the system. Through decomposing the manipulator end-effector wrenches into resultant and internal force components, control laws are derived such that the internal forces are used to stabilize the system zero dynamics. Consequently, nonlinear inversion control can be applied for the object trajectory tracking control. Numerical simulations are performed to illustrate the performance of the control strategy developed in the paper.

Nonlinear Control VIA Generalized Feedback Linearization Using Neural Networks

ABSTRACTA novel approach to nonlinear control, called Generalized Feedback Linearization (GFL), is presented. This new strategy overcomes one important drawback of the well known Feedback Linearization strategy, in the sense that it is able to handle a broader class of nonlinear systems, namely those having unstable zero dynamics. It is shown that the use of a nonlinear predictor for the system output is a key feature in the derivation of the control strategy. For certain types of systems this predictor can be found as a nonlinear function of the system input and output, allowing an output feedback control solution. The use of Artificial Neural Networks (ANN) to directly parameterize the predictor of the controlled variable when an explicit model for the system is not available, is investigated via computer simulations. This approach is based on the functional approximation capability of multi layer ANN.

Feedback limitations in nonlinear systems: from Bode integrals to cheap control

Feedback limitations of nonlinear systems are investigated using the cheap control approach. The main result is that in the limit, when the control effort is free, the smallest achievable L/sub 2/ norm of the output is equal to the least amount of control energy (L/sub 2/ norm) needed to stabilize the unstable zero dynamics. This nonlinear result is structurally similar to an earlier linear result by Qiu and Davison (1993), which, in turn, is connected with a Bode-type integral derived by Middleton (1991).

Further results on almost disturbance decoupling with global asymptotic stability for nonlinear systems

As a complement to some new breakthroughs on global almost disturbance decoupling problem with stability for nonlinear systems, in a recent note, we identified a class of unstable zero dynamics that are allowed to be affected by disturbances. The class of the unstable zero dynamics identified in that note is linear and have all the poles at the origin. In this paper, we enlarge such a class of zero dynamics to include any linear system with all its poles in the closed left-half plane. The condition on the way the disturbance affects this part of zero dynamics is also identified. This enlargement is due to a new scaling technique that views each pair of jw axis zeros as a “generalized integrator” and transforms the zero dynamics into a number of chains of “generalized integrators”.

Block Control Principle for Mechanical Systems

In this paper, a generalized design procedure for sliding mode control of nonlinear mechanical systems is proposed. The design approach combines the essential idea of the block control principle, utilizing some of the components of the state vector as a virtual control, with the basic concept of zero dynamics. For mechanical systems governed by a set of interconnected second-order equations, the block control principle cannot be directly applied. To facilitate the controller design, we assume that control systems can be transformed into a regular form consisting of second-order equations. The proposed design approach consists of reducing the original plant into the regular form, constructing a switching manifold, and enforcing sliding mode in the manifold such that the reduced order system in sliding mode has desired dynamics. Stabilization of the mechanical system with unstable zero dynamics is taken into consideration. It is shown that the approach has the advantage of decomposing the original problem into subproblems of lower dimensions, and each of them can be handled independently. As an example, control of a rotational inverted pendulum system is examined. The performance of the proposed approach is validated by both numerical and experimental results. [S0022-0434(00)01601-4]

Almost disturbance decoupling with global asymptotic stability for nonlinear systems with disturbance-affected unstable zero dynamics

The problem of global almost disturbance decoupling problem with stability for nonlinear system is revisited and is shown to be solvable for a class of nonlinear systems whose zero dynamics contains a chain of integrators affected by disturbance. This result complements some recent work on the topic, where unstable zero dynamics is not allowed to be affected by the disturbances.

Lyapunov stability analysis of daisy chain control allocation

A demonstration that feedback control of systems with redundant controls can be reduced to feedback control of systems without redundant controls and control allocation is presented. It is shown that control allocation can introduce unstable zero dynamics into the system, which is important if input/output inversion control techniques are utilized. The daisy chain control allocation technique for systems with redundant groups of controls is also presented. Sufficient conditions are given to ensure that the daisy chain control allocation does not introduce unstable zero dynamics into the system. Aircraft flight control examples are given to demonstrate the derived results.

Global almost disturbance decoupling with stability for non minimum-phase single-input single-output nonlinear systems

The so-called problem of almost disturbance decoupling with internal stability (ADDPS) is the following one. Given a system and an (arbitrarily small) number γ > 0, find a feedback law yielding a closed loop system which is stable and in which the gain (in the L 2 sense) between the exogenous input and the regulated output is less than or equal to γ. The complete solution of this problem has been known since a long time in the case of linear systems. In the case of nonlinear systems, the only global results available so far in the literature were about SISO systems having an asymptotically stable zero dynamics. In this paper, a new set of results are presented, dealing with nonlinear SISO systems having a possibly unstable zero dynamics, which include the (general) class of linear SISO systems as a special case.

Stable inversion of nonlinear non-minimum phase systems

Output tracking control of non-minimum phase systems is a highly challenging problem encountered in the control of flexible manipulators, space structures, and elsewhere. Classical inversion provides exact output tracking but leads to internal instability, while recent nonlinear regulation provides stable asymptotic tracking but admits large transient errors. As a first step to solve this problem, this paper addresses the stable inversion of non-minimum phase nonlinear systems. Using the notions of zero dynamics and stable/unstable manifolds, an invertibility condition is established for a class of systems. A stable but non-causal inverse is obtained offline that can be incorporated into a stabilizing controller for dead-beat output tracking. This inverse contrasts with the causal inverse proposed by Hirschorn where unstable zero dynamics result in unbounded inverse solutions. Our results reduce to those of Hirschorn for minimum phase systems, however. In a numerical example, the stable inverse has achiev...

Inverse Force and Motion Control of Constrained Elastic Robots

We treat the question of position and force control of a three-axis elastic robotic system on a constraint surface based on nonlinear inversion of an input-output map and linear feedback stabilization. Unlike the rigid robots, the feedback linearizing control of end point motion gives rise to unstable zero dynamics. Instability of zero dynamics is avoided by controlling a parameterized output vector corresponding to a point close to the end point of the arm. Zero dynamics are stable or almost stable as long as the parameter in the output vector does not exceed a critical value. Using the inverse controller, the control of the force and the position of the end point is possible while the end effector moves on the constraint surface. However, this excites the elastic modes. For the final capture of the terminal state and vibration suppression, a linear stabilizer is designed. Simulation results are presented to show that in the closed-loop system trajectory and force control on the constraint surface is accomplished.

Sliding mode force, motion control, and stabilization of elastic manipulator in the presence of uncertainties

AbstractThis article considers the question of position and force control of three‐link elastic robotic systems on a constraint surface in the presence of robot parameter and environmental constraint geometry uncertainties. The approach of this article is applicable to any multi‐link elastic robot. A sliding mode control law is derived for the position and force trajectory control of manipulator. Unlike the rigid robots, sliding mode control of an end point gives rise to unstable zero dynamics. Instability of the zero dynamics is avoided by Controlling a point that lies in the neighborhood of the actual end point position. The sliding mode controller accomplishes tracking of the end‐effector and force trajectories on the constrained surface; however, the maneuver of the arm causes elastic mode excitation. For point‐to‐point control on the constraint surface, a stabilizer is designed for the final capture of the terminal state and vibration suppression. Numerical results are presented to show that in the closed‐loop system position and force control is accomplished in spite of payload and constraint surface geometry uncertainty. © 1995 John Wiley & Sons, Inc.

Approximate output tracking for nonlinear non‐minimum phase systems with an application to flight control

AbstractAn unstable zero‐dynamics is a known obstruction to inducing exact asymptotic tracking for an open set of output trajectories with internal stability. This paper proposes a procedure for achieving approximate tracking for a nonlinear system whose linearization possesses real right‐half plane zeros. The method is guaranteed to remove the right‐half plane zeros while the other zeros remain in their previous location; moreover, it provides information on the class of signals for which good approximate tracking can be obtained. With other methods, the right‐half plane zeros are eliminated but the final location of the remaining zeros is not known a priori. The design procedure is illustrated on a trajectory control problem of an aircraft in rapid manoeuvres. Simulations illustrate the computations involved and show that precise lateral and longitudinal manoeuvres can be performed, even in the presence of uncertainties.

Variable structure trajectory control of an elastic robotic arm

AbstractThis article introduces a variable structure system scheme to control the end effector trajectory of a two‐link flexible robotic arm. Because control of the actual tip position leads to unstable zero dynamics, control of points in the neighborhood of the tip is considered. An output is chosen as the sum of the joint angle and tip elastic deformation times a constant factor for each link. For the chosen output, a discontinuous output control law is derived based on the variable structure theory. The control law thus derived accomplishes the desired trajectory tracking of the output. A linear stabilizer is designed using the pole assignment technique for the final capture of the terminal state and stabilization of the elastic modes. Simulation results are presented to show that in the closed‐loop system large maneuvers can be performed in the presence of payload uncertainty, thereby exhibiting the robustness of the controller.