Velocity-Based Methods


Nonlinear dynamic systems are everywhere, but tools for the analysis/design of nonlinear systems are poorly developed.  No system is, in reality, linear but methods for linear systems are well developed and a wealth of practical experience with them has been accumulated.  Hence, it is attractive to adopt a divide and conquer approach whereby the analysis/design task for a nonlinear system is decomposed into a number of linear tasks.

Conventional gain-scheduling

Gain-scheduling is a divide and conquer approach for the design of nonlinear control systems which has been applied in fields ranging from aerospace to process control.  The conventional gain-scheduling design approach typically involves

  1. linearise the nonlinear plant about a number of equilibrium points
  2. design a linear controller for each of the plant linearisations
  3. combine the linear controllers to obtain a nonlinear controller

(See survey of gain-scheduling methods).

Limitations of conventional gain-scheduling

Conventional gain-scheduling controllers are generally confined to near equilibrium operation (because they are designed on the basis of the plant equilibrium linearisations).  Moreover, although gain-scheduled controllers are widely applied, the underlying theory is poorly developed. 

Velocity-based gain-scheduling

The velocity-based framework resolves many of the deficiencies of conventional gain-scheduling. 
A linear system (the 'velocity-based linearisation') is associated with every operating point of a nonlinear system (not just the equilibrium points).   A family of velocity-based linearisations is therefore associated with the nonlinear system.  This family embodies the entire dynamics of the nonlinear system and so is an alternative representation.  It is emphasised that this representation is valid globally and does not involve any restriction to the vicinity of the equilibrium points.  Large transients and sustained non-equilibrium operation can both be accommodated.  This suggests the following velocity-based gain-scheduling design procedure.

  1. Determine the velocity-based linearisation family of the plant
  2. Design a linear controller for each member of the plant family. 
  3. Realise a nonlinear controller with velocity-based linearisation family corresponding to the linear controller family
    designed at step 2.

The gain-scheduled controller is not inherently confined to near equilibrium operation or subject to any slow variation constraint.  As a concrete illustration of this lack of restrictions, the velocity-based gain-scheduling approach can be used to design a dynamic inversion controller which is valid globally and does not involve any slow variation constraints whatsoever.   This freedom is achieved while still retaining the divide and conquer approach and continuity with linear methods which is the principal advantage of the conventional gain-scheduling approach.

An extended summary of velocity-based modelling and control is also available (.pdf, 55Kb.  Requires Adobe Acrobat Reader 3.1 or better to view).

Input-Output Linearisation/Dynamic Inversion

The velocity-based gain-scheduling approach is quite general and directly supports the design of feedback configurations for which the closed-loop dynamics are nonlinear. Dynamic inversion corresponds to the special case where the closed-loop dynamics are linear. The velocity-based approach to dynamic inversion is quite distinct from (and indeed in many ways complementary to) standard input-output linearisation techniques based on differential geometric methods.  In particular, the velocity-based approach

  • is a direct generalisation to nonlinear systems of the classical frequency-domain pole-zero inversion approach. (cf. conventional methods are a generalisation of Silverman's work on state-space inversion techniques).
  • requires, in general, only a measurement or estimate of the scheduling variable, r . Frequently, r depends on only a small number of elements of the state and/or input vectors. Indeed, in the case of purely linear systems, there is no scheduling variable and, consequently, plant measurements are not required to implement the pole-zero inverse. (cf. the full state-feedback required in conventional approaches, even in the linear case)
  • decomposes the nonlinear design task into a number of straightforward linear sub-problems; that is, the methodology supports the divide and conquer philosophy and maintains continuity with well established linear methods. In this sense, it is closely related to the gain-scheduling methodology. However, it is emphasised that the velocity-based approach does not necessitate a slow variation requirement.

LPV gain-scheduling & velocity-based methods

In addition to velocity-based gain-scheduling methods, a number of other approaches have recently been developed.   These are widely referred to as LPV gain-scheduling methods owing to their use of a quasi-LPV/LPV representation of the plant and controller.  A considerable body of results now exists relating to the design of controllers for plants which are in LPV or quasi-LPV form. However, the literature typically takes the existence of a plant in LPV/quasi-LPV form as its starting point, largely neglecting the critical issue of how general nonlinear dynamics might be transformed to LPV/quasi-LPV form.   It is important to emphasise the importance of this issue since the rigorous basis of LPV methods is removed if the plant is not placed in LPV/quasi-LPV form using soundly-based techniques.

Apparently lacking practical, generally applicable methods for carrying out such a transformation, a number of ad hoc approaches have been proposed in the literature. Although these might lead to acceptable control designs on some occasions, this need not be the case in general.  E.g. for one such popular method at least it is straightforward to devise counter-examples where the control design fails (closed-loop is unstable). 

The velocity-based framework provides very general and soundly-based methods for transforming systems into LPV/quasi-LPV form.

MATLAB velocity-based gain-scheduling demo

Frequently Asked Questions