Overview of Non-linear Control Methods

Schweizerische Gesellschaft fur Automatik

Association Suisse pour l’Automatique

Associazione Svizzera di Controllo Automatico

Swiss Society for Automatic Control

Overview of non-linear control methods

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Scope Get an overview of design methods for non-linear control systems.

Keywords Linearization, compensation of static non-linearities linearizationthrough local feedback, exact feedback linearization, Lyapunovredesign, backstepping control, sliding mode control, relay con-trol, gain scheduling, adaptive control, neural networks,fuzzylogic.

Prerequisites LTI systems, properties of non-linear systems

Contact Peter Gruber ([email protected]) andSilvano Balemi ([email protected])

Version 1.0

Date May 28, 2010

, Overview of non-linear control methods SGA-ASSPA-SSAC

2 Peter Gruber and Silvano Balemi May 28, 2010

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Linearization around an operating point . . . . . . . . . . . . . . . . 53 Compensation of static nonlinearities . . . . . . . . . . . . . . . . . . 84 Linearization by local feedback . . . . . . . . . . . . . . . . . . . . . 95 Input-to-state linearization . . . . . . . . . . . . . . . . . . . . . . . . 106 Input-to-output linearization . . . . . . . . . . . . . . . . . . . . . . . 107 Lyapunov redesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Sliding-mode control . . . . . . . . . . . . . . . . . . . . . . . . . . . 1610 Relay control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1811 Gain scheduling (open-loop adaptive scheme) . . . . . . . . . . . . . . 2012 Adaptive Control (closed-loop adaptive scheme) . . . . . . . . . . . . 2113 Neural network control . . . . . . . . . . . . . . . . . . . . . . . . . . 2414 Fuzzy control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29B.1 Sliding-mode control of an inverted pendulum . . . . . . . . . . . . . 30B.2 Nonlinear feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

May 28, 2010 Peter Gruber and Silvano Balemi 3


1 Introduction

When real systems present significant non-linear behavior or when linear techniquesdo not offer sufficient performance and stability levels, one is forced to apply non-linear techniques. Non-linear control is advised in particular in the following cases:

• The non-linearity is dominating and cannot be compensated for.

• The range of operation is large.

• Saturation, hysteresis effects or other constraints are crucial (hard non-linearities)and degrade the linear design.

• The non-linearity of the system has a beneficial effect on the system behaviourand its compensation is therefore not wanted.

• The controller has switching elements or is piecewise linear.

This section only gives a very brief overview of techniques dealing with non-linearsystems and control and does not claim to be complete. As there is no general non-linear control theory, many techniques have been developed.

One main issue with a lot of these techniques is the a priori knowledge onehas about the system to be controlled. If the system can be modeled by as set ofnon-linear differential and algebraic equations with given structure and parameters,techniques which require a precise mathematical description can be used. Because inreality many systems are not known in such detail the question of the applicabilityof these methods poses a major challenge.

The “best” controller can be found by addressing in sequence the followingpoints:

1. Find a control law for stabilization of the non-linear system under given orassumed uncertainties.

2. Find a way that ensures that the constraints can be met and are not violated.

3. Find a optimization procedure to modify the control law of of 1. and 2. suchthat a performance criteria is additionally minimized.

The different methods listed in the following weigh the three control goals dif-ferently. Each method is represented by its idea, its advantages and disadvantages.

For interested readers books are recommended at the end of the document.


SGA-ASSPA-SSAC , Overview of non-linear control methods

2 Linearization around an operating point

Idea

The method can be used when the range of operation is small or the non-linearitiesare weak around the operating point. A linear approximation of the process be-haviour is determined either analytically or numerically. The controller is then de-signed for the linear approximation, under the assumtion that this controller workswell with the real non-linear process.

Starting from a non.linear system model given by n first-order non-linear differ-ential equations with p inputs represented in the state plane by

x = f (x(t), u(t)) (1.1)

it is possible to linearize the process around a choosen operating point or also arounda choosen trajectory by performing a Taylor series expansion and by eliminating thehigher order terms.

The linearization of the model 1.1 arnound the state trajectory x0(t) with thecorresponding input u0(t) gives

xi =dxi

dt≈ fi (x0, u0)+

n∑

j=1

∂fi(x, u)

∂xj

∣∣∣∣∣x0, u0

·(xj−(x0)j)+

p∑

j=1

∂fi(x, u)

∂uj

∣∣∣∣∣x0, u0

·(uj−(u0)j)

With the notation

∆xj = xj − (x0)j, ∆uj = uj − (u0)j,

and the with∆xi = xi − (x0)i = xi − fi (x0, u0)

the final expression for the linearized model is

∆xi =n∑

j=1

∂fi(x, u)

∂xj

∣∣∣∣∣x0, u0

· ∆xj +

p∑

j=1

∂fi(x, u)

∂uj

∣∣∣∣∣x0, u0

· ∆uj

This equation can be rewritten in the standard linear state-space form

∆x = A · ∆x + B · ∆u

where A and B are the Jacobian matrices

A =

∂f1

∂x1

∂f1

∂x2

. . .∂f1

∂xn

∂f2

∂x1

∂f2

∂x2

. . .∂f2

∂xn

......

. . ....

∂fn

∂x1

∂fn

∂x2

. . .∂fn

∂xn

∣∣∣∣∣∣∣∣∣∣∣∣∣∣x0, u0

and B =

∂f1

∂u1

∂f1

∂u2

. . .∂f1

∂up

∂f2

∂u1

∂f2

∂u2

. . .∂f2

∂up...

.... . .

...∂fn

∂u1

∂fn

∂u2

. . .∂fn

∂up

∣∣∣∣∣∣∣∣∣∣∣∣∣∣x0, u0

(1.2)



Note that the matrices A and B are constant if the linearization is performed aroundthe operating point x0 and time-varying if around a trajectory x0(t).

Example 1 Given is the system{

x1 = f1(x, u) = x22 + u − 1

x2 = f2(x, u) = x1

First the system is linearized arounf the equilibrium point by setting the deriva-tives equal to zero (x1 = x2 = 0). Then

{0 = x2

2 + u − 10 = x1

The equilibrium point for the null input u0(t) = 0 becomes

x0 =

[x1

x2

]

=

[0

±√

1 − u

]∣∣∣∣u0=0

=

[0±1

]

with two possible values. With the arbitrary choice of x0 = [0, 1]′, the linearizedsystem is

∆x = x − f(x0, u0) =

[∂f1(x,u)

∂x1

∂f1(x,u)∂x2

∂f2(x,u)∂x1

∂f2(x,u)∂x2

]∣∣∣∣∣x0,u0

· ∆x +

[∂f1(x,u)

∂u∂f2(x,u)

∂u

]∣∣∣∣∣x0,u0

· ∆u

=

[0 2 · x2

1 0

]∣∣∣∣x0,u0

· ∆x +

[10

]∣∣∣∣x0,u0

· ∆u

=

[0 21 0

]

· ∆x +

[10

]

· ∆u

Consider now the linearization around the trajectory x0(t) = [1, t]′. From thestate equation the required input

u0(t) = 1 − x22(t) + x1(t) = 1 − t2

is obtained giving the linearized model

∆x = x − f(x0(t), u0(t)) =

[0 2 · x2

1 0

]∣∣∣∣x0(t),u0(t)

· ∆x +

[10

]∣∣∣∣x0(t),u0(t)

· ∆u

=

[0 2 · t1 0

]

· ∆x +

[10

]

· ∆u

which time-variant. Equating the characteristic polynomial

p(s) = det(sI − A + B · K) = det

[s + k1 −2t + k2

−1 s

]

= s2 + k1 · s + k2 − 2t

with the desired polynomials2 + 2s + 1

the time-varying state-feedback controller K(t) = [2, 1 + 2t]′ is obtained.



Advantages

Linear techniques can be fully applied.

Disadvantages

Operating range is limited. Possible critical behavior in the presence of large un-measured disturbances.



3 Compensation of static nonlinearities

Idea

A single-input, single-output static non-linearity is compensated in serie using theinverse function of the non-linearity. Figure 1.1 shows this method. The compen-sated system behaves linearly, so that it is possible to design a linear controller forthe compensated system.

e

+--

6- -- -

6-

��

��r 6

-

��u

G(s)y

C(s)e

−

controller process

Figure 1.1: Compensation of static non-linearity: C(s) is a linear controller for thelinear part G(s) of the process.

Advantages

Can be applied for actuators (precompensation) and sensors (processing the mea-sured signal). Linearization is valid over a large range. Linear control design tech-niques can be applied.

Disadvantages

Nonlinearity must be known and must be invertible.



4 Linearization by local feedback

Idea

The non-linearity is isolated and controlled by a fast local loop (often a P-typecontroller can be used): the controlled local loop can be treated as a gain by theouter loop controller If the inner loop dynamic is fast enough it can be replaced bya constant gain close to one. If inner loop has also an integral part in the controlleror process part (e.g. integrating actuator), the gain is 1.

The inner local loop must be faster (5 to 10 times) as the outer dynamics.

Advantages

Nonlinearity must not be known exactly. Inner disturbances are reduced at the sametime. Even if not completely eliminated, the nonlinearity is greatly reduced.

Disadvantages

Additional measurement is needed. Time scale difference of inner and outer loopmust be large enough (inner loop must be much faster than desired overall closed-loop dynamics).



5 Input-to-state linearization

Idea

The approach involves coming up with a transformation of the non-linear systeminto an equivalent linear system through a change of variables and a suitable controlinput. After this, a stabilizing outer state feedback loop is designed. Sometimes theinner feedback loop acts like a dynamic non-linear parallel compensation loop.

Example 2 Given is the nonlinear system

x1 = a · sin x2

x2 = −x21 + u y = x2

The choice z1 = x1, z2 = a · sin(x2), u = x21 + a

a·cos x2

· vyields

z1 = z2

z2 = v y = arcsin z2

a

which is linear from the input to the states. A controller can be easily designed tocontrol the states z1 and z2.

Advantages

Often useful in simple cases. It is the non-linear version of ”pole-zero cancellation”.

Disadvantages

Only possible for stable processes. The variable change must be invertible. Impor-tant intuition may be lost. Sensitive on parameters. All state variables must bemeasurable. The state equation is linear, whereas the output is nonlinear: it is stillcomplicated to solve the control problem. Nonlinear observer needed, if not all statevariables are measurable.

6 Input-to-output linearization

Idea

The approach involves coming up with a transformation of the non-linear systeminto an equivalent linear system from the input to the output, through a change ofvariables and a suitable control input. After this, a stabilizing outer state feedbackloop is designed.

Example 3 Given is the nonlinear system

x1 = a · sin x2

x2 = −x21 + u y = x2



The choice v = −x21 + u

yieldsx1 = a · sin x2

x2 = v y = x2

which is linear from the “artificial” input v to the output.

Example 4 Inverse dynamicsThe technique is particularly used in robotics. Consider the model

Mn×n(q) · q + Hn(q, q) = Qn (1.3)

for a general mechanical system, where M is the generalized inertia matrix, Q theactuation vector (force or torque), H the vector of coupled dynamics (including grav-itational, Coriolis and centrifugal forces) and n is the number of degrees of freedomof the system.

With the introduction of the new “input” ac defined implicitely by xsthe expression

Mn×n(q) · ac + Hn(q, q) = Qn

the equation for the dynamics becomes

q = ac

with the system now observable and controllable. The controller only has to controla set of decoupled integrators. Note that the procedure implies the invertibility of thegeneralized inertia matrix, which could be singular for some values of the variables.

Problems of this procedure are:

1. all measurements must be available (q and q in the example)

2. unmodeled dynamics (model approximation) and or varying or uncertain pa-rameters (i.e. weight, inertia,...) may cause unsatisfactory behavior.

Advantages

Often useful in simple cases. Non-linear version of ”pole-zero cancellation”. Theimplementation of the complete controller might be simpler than for the input-to-state linearization.

Disadvantages

Only possible for stable processes. Nonlinearities must be invertible. Importantintuition may be lost. Sensitive on parameters. All state variables must be mea-surable. Non-linear observer needed, if not all state variables are measurable. Thelinearized input-output map may not account for all the dynamics of the system(i.e. x1). Some state become unobservable (see x1 above): such state must stay wellbehaved (stable or bounded).



7 Lyapunov redesign

Idea

A Lyapunov function can be exploited for the synthesis of nonlinear control systems.First a Lyapunov function V must be found for the closed-loop system and then acontrol law designed which makes the derivative V negative for the required region ofattraction (for all possible initial conditions, disturbances and other uncertainties).

The resulting control law is usually a nonlinear state feedback law.Consider a non-linear system which is affine (linear) in the input, i.e.

x = f(x) + g(x) · u, f(0) = 0

Also suppose that a Lyapunov function V (x) has been found for this system.Then a stabilising feedback control law, u(x) is given by Sontag’s formula

u(x) =

− ∂V

∂x·f+

q

( ∂V

∂x·f)

2

+( ∂V

∂x·g)

4

( ∂V

∂x·g)

for ∂V∂x

· g 6= 0

0 for ∂V∂x

· g = 0

Note that this still does not help in the selection of the Lyapunov function!

Example 5 Scalar exampleConsider a scalar system of the form

x = −x3 + u

This system can be stabilized by setting

u = x3 − k · x

The function V (x) = 12x2 is such that

1. V (x) is positive definite

2. d V (x)d t

= x(

d xd t

)= −k · x2 ≤ 0 (i.e. the derivative is negative definite)

hence the system is now globally asymptotically stable.However it is not necessary to cancel out the x3 term from the system using the

control. For example, with the choice

u(x) = −k · x

and the same Lyapunov function above, its the derivative becomes

V = x · x = −x4 − k · x2 ≤ −x4 (1.4)

which is negative definite. This result has two advantages

1. The control signal magnitude increases only linearly with x



2. The Lyapunov function decreases faster from large x

Finally let’s apply Sontag’s formula. The assumption f(0) = 0 is satisfied. Thenalso f(x) = −x3 and g(x) = 1. The choice of the Lyapunov function V (x) = 1

2x2

gives∂V

∂x· g = x,

∂V

∂x· f = −x4

yielding finally the state-dependent input

u(x) = x3 − x√

x4 + 1

The control signal found with Sontag’s formula is even smaller than that from thetwo previous controllers.

Advantages

Closed-loop stability can be guaranteed. Can be designed trading robustness againstperformance.

Disadvantages

Difficulty in finding a Lyapunov function. All state variables must be measurable.Non-linear observer needed if not all state variables are measurable.



8 Backstepping

Idea

The method is based on a Lyapunov function and proposes a stepwise control designof the system. First the design of a first-order subsystem is performed, then anadditional state is considered and the design for the second-order system is performedfor incremental orders until the whole system is controlled

The backstepping procedure can be applied as follows to a system of the form

{

x = fx(x) + gx(x) · z1

z1 = f1(x, z1) + g1(x, z1) · u1

(1.5)

where

• x = [x1, x2, . . . , xn]T

• z1 and u1 are scalars

• for all x and z1, g1(x, z1) 6= 0

It is assumed that a control input ux(x) is known for which the Lyapunov Vx(x)function proves the stability of the upper system x = fx(x) + gx(x) · ux(x) .

First, rather than designing the feedback-stabilizing control u1 directly, a newcontrol input ua1 (to be designed later) is introduced together with the variablesubstitution

u1(x, z1) =ua1 − f1(x, z1)

g1(x, z1)

which is possible because g1 6= 0. Then the system in (1.5) becomes

x = fx(x) + gx(x) · z1

z1 = f1(x, z1) + g1(x, z1) ·ua1 − f1(x, z1)

g1(x, z1)︸︷︷︸

u1(x,z1)

= ua1 (1.6)

Now, the input ua1 is designed according to the law

ua1(x, z1) = −∂Vx

∂x· gx(x) − k1(z1 − ux(x)) +

∂ux

∂x· (fx(x) + gx(x) · z1) (1.7)

with gain k1 > 0. So the final feedback-stabilizing control law is

u1(x, z1) =

ua1(x,z1)︷︸︸︷

−∂Vx

∂xgx(x) − k1(z1 − ux(x)) +

∂ux(x)

∂x(fx(x) + gx(x)z1)−f1(x, z1)

g1(x, z1)



This input in fact stabilizes the whole system as the new Lyapunov function

V1(x, z1) = Vx(x) +1

2(z1 − ux(x))2

proves stability of the system with the design of ua1(x, z1) and u1(x, z1) above.Proof: The Lypaunov function V1(x, z1) is positive definite. Its derivative isnegative semi-definite as shown by the next steps.

V1(x, z1) = Vx(x) + (z1 − ux(x)) · (z1 − ux(x))

= ∂Vx(x)∂x

· x + (z1 − ux(x)) ·(

ua1(x, z1) − ∂ux(x)∂x

· x)

= ∂Vx(x)∂x

· (fx(x) + gx(x) · z1)

+(z1 − ux(x)) ·(

ua1(x, z1) − ∂ux(x)∂x

· (fx(x) + gx(x) · z1))

= ∂Vx(x)∂x

· (fx(x) + gx(x) · ux(x)) + ∂Vx(x)∂x

· (gx(x) · (z1 − ux(x)))

+(

ua1(x, z1) − ∂ux(x)∂x

· (fx(x) + gx(x) · z1))

· (z1 − ux(x))

= ∂Vx(x)∂x

· (fx(x) + gx(x) · ux(x))

+(

ua1(x, z1) + ∂Vx(x)∂x

· gx(x) − ∂ux(x)∂x

· (fx(x) + gx(x) · z1))

·(z1 − ux(x))

and by replacing ua1 from equation (1.7) the final expression is obtained

V1(x, z1) =∂Vx(x)

∂x· (fx(x) + gx(x) · ux(x))

︸︷︷︸

≤0 by definition of Vx

−k · (z1 − ux(x)) · (z1 − ux(x))︸︷︷︸

≤0

and thus V1(x, z1) ≤ 0.

Because this strict-feedback system has a feedback-stabilizing control and a cor-responding Lyapunov function, it can be cascaded as a part of a larger strict-feedbacksystem. The procedure can be repeated to find the surrounding feedback-stabilizingcontrol.

Advantages

Can be applied recursively in terms of the extension of the dimension of the system.

Disadvantages

Difficulty in finding a Lyapunov function. All state variables must be measurable.Nonlinear observer needed, if not all state variables are measurable. Sensitive toparameter variation.



9 Sliding-mode control

Idea

The control is divided into to phases in time: In a reaching phase the n-th ordersystem is first driven by a stabilizing control law (derived e.g. by a Lyapunovfunction) to a stable manifold of order n − 1. In the sliding phase the systemslides along the manifold towards the equilibrium. The overall control law switchesbetween several controllers, depending on the phase.

Example 6 Consider the second order system

{x1 = x2

x2 = f(x) + g(x) · u

where

f(x) and g(x) are non-linear

g(x) > 0

f(x) and g(x) can have discontinuities

Reaching phase dynamicsthe variable x1 to 0 if x1 = −a · x1 with a > 0. Then the coordinate s indicating thedistance from this manifold is s = x1 + a · x1 = x2 + a · x1 So x1 converges to 0 iss = 0.Sliding phase dynamicsThe derivative of s is s = x2 + a · x1 = f(x) + g(x) · u + a · x2

The derivative of the Lyapunov function V = 12· s2 is dotV = s · s = s · [f(x) +

g(x) · u + a · x2], which is is negative definite if

f(x) + g(x) · u + a · x2 < 0 for s > 0f(x) + g(x) · u + a · x2 = 0 for s = 0f(x) + g(x) · u + a · x2 > 0 for s < 0

Thus stability in the reaching phase (s converges to 0, i.e. the trajectory convergesto the manifold above if

u < β(x) for s > 0u = β(x) for s = 0u > β(x) for s < 0

where β(x) = −f(x)+a·x2

g(x). This condition can be ensured (for example) using the

control law u = β(x) − K · sign(s) = β(x) − K · sign(x2 + a · x1) with k > 0.

Advantages

Can be applied to a wide range of non-linear systems (also of higher order). Can becombined with PID controllers. Robust to parameter uncertainties



Disadvantages

Restricted to single input systems. Control law is discontinuous. Fast switching(chattering) is possible.



10 Relay control

Idea

The use of simple discontinuous non-linear control elements (e.g. on-off relais) forcesthe system to oscillate in a limit cycle around the required operating point. For alinear plant, the controlled system is piecewise linear with unknown switching times.

The control may be applied also to non-linear plants, in which case the systemmight behave differently for the different modes (on-off) of the controller (e.g., itmay present different time constants or process gains).

Example 7 Given is the first order process

G(s) =1

τ · s + 1

controlled by the relay controller of figure 1.2 (resulting output in figure 1.3) Such amodel could represent a room controlled by a switched heating system.

20

ref temp ScopeRelay

K

tau.s+1

Plant

Figure 1.2: Relay control of linear system (e.g. temparature control of a room).

500 550 600 650 700 750 800 850 900 950 100019.4

19.6

19.8

20

20.2

20.4

20.6

Figure 1.3: Output of the system with relay control(see Figure 1.2).



Advantages

Linear theory applicable for the different modes of the controller in presence of alinear plant. Controller implementation very simple. Systems with delay elementscan be treated easily as well. Robust with respect to plant parameter variations.Describing function technique can be used for the existence and estimation of limitcycles.

Disadvantages

Oscillation and switching occurs. Steady-state never reached.



11 Gain scheduling (open-loop adaptive scheme)

Idea

The process is operated over a large range of operating conditions, which can changecontinuously over time or assume discrete values.

Linear controllers are designed for these operating conditions. The use of thedesigned controllers is scheduled according to variables which are measured (dis-turbances, controller output, process output, state variable) or otherwise known(reference value, known environment).

In particular, the linear controllers are parametrized in function of the schedulervariables.

Figure 1.4 shows a generic gain scheduling block diagram.

Figure 1.4: Gain scheduling block diagram.

Typical applications of the gain scheduling approach are in the aerospace in-dustry. The behavior (and thus the dynamic model) of an airplance dramaticallychanges in function of the speed: the classical solution consists in changing thecontroller in function of the speed.

Advantages

Linear techniques are applicable. The method can be understood as a feedforwardcontrol for certain scheduler variables. Often a small distinct set of control parametervalues is sufficient.

Disadvantages

Some engineering effort needed for finding the dependency between parameter val-ues and scheduled variable. Slow operating points variations leads to nonlinearbehaviour of the controlled system. Proof for stability might pose a problem.



12 Adaptive Control (closed-loop adaptive scheme)

Idea

An adaptive controller “adapts” its parameters continuously to accommodate rela-tively slow changes in the process dynamics and disturbances. Adaptation can beapplied both to feedback and feedforward control parameters.

There are two classes of adaptive controllers based on direct and indirect meth-ods. In direct methods, controller parameters are adjusted directly from data mea-sured during closed-loop operation (e.g. closed-loop model reference adaptive con-trol). In indirect methods, first process model parameters are determined on-line byrecursive parameter estimation and then the control parameters are derived fromthe process parameter estimates (e.g. self-tuning regulator). Indirect methods canthus be seen as the repeating sequence of an identification and a controller designstep.

Figure 1.5 shows an adaptive control block diagram.

Figure 1.5: Adaptive control scheme.

In the literature is possible to find several closed-loop adaptive control schemes:

• Model reference adaptive system

• Model identification adaptive system

• Adaptive pole placement

• Several auto tuning methods

Several adaptive laws for the controller parameters exist

• Sensitivity and M.I.T methods

• Positivity and Lyapunov design

• Gradient method and least-square methods



Example 8 Simple feedforward controllerFigure 1.6 shows a simple feedforward controller. The parameter a is known but thegain kp of the plant is unknown (however it is known to be positive). The controlobjective is to get the plant output to match a model output ym thanks to the use ofa gain compensation θ to be determined.

r e0Variable Gain

theta

SubtractPlant

kp

s+a

Model

1

s+a

Ym

Yp

Figure 1.6: Simple feedforward controller

The plant and the reference model are described by

yp = −a · yp + kp · θ · rym = −a · ym + r = −a · ym + kp · θ∗ · r

where θ∗ denotes the “nominal” value of parameter θ. Subtraction of the first equa-tion from the second one gives the following expression for the evolution of the outputestimate error ey = yp − ym

ey = −a · ey + kp · (θ − θ∗)︸︷︷︸

eθ

·r

The parameter error is eθ = θ − θ∗. The Lyapunov redesign approach consists infinding an update law so that the Lyapunov function

V (ey, eθ) = e2y + kp · e2

θ

(kp is positive) is decreasing along trajectories of the error system. The objective isthus to find an update law for θ such that the derivative of V

V (ey, eθ) = 2 · eo · eo + 2 · kp · eθ · eθ

= −2 · a · e2y + 2 · kp · ey · eθ · r + 2 · kp · eθ · eθ

is negative semi-definite. Note that since θ∗ is fixed (though unknow), eθ = θ. Choiceof the update law

θ = eθ = −ey · ryields

V (ey, eθ) = −2 · a · e2y ≤ 0

thereby guaranteeing that V = e2y + kp · e2

θ is decreasing. Thus the integration of−ey · r delivers the compensation parameter θ. Note that this proof holds for anunkown but constant value of kp; however, the method can be used also if kp changesslowly enough.



Advantages

Wide operation range possible. Nonlinearities due to parameter changes can beeliminated or reduced.

Disadvantages

Needs enough excitation (otherwise there might be a wind-up problem in parameterestimation). Adaptation loop must be slower than control loop. Leads often tononlinear observer problems. Proof of stability difficult, in particular in case ofchanging process parameters.



13 Neural network control

Idea

A feedforward neural network controller is nothing else than a nonlinear function ofseveral input and output variables with a parameter identification procedure called“training”. The neural network is composed of many identical elements (calledneurons) performing the same nonlinear operation, with the inputs of one elementconnected to the output of other elements. Typically, the neurons are organized inlayers (inputs of neurons of one layer are outputs of neurons from another layer).

The choice of the operation implemented in the neuron (sigmoid, simple thresh-old, ...) as well as the number of layers and number of neurons per layer are themain design parameters.

As input variables for the whole network the control error, its derivative, the setpoint and disturbances can be used; as output variables the control inputs to theplant are used (as for fuzzy controllers).

Neural networks are very useful in classification tasks or in the modelling ofnonlinear plants.

Advantages

Typically data driven. Needs no precise model. Used for feedforward control possi-ble.

Disadvantages

Many parameters must be adjusted. Choice of structure and size critical. Can onlybe used in the region where it has been trained. No stability is guaranteed. Shapeof the function must be checked for all admissible inputs.



14 Fuzzy control

Idea

Knowledge is exploited in form of rules applied to control the system.In a first “fuzzification” step the inputs and the measurements are assigned to a

class with an indication of how “true” the inputs and measurements belong to theclass (e.g. it is 50% true that the input signal is large).

Based on the input classification, the application of rules (called “inference”)translates the belonging to an input class to the belonging to an output class (e.g.if the input signal is large, the output signal is small).

Finally, based on the classification according to the output classes, a quantitativevalue for the ouput is determined (“defuzzyfication”).

The rules are typically provided by experts or operators. The resulting controllaw is just a nonlinear static function between several inputs and controller out-puts. Inputs are usually the control error, the derivative of control error and certainexternal variables (disturbances). Integral action of controller can be added.

Advantages

Intuitive design. Rules document the control action. No precise model needed.Controller can be built up by experience and new rules can be added.

Disadvantages

Problem of completeness of rules and their consistency. No stability is guaranteed.Optimization by trial and error. Care must be given for the control of criticalsystems. Many tuning parameters (fuzzification, inference, defuzzification).


Bibliography

[1] W. S. Levine, editor. The control handbook. CRC Press, 1996. Very broad forevery aspects of control, covers many methods, differs from author to author inthe theoretical background needed to understand the individual contribution.

[2] H. K. Khalil, editor. Nonolinear system. Prentice-Hall, 2002. Standard bookin many courses on nonlinear system and control, theoretically oriented.

[3] D. Bertsekas, editor. Dynamic programming, deterministic and stochastic mod-els. Prentice-Hall, 1987. Together with Dreyfus book on dynamic programmingthe reference book for optimization and control of nonlinear discrete time dy-namical systems with and without uncertainties by the dynamic programmingmethod. Medium to high level.

[4] D. W. Jordan and Smith, editors. Nonlinear ordinary differential equation.Clarendon Press Oxford, 1989. Thorough treatment of analysis methods fordynamical system or low order. Medium level.

[5] O. Foellinger, editor. Nichtlineare Regelung I und II. Oldenburg, 1969, 1980.Classic books in german for the analysis and design of nonlinear system viadescribing function method, phase plane methods, Lyapunov function. Veryreadable.

[6] H. Schwarz, editor. Nichtlineare Regelungsysteme. Oldenburg, 1991. Coversfeedback linearization and Volterra series expansion techniques. Voluminous,not easily readable.

[7] K.H. Seitz, editor. Regelung mit Zwei- und Dreipunktregeln. Oldenburg, 1986.Covers many type of discontinuous controller and linear system combinations.Practically oriented.

[8] VDI-Tagung: Nichtlineare Regelung, Methoden, Werkzeuge, Anwendungen.In Bericht, number 1026. VDI Verlag, 1993. Interesting book as it comparesdifferent methods (including fuzzy control) of designing a nonlinear controllerfor three given processes: a mixing tank, an electromechanical actuator an ahydraulic drive.



[9] Anders Robertson. Nonlinear control theory. Technical report, Lund University,2006. Lecture notes, very good summary of important results for nonlinearsystems and control.

[10] Michael D. Lemmon. Nonlinear control. Technical report, Clemson University,South Carolina, 2004. Lecture notes, very good summary of important resultsfor nonlinear systems and control.



Exercises

A.1 Linearization

Calculate the equilibria of the following nonlinear system (Lorentz equations formodelling the weather dynamics):

x = −ax + ay

y = bx − y − xz

z = −cz + xy

and a, b, c > 0. Determine the linearized systems around the operating points.



Solved exercises

B.1 Sliding-mode control of an inverted pendulum

Given are the inverted pendulum equations

x1 = x2

x2 =−g

l· [sin (x1 + δ) − sin δ] − d

m· x2

︸︷︷︸

f(x)

+1

m · l2︸︷︷︸

g(x)

·u

Design a sliding mode control according to the description in the correspondingsection of this document. Write the expression of β, choose a control form and writethe control law.

Solution: Solution:

The expression for β(x) is

β(x) = −f(x) + a · x2

g(x)= g · m · l · sin(x1 + α) + d · m · l2 · x2 − m · l2 · a · x2

With the choice of the controller K

K = k1 + k2 · s2 k1 , k2 > 0

the control law becomes

u = β(x) − K · sign(s)

= g · m · l · (sin(x1 + α) − sin(α)) + d · m · l2 · x2 − m · l2 · a · x2

−(k1 + k2 · s2) · sign(s)

where α is the reference input.

The simulation results with the input reference = π2

is shown in Figure 1.7 wherethe reaching phase and the sliding phase are clearly identifiable. Figure 1.8 showsthe actuation signal.



−0.5 −0.4 −0.3 −0.2 −0.1 0−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x(1) (rad)

x(2)

(ra

d/s)

Figure 1.7: States evolution x2 = f(x1) (phase portrait) for the inverted pendulumcontrolled wth a sliding mode controller.

0 0.5 1 1.5 2 2.5 3−15

−10

−5

0

5

10

15

tempo (s)

u(t)

Figure 1.8: Actuation signal for the sliding mode control of the inverted pendulum.



B.2 Nonlinear feedback

Given is the mechanical system represented in Figure 1.9 (see [1]). Write the non-linear differential equation of motion, the equivalent block diagram and design afeedback scheme for the linearization of the system and impose a desired transferfunction from the reference angle to the actual angle.

Figure 1.9: Mechanical system.

Solution: The equation of motion is

m · r2g · θ = Ce − cf · θ − m · g · rg · cosθ (1.8)

Figure 1.10 shows the mechanical system model. The choice made here is to forcethe behaviour from the reference angle θc and the actual angle θ according to thetransfer function

θ(s)

θc(s)=

b0

s2 + a1 · s + a0

Note that if θ = θc at steady-state, then

limt→∞

θ(t)

θc(t)= lim

s→0

θ(s)

θc(s)= lim

s→0

b0

s2 + a1 · s + a0

=b0

a0

Thus b0 = a0 andθ + a1 · θ + a0 · θ = a0 · θc

Substitution of this expression in equation (1.8) leads to the final result

Ce = m · r2g · (a0 · (θc − θ) − a1 · θ) + cf · θ + m · g · rg · cosθ



Figure 1.10: Model

Figure 1.11: System with non linear feedback.


Overview of Non-linear Control Methods

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