Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for
Nonlin
Nonlinear Control Lecture 7: Nonlinear Control System Design
Farzaneh Abdollahi
Department of Electrical Engineering Amirkabir University of Technology
Fall 2010
Nonlinear Control Lecture 7 1/26
Farzaneh Abdollahi
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Nonlinear Control Problems Stabilization Problems
Feedback Control
Tracking Problems
Tracking Problem in Presence of Disturbance Tracking Problem in Presence of Disturbance
Specify the Desired Behavior Some Issues in Nonlinear Control Modeling Nonlinear Systems Feedback and FeedForward Importance of Physical Properties Available Methods for Nonlinear Control
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
2/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Nonlinear Control Problems
Objective of Control design: given a physical system to be controlled and specications of its desired behavior, construct a feedback control law to make the closed-loop system display the desired behavior. Control problems:
1. 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. 2. Tracking (Servo): makes the system output tracks a given time-varying trajectory such as aircraft y along a specied path, a robot manipulator draw straight lines. 3. Disturbance rejection or attenuation: rejected undesired signals such as noise 4. Various combination of the three above
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
3/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Stabilization Problems
Asymptotic Stabilization Problem: Given a nonlinear dynamic system: x = f (x , u , t ) nd a control law, u, s.t. starting from anywhere in region x 0 as t . 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 controller. Dynamic control law: the control law depends on the measurement through a dierential Eq, such as lag-lead controller
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
4/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Feedback Control
State feedback: for system x = f (t , x , u ) Output feedback for the system x y = f (t , x , u ) = h (t , x , u )
The measurement of some states is not available. an observer may be required
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
5/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Feedback Control
State feedback: for system x = f (t , x , u ) Output feedback for the system x y = f (t , x , u ) = h (t , x , u )
The measurement of some states is not available. an observer may be required
Static control law:
u = (t , x )
Dynamic control law:
u = (t , x , z ) z is the solution of a dynamical system driven by x : z = g (t , x , z ) The origin to be stabilize is x = 0, z = 0
Nonlinear Control Lecture 7 5/26
Farzaneh Abdollahi
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
For linear systems
When is stabilized by FB, the origin of closed loop system is g.a.s
For nonlinear systems
When is stabilized via linearization the origin of closed loop system is a.s
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
6/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
For linear systems
When is stabilized by FB, the origin of closed loop system is g.a.s
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
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
6/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
For linear systems
When is stabilized by FB, the origin of closed loop system is g.a.s
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 dened, FB provides regional stabilization
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
6/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
For linear systems
When is stabilized by FB, the origin of closed loop system is g.a.s
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 dened, FB provides regional stabilization If g.a.s is achieved, FB provides global stabilization
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
6/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
For linear systems
When is stabilized by FB, the origin of closed loop system is g.a.s
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 dened, 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
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
6/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Example
Consider the system x = x2 + u Linearize at the origin x =u Stabilize by u = kx , k > 0 the closed loop system x = kx + x 2 RoA is x < k It is regionally stabilized Given any compact set Br = {|x | r }, we can choose k > r FB achieves semiglobal stabilization.
Once k is xed and the controller is implemented, for x0 < k a.s. is guaranteed
Global stabilization is achieved by FB: u = x 2 kx
Farzaneh Abdollahi Nonlinear Control Lecture 7 7/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Example: Stabilization of a Pendulum
Consider the dynamics of the pendulum: mgl sin = J Objective: take the pendulum from a large initial angel ( = 60o ) to the vertical up position A choice of stabilizer:
a feedback part for stability (PD)+ a feedforward part for gravity compensation: kp mgl sin = kd kd and kp are pos. constants. globally stable closed-loop dynamics:prove it + kd + kp = 0 J In this example feedback (FB) and feedforward (FF) control actions modify the plant into desirable form.
Farzaneh Abdollahi Nonlinear Control Lecture 7 8/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Example: Stabilization of a Inverted Pendulum with Cart
Consider the dynamics of the inverted pendulum shown in Fig.: ml sin 2 = u (M + m) x + ml cos ml x sin + mg sin = 0 mx cos + ml mass of the cart is not negligible Objective: Bring the inverted pendulum from vertical-down at the middle of the lateral track to the vertical-up at the same lateral point. It is not simply possible since degree of freedom is two, # inputs is one (under actuated).
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
9/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Tracking Problems
Asymptotic Tracking Problem: Given a nonlinear dynamics: x = f (x , u , t ) y = h(x , u , t )
and a desired output, yd , nd a control law for the input u s.t. starting from any initial state in region , the tracking error y (t ) yd (t ) goes to zero, while whole state x remain bounded. A practical point: Sometimes x can just be remained reasonably bounded, i.e., bounded within the range of system model validity. Perfect tracking: proper initial states imply zero tracking error for all time: y (t ) yd (t ) t 0; in asymptotic/ exponential tracking perfect tracking is achieved asymptotically/ exponentially Assumption throughout the rest of the lectures: yd and its derivatives up to a suciently high order ( generally equal to the systems order) are cont. and bounded. yd and its derivatives available for on-line control computation yd is planned ahead
Nonlinear Control Lecture 7 10/26
Farzaneh Abdollahi
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Sometimes derivatives of the desired output are not available. A reference model is applied to provide the required derivative signals
Example: For tracking control of the antenna of a radar, only the position of the aircraft ya (t ) is available at a given time instant (it is too noisy to be dierentiated numerically). desired position, velocity and acceleration to be tracked is obtained by y d + k1 y d + k2 yd = k2 ya (t ) k1 and k2 are pos. constants following the aircraft is translated to the problem of tracking the output yd of the reference model The reference model serves as providing the desired output of the tracking system in response to the aircraft position generating the derivatives of the desired output for tracker design. (1) Should be fast yd to closely approximate ya
Nonlinear Control Lecture 7 11/26
(1)
Farzaneh Abdollahi
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Tracking Problem
Perfect tracking and asymptotic tracking is not achievable for non-minimum phase systems. Example: Consider y + 2y + 2y = u + u. It is non-minimum phase since it has zero at s = 1. Assume the perfect tracking is achieved. u u = ( yd + 2 y d + 2yd ) u = s
2 +2s +2
s 1
yd
Perfect tracking is achieved by innite control input. Only bounded-error tracking with small tracking error is achievable for desired traj. Perfect tracking controller is inverting the plant dynamics
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
12/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Tracking Problem in Presence of Disturbance
Asymptotic disturbance rejection:Given a nonlinear dynamics: x = f (x , u , w , t ) y = h(x , u , w , t )
and a desired output, yd , nd a control law for the input u s.t. starting from any initial state in region , the tracking error y (t ) yd (t ) goes to zero, while whole state x remain bounded.
When the exogenous signals yd and w are generated by a known model, asymptotic output tracking and disturbance rejection can be achieved by including such model in the FB controller.
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
13/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
For T.V disturbance w (t ), achieving asymptotic disturbance rejection may not be feasible. look for disturbance attenuation:
achieve u.u.b of the tracking error with a prescribed tolerance: e (t ) < , t > T ,, is a prespecied (small) positive number. OR consider attenuating the closed-loop input-output map from the disturbance input w to the tracking error e = y yd
e.g. considering w as an L2 signal, goal is min the L2 gain of the closed-loop I/O map from w to e
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
14/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
For T.V disturbance w (t ), achieving asymptotic disturbance rejection may not be feasible. look for disturbance attenuation:
achieve u.u.b of the tracking error with a prescribed tolerance: e (t ) < , t > T ,, is a prespecied (small) positive number. OR consider attenuating the closed-loop input-output map from the disturbance input w to the tracking error e = y yd
e.g. considering w as an L2 signal, goal is min the L2 gain of the closed-loop I/O map from w to e
For tracking problem one can design:
Static/Dynamic state FB controller Static/Dynamic output FB controller
Tracking may achieve locally, regionally, semiglobally, or globally:
These phrases refer not only to the size of the initial state, but to the size of the exogenous signals yd , w Local tracking means tracking is achieved for suciently small initial states and suciently small exogenous signals Global tracking means tracking is achieved for any initial state and any yd , w
Nonlinear Control Lecture 7 14/26
Farzaneh Abdollahi
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Relation between Stabilization and Tracking Problems
Tracking problems are more dicult to solve than stabilization problems In tracking problems the controller should
not only keep the whole state stabilized but also drive the system output toward the desired output
However, for tracking problem of the plant: y + f (y , y , u) = 0
e (t ) = y (t ) yd (t ) goes to zero It is equivalent to the asymptotic stabilization of the system e d + f (e , e , u , yd , y d , y d ) = 0 with states e and e
(2)
tracking problem is solved if we can design a stabilizer for the non-autonomous dynamics (2) On the other hand, stabilization problems can be considered as a special case of tracking problem with desired trajectory being a constant.
Nonlinear Control Lecture 7 15/26
Farzaneh Abdollahi
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Specify the Desired Behavior
In Linear control, the desired behavior is specied 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 specications of the closed-loop system is dened, the a controller is synthesized to meet the specications For nonlinear systems the system specication of nonlinear systems is less obvious since
response of the nonlinear system to one command does not reect the response to an other command a frequency description is not possible
In nonlinear control systems some qualitative specications of the desired behavior is considered.
Nonlinear Control Lecture 7 16/26
Farzaneh Abdollahi
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Some desired qualitative specications of nonlinear system:
Stability must be guaranteed for the nominal model, either in local or global sense. In local sense, the region of stability and convergence are of interest.
stability of nonlinear systems depends on initial conditions and only temporary disturbances may be translated as initial conditions
Robustness is the sensitivity eect which are not considered in the design like persistent disturbance, measurement noise, unmodeled dynamics, etc. Accuracy and Speed of response for some typical motion trajectories in the region of operation. For instance, sometimes appropriate control is desired to guarantee consistent tracking accuracy independent of the desired traj. Cost of a control which is determined by # and type of actuators, sensors, design complexity.
The mentioned qualitative specications are not achievable in a unied design. A good control can be obtained based on eective trade-os of them.
Nonlinear Control Lecture 7 17/26
Farzaneh Abdollahi
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Nonlinear Control Problems
A Procedure of designing control
1. 2. 3. 4. 5. Specify the desired behavior and select actuators and sensors model the physical plant by a set of dierential Eqs design a control law analyze and simulate the resulting control system implement the control system in hardware
Experience and creativity of 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 dierential Eqs.) for the physical system to be controlled.
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
18/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Modeling Nonlinear Systems
Two points in modeling:
1. To obtain 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 eects and discard insignicant eects in operating range of interest. 2. 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 in robust control, adaptive design or simulation. Model uncertainties: dierence between the model and real physical system parametric uncertainties: uncertainties in parameters Example: model of controlled mass: mx =u Uncertainty in m is parametric uncertainty neglected motor dynamics, measurement noise, and sensor dynamics are non-parametric uncertainties. Parametric uncertainties are easier to characterize; 2 m 5
Nonlinear Control Lecture 7
Farzaneh Abdollahi
19/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
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 eect 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 specied motion traj and canceling the eect of known disturbances FB to stabilize the tracking error dynamics.
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
20/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Example
Consider a minimum-phase system A(s )y = B (s )u
where A(s ) = a0 + a1 s + ... + an1 s n1 + s n , B (s ) = b0 + b1 s + ... + bm s m Objective: make the output y (t ) follow a time-varying traj yd (t ) 1. To achieve y = yd , input should have a FF term of A(s ) u=v+ yd B (s )
A(s ) B (s ) :
(3)
(4)
Substitute (4) to (3): A(s )e = B (s )v , where e (t ) = y (t ) yd (t )
2. Use FB to stabilize the system:
C (s ) v= D e closed loop system (AC + BD )e = 0. (s ) Choose D and C to poles in desired places
u=
A B yd
C De (i )
e (t ) is zero if initial conditions y (i ) (0) = yd (0), i = 1, ..., r ,otherwise exponentially converges to zero
Nonlinear Control Lecture 7 21/26
Farzaneh Abdollahi
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Example Contd
If some derivatives of yd are not available, one can simply omit them from FF only bounded tracking error is guaranteed, This method is not applicable for non-min phase systems.
low freq. components of desired traj in FF, good tracking in freq lower than the LHP zeros of plant By dening FF term as D B A e (t ) = AD B 1 yd BC [ B1 1]Ayd If B1 eliminates the RHP zeros of B good tracking for desired traj with frequencies lower than the RHP zeroes (but we may not have internal stability)
Farzaneh Abdollahi Nonlinear Control Lecture 7 22/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
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 manipulator was long recognized to be far of reach. Because robots 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.
Farzaneh Abdollahi
Nonlinear Control
Lecture 7
23/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Available Methods for Nonlinear Control
There is no general method for designing nonlinear control Some alternative and complementary techniques to particular classed of control problem are listed below:
Trail-and Error: The idea is using analysis tools such a phase-plane methods, Lyapunov analysis , etc, to guide searching a controller which can be justied 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
Nonlinear Control Lecture 7 24/26
Farzaneh Abdollahi
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Available Methods for Nonlinear Control
Robust Control is deigned 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 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 parameterizes 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.
Nonlinear Control Lecture 7 25/26
Farzaneh Abdollahi
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlin
Available Methods for Nonlinear Control
Gain Scheduling Employs the well developed linear control methodology to 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.
The main problems of gain scheduling:
provides limited theoretical guarantees of stability in nonlinear operation The system should satisfy some conditions:
the scheduling variables should change slowly The scheduling variables should capture the plants nonlinearities.
Due to the necessity of computing many linear controllers, this method involves lots of computations.
Nonlinear Control Lecture 7 26/26
Farzaneh Abdollahi
