Design issues in non linear control system
- 2. Content : • Nonlinear Control Problems • Stabilization Problems • Feedback Control • Specify the Desired Behavior • Some Issues in Nonlinear Control • Modeling Nonlinear Systems • Feedback and FeedForward • Importance of Physical Properties • Available Methods for Nonlinear Control
- 3. Nonlinear Control Problems ➤ Objective of Control design: given a physical system to be controlled and specifications of its desired behavior, construct a feedback control law to make the closed-loop system display the desired behavior. ➤ Control problems: • Stabilization (regulation): stabilizing the state of the closed-loop system around an Equ. point, like: temperature control, altitude control of aircraft, position control of robot manipulator. • Tracking (Servo): makes the system output tracks a given time-varying trajectory such as aircraft y along a specified path, a robot manipulator draw straight lines. • Disturbance rejection or attenuation: rejected undesired signals such as noise.
- 4. Stabilization Problems ➤ Asymptotic Stabilization Problem: Given a nonlinear dynamic system: _ x = f (x; u; t) and a control law, u, s.t. starting from anywhere in region x ! 0 as t ! 1. ➤If the objective is to drive the state to some nonzero set-point xd , it can be simply transformed into a zero-point regulation problem x --- xd as the state. ➣ Static control law: the control law depends on the measurement signal directly, such as proportional controllers. ➣ Dynamic control law: the control law depends on the measurement through a differential Eq, such as lag-lead controller.
- 5. Feedback Control ➤ State feedback: for system _ x = f (t; x; u) ➤ Output feedback for the system _ x = f (t; x; u) y = h(t; x; u) ➣ The measurement of some states is not available. ➣an observer may be required. ➤ For linear systems When is stabilized by FB, the origin of closed loop system is g.a.s
- 6. Feedback Control ➤ For nonlinear systems ➣When is stabilized via linearization the origin of closed loop system is a.s • If RoA is unknown, FB provides local stabilization • If RoA is denied, FB provides regional stabilization • If g.a.s is achieved, FB provides global stabilization • If FB control does not achieve global stabilization, but can be designed s.t. any given compact set (no matter how large) can be included in the RoA, FB achieves semiglobal stabilization.
- 7. Specify the Desired Behavior ➤In Linear control, the desired behavior is specified in ➣ time domain: rise time, overshoot and settling time for responding to a step command ➣ frequency domain: the regions in which the loop transfer function must lie at low and high frequencies ➤So in linear control the quantitative specifications of the closed-loop. system is denied, the controller is synthesized to meet the specifications. ➤ For nonlinear systems the system specification of nonlinear systems is less obvious since • response of the nonlinear system to one command does not reflect the response to another command • a frequency description is not possible • In nonlinear control systems some qualitative specifications of the desired behavior is considered.
- 8. Nonlinear Control Problems ➤A Procedure of designing control 1. Specify the desired behavior and select actuators and sensors 2. model the physical plant by a set of differential Eqs 3. design a control law 4. analyze and simulate the resulting control system 5. implement the control system in hardware • Experience and creativity are important factor in designing the control • Sometimes, addition or relocation of actuators and sensors may make control of the system easier. • Modeling Nonlinear Systems • Modeling is constructing a mathematical description (usually as a set of differential Eqs.) for the physical system to be controlled.
- 9. Modeling Nonlinear Systems ➤ Two points in modeling: • To obtain a tractable yet accurate model, good understanding of system dynamics and control tasks requires. ➣ Note: more accurate models are not always better. They may require unnecessarily complex control design and more computations. • Keep essential effects and discard insignificant effects in the operating range of interest. • In modeling not only the nominal model for the physical system should be obtained, but also some characterization of the model uncertainties should be provided for using robust control, adaptive design or simulation.
- 10. Model uncertainties • Difference between the model and real physical system ➣ Parametric uncertainties: uncertainties in parameters Example: model of controlled mass: m•x = u ➣ Neglected motor dynamics, measurement noise, and sensor dynamics are non- parametric uncertainties. ➣ Parametric uncertainties are easier to characterize; 2<m<5
- 11. Feedback and FeedForward • Feedback (FB) plays a fundamental role in stabilizing the linear as well as nonlinear control systems. • Feedforward (FF) in nonlinear control is much more important than linear control. • FF is used to • cancel the effect of known disturbances. • provide anticipate actions in tracking tasks. ➤ For FF a model of the plant (even not very accurate) is required. • Many tracking controllers can be written in the form: u = FF+ FB • FF: to provide necessary input to follow the specified motion traj and cancelling the effect of known disturbances. • FB to stabilize the tracking error dynamics.
- 12. Importance of Physical Properties ➤ In nonlinear control design, explanation of the physical properties may make the control of complex nonlinear plants simple. • Example: Adaptive control of robot manipulators was long recognized to be far from reach. • Because robot's dynamics is highly nonlinear and has multiple inputs ➤ Using the two physical facts: ➣ pos. def. of inertia matrix ➣possibility of linearly parameterizing robot dynamics • yields adaptive control with global stability and desirable tacking convergence.
- 13. Available Methods for Nonlinear Control ➤ There is no general method for designing nonlinear control ➤ Some alternative and complementary techniques to particular classes of control problem are listed below: ➣ Trial-and Error: The idea is using analysis tools such as phase-plane methods, Lyapunov analysis , etc, to guide searching a controller which can be justified by analysis and simulations. • This method fails for complex systems ➣Feedback Linearization: transforms original system models into equivalent models of simpler form (like fully or partially linear) • Then a powerful linear design technique completes the control design • This method is applicable for input-state linearizable and minimum phase systems • It requires full state measurement • It does not guarantee robustness in presence of parameter uncertainties or disturbances. • It can be used as model-simplifying for robust or adaptive controllers.
- 14. Available Methods for Nonlinear Control • Robust Control: It is designed based on consideration of nominal model as well as some characterization of the model uncertainties. ➣ An example of robust controls is sliding mode control ➣They generally require state measurements. ➣ In robust control design tries to meet the control objective for any model in the "ball of uncertainty." • Adaptive Control: It deals with uncertain systems or time-varying systems. ➣ They are mainly applied for systems with known dynamics but unknown constant or slowly-varying parameters. ➣They parameterize the uncertainty in terms of certain unknown parameters and use feedback to learn these parameters on-line , during the operation of the system. ➣In a more elaborate adaptive scheme, the controller might be learning certain unknown nonlinear functions, rather than just learning some unknown parameters.
- 15. Available Methods for Nonlinear Control ➤ Gain Scheduling Employs the well developed linear control methodology of the control of nonlinear systems. • A number of operating points which cover the range of the system operation is selected. • Then, at each of these points, the designer makes a linear TV approximation to the plant dynamics and designs a linear controller for each linearized plant. • Between operating points, the parameters of the compensators are interpolated, ( scheduled), resulting in a global compensator. • It is simple and practical for several applications.
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