Feedback linearization: pros and cons. Pros: Used for both stabilization and tracking control problems, SISO and MIMO systems. Successfully applied to a ...
Local methods for nonlinear control: a survey
Gianluca Bontempi IRIDIA Universite´ Libre de Bruxelles, Belgium
http://iridia.ulb.ac.be/˜gbonte/
Outline
Linearization Gain scheduling Feedback linearization Fuzzy (LMN) controllers 1. Takagi Sugeno controller 2. Fuzzy gain scheduler controller 3. Fuzzy self-tuning controller Lazy learning
Notation
Nonlinear autonomous system
Equilibrium
Trajectory
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Linearization
Equilibrium about a point: linear time-invariant dynamics
(LTI) Equilibrium about a trajectory: linear time-varying dynamics
(NL non aut.)
(LTV)
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Linearization and stability
about an equilibrium point: Lyapunov’s linearization method – linearized system strictly stable
equilibrium point asymptotically stable for the
nonlinear system – linearized system unstable
equilibrium point unstable for the nonlinear system
– linearized system is marginally stable
one cannot conclude anything
about a trajectory: linearization methods for NL non autonomous systems – linearized system uniformly asymptotically stable
equilibrium point of the original
non-autonomous system uniformly asymptotically stable – no relation between the instability of LTV and that of the nonlinear system
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Shortcomings of Linearization
Control design based on the linearized dynamics could have no good performance or be not stabilizing when operating away from the equilibrium or trajectory
Equilibrium points or trajectories must be known in advance. This knowledge is often not available.
Gain scheduling to address the restrictions of linearization
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Gain scheduling
Family of equilibrium points
i.e.
parametrized by the scheduling variable . Choice of scheduling variable – Exogenous variables: state variables in a more complex model representation – State variables – Reference state trajectories: assumption that the system state is near to the reference command.
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Gain scheduling design
Frozen parameter design: controllers designed at a finite number of operating points indexed by the set
Variable
used to design the nearest operating point
Scheduling of controllers – Discontinuous (switching) – Smooth interpolation
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Gain scheduling and stability (Shamma, 1988)
Linear Parameter Varying formalism
Time-varying closed loop dynamics
Frozen closed loop
stable for a set of :
Problem: frozen time stability does not imply time varying stability |x| |x|
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Gain scheduling and stability (Shamma, 1988) Assumption 1. The dynamics matrix continuous with constant
is bounded and globally Lipschitz
, i.e.
Theorem 1. Consider the closed loop linear system under the above assumption. Assume that at each instant (1)
is stable and (2) there exist
Under these conditions, given any
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and
such that
Considerations on gain scheduling Pros: Extend linear methods to non linear control Control on greater operating regions than single equilibrium Solve the problem of introducing time variations in the overall control systems. Cons: Linearization about equilibrium points only Designer must know a priori the distribution of the equilibrium points State of the nonlinear system assumed to be close to one of the equilibrium points
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Feedback linearization Idea: transform the nonlinear system model into a fully, or partially, linear model so that linear control techniques can be applied canonical form:
input-state linearization:
r differentiation
input-output linearization
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Feedback linearization and stability
No problems of stability for the canonical form and the input-state linearization
Input-output linearization decomposes dynamics into an external I/O part and an internal part, not observable
Difficult stability analysis of the internal dynamics
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Feedback linearization: pros and cons Pros: Used for both stabilization and tracking control problems, SISO and MIMO systems Successfully applied to a number of practical nonlinear control problems. Cons: It cannot be used for all nonlinear systems (singularity) State has to be measured No robustness is guaranteed in the presence of parameter uncertainty or unmodeled dynamics.
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Model based fuzzy control
1. Conventional Takagi Sugeno (Takagi & Sugeno, 1985)
gain scheduling on the state
2. Fuzzy gain scheduler (Palm & Rehfuess, 1997; Palm et al., 1997) the operating point
3. Fuzzy self-tuning
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approximate feedback linearization
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gain scheduling on
Conventional Takagi Sugeno (Takagi & Sugeno, 1985)
Fuzzy model: partition on the state variable domain if
then
Fuzzy controller if
then
Closed loop
nonlinear dynamics
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Stability of Takagi Sugeno (Tanaka & Sugeno, 1992) Close loop dynamics:
Phenomenon of interference (local controller
interacts with local model
)
Theorem 2 (Sufficient condition). The equilibrium of the TS fuzzy system is globally asymptotically stable if there exists a common positive definite matrix P such that
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Design of Takagi Sugeno (Tanaka, 1995)
Design problem: Given
find
theorem is satisfied. Iterative procedure 1. Find the controllers
that stabilize locally
2. Check necessary condition 3. Check of sufficient condition (LMI, different forms of P)
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such that the stability
Fuzzy gain scheduler
Fuzzy model: partition in the space of equilibrium points if
then
Fuzzy controller if
then
Closed loop
linear
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Fuzzy gain scheduler stability (Palm et al., 1997) Consider:
Let
be the unique solution of the Lyapunov equation
Theorem 3. The linear system (18) is asymptotically stable if
with 18
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Fuzzy gain scheduler design (Palm et al., 1997) Design
By putting
Compute the set of singular values for each design otherwise the gain 19
. If this condition is satisfied we have a stable
have to be redesigned over and over until it is satisfied. c 1998 G. Bontempi
Fuzzy self-tuning
Fuzzy model: partitioning on the state space if
then
Combination made at the modeling level and not at the controller level (LPV)
Linearization also in configurations which are far from the equilibrium locus System dynamics linearized by the fuzzy model in the neighborhood of the current state Indirect controller: linear parameters update controller
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Stability Fuzzy self-tuning If 1. there exists a LPV model which represents in a sufficiently accurate manner the nonlinear system (certainty equivalence principle) 2. the hypothesis of complete controllability and observability is satisfied (stable internal dynamics) 3. the pole placement design makes the closed loop constant and stable
then the closed loop system system is stable
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Lazy self-tuning controller
Lazy learning estimator returns the local linearization (LPV)
Linearization also in configurations which are far from the equilibrium locus System dynamics linearized by the lazy model in the neighborhood of the current state Indirect controller: linear parameters update controller
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Overview LINEARIZATION
GAIN SCHEDULING
FEEDBACK LINEARIZATION COMPLETE LINEARIZATION
STABILITY ABOUT EQUILIBRIUM INTERNAL DYNAMICS
STABILITY OF LPV
LINEAR CONTROLLER FOR NONLINEAR SYSTEMS SMOOTH INTERPOLATION
ADAPTIVITY
MODEL ESTIMATION
LAZY MODEL
FUZZY SCHEDULER
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REFERENCES
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References R. Palm, H. Hellendoorn, & D. Driankov. 1997. Model Based Fuzzy Control. Springer. T. Palm, & U. Rehfuess. 1997. Fuzzy controllers as gain scheduling approximators. Fuzzy Sets and Systems, 85, 233–246. J.S. Shamma. 1988. Analysis and Design of Gain Scheduled Control Systems. Ph.D. thesis, Lab. for Information and Decision Sciences, MIT,, Cambridge, MA. T. Takagi, & M. Sugeno. 1985. Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on System, Man and Cybernetics, 15(1), 116–132. K. Tanaka. 1995. Stability and Stabilizability of Fuzzy-Neural-Linear Control Systems. IEEE Transactions on Fuzzy Systems, 3(4), 438–447. K. Tanaka, & M. Sugeno. 1992. Stability analysis and design of fuzzy control systems. Fuzzy Sets and Systems, 45, 135–156.
