NONLINEAR ADAPTIVE NONLINEAR ADAPTIVE CONTROLCONTROL

RICCARDO MARINORICCARDO MARINO

UNIVERSITA DI ROMA TOR UNIVERSITA DI ROMA TOR VERGATAVERGATA

NONLINEAR ADAPTIVE NONLINEAR ADAPTIVE CONTROLCONTROL

• ADAPTIVE CONTROL ADAPTIVE CONTROL OF LINEAR SYSTEMSOF LINEAR SYSTEMS

• ADAPTIVE CONTROL ADAPTIVE CONTROL OF NONLINEAR OF NONLINEAR SYSTEMSSYSTEMS

• ADAPTIVE CONTROL ADAPTIVE CONTROL OF TIME-VARYING OF TIME-VARYING SYSTEMSSYSTEMS

• ADAPTIVE ADAPTIVE DISTURBANCE DISTURBANCE ATTENUATION/ ATTENUATION/ REJECTIONREJECTION

• ADAPTIVE ADAPTIVE REGULATIONREGULATION

• LEARNING LEARNING CONTROL CONTROL

Adaptive control of linear Adaptive control of linear systemssystems

• Given a family of linear time-invariant Given a family of linear time-invariant systems (minimum phase, known upper systems (minimum phase, known upper bound on system order, known relative bound on system order, known relative degree, known sign of high frequency gain), degree, known sign of high frequency gain), design an output feedback tracking control design an output feedback tracking control such that ANY smooth bounded output such that ANY smooth bounded output reference signal is asymptotically tracked reference signal is asymptotically tracked from any initial condition with transient from any initial condition with transient specifications: the resulting control is specifications: the resulting control is NONLINEAR.NONLINEAR.

Adaptive control of Adaptive control of nonlinear systemsnonlinear systems

• Given a family of systems with KNOWN Given a family of systems with KNOWN nonlinearities, design an output feedback nonlinearities, design an output feedback control which solves an output tracking control which solves an output tracking problem with transient specifications.problem with transient specifications.

• If the nonlinearities depend on the If the nonlinearities depend on the measured output, the problem has been measured output, the problem has been solved for a class of nonlinear minimum solved for a class of nonlinear minimum phase systems which strictly contains the phase systems which strictly contains the linear ones.linear ones.

• The model is required to be LINEARLY The model is required to be LINEARLY parametrized with respect to unknown parametrized with respect to unknown constant parameters: while this is constant parameters: while this is natural for linear systems, it may be a natural for linear systems, it may be a strong assumption for nonlinear systems.strong assumption for nonlinear systems.

• The control contains parameter The control contains parameter estimates which are adapted on the basis estimates which are adapted on the basis of the tracking error. They may or may of the tracking error. They may or may not converge to the true values not converge to the true values depending on persistency of excitation. depending on persistency of excitation. Nevertheless, asymptotic tracking is Nevertheless, asymptotic tracking is achieved.achieved.

Adaptive control of time- Adaptive control of time- varying systemsvarying systems

• Adaptation really means to adapt with Adaptation really means to adapt with respect to system’s time variations and to respect to system’s time variations and to changes from the environment.changes from the environment.

• From a technical viewpoint this implies time From a technical viewpoint this implies time varying parameters and time varying varying parameters and time varying disturbances and poses many difficulties in disturbances and poses many difficulties in adaptive control design: what does adaptive control design: what does adaptation mean if asymptotic tracking adaptation mean if asymptotic tracking cannot be achieved?cannot be achieved?

Adaptive disturbance Adaptive disturbance attenuation/rejectionattenuation/rejection

• Given an additive sinusoidal disturbance Given an additive sinusoidal disturbance of unknown frequency acting on a of unknown frequency acting on a stable/stabilizable system, design an stable/stabilizable system, design an output feedback control which output feedback control which asymptotically rejects the influence of asymptotically rejects the influence of the disturbance on the output.the disturbance on the output.

• More generally the disturbance is the More generally the disturbance is the sum of a bias and unknown sinusoids.sum of a bias and unknown sinusoids.

Adaptive regulationAdaptive regulation

• Given a controllable and observable Given a controllable and observable system with additive disturbances system with additive disturbances and output reference generated by and output reference generated by a linear stable exosystem with a linear stable exosystem with UNKNOWN parameters, design an UNKNOWN parameters, design an output feedback regulator which output feedback regulator which achieves asymptotic regulation and achieves asymptotic regulation and disturbance rejection.disturbance rejection.

Learning controlLearning control

• Learning control deals with repetitive Learning control deals with repetitive tasks, i.e. asymptotic tracking of tasks, i.e. asymptotic tracking of PERIODIC reference signals (and not any PERIODIC reference signals (and not any smooth bounded as adaptive control smooth bounded as adaptive control does).does).

• The goal of learning control is to learn The goal of learning control is to learn the unknown reference (periodic) input the unknown reference (periodic) input and not to identify systems parameters.and not to identify systems parameters.

• OPEN PROBLEMS IN NONLINEAR OPEN PROBLEMS IN NONLINEAR ADAPTIVE/LEARNING CONTROL: ADAPTIVE/LEARNING CONTROL: If the unknown nonlinearities are If the unknown nonlinearities are not linearly parametrized, can we not linearly parametrized, can we still track a time varying (for still track a time varying (for instance periodic) reference signal? instance periodic) reference signal? Can we learn the required reference Can we learn the required reference input? How do we input? How do we hold the system in the hold the system in the adaptive/learning phase?adaptive/learning phase?