Robust Control Theory and Applications Part 13 doc

The nonlinear inputs-to-flatoutputs representation is viewed as a linear perturbed system in which only the orders ofintegration of the Kronecker subsystems and the control input gain mat

linearization, we introduce, as an extended auxiliary control input, the time derivative of u1.

We have:

˙u1= ˙x1¨x1+ ˙x2¨x2

˙x2

1+ ˙x2This control input extension yields now an invertible control input-to-flat outputs highestderivatives relation, of the form:

4.2 Observer-based GPI controller design

Consider the following multivariable feedback controller based on linear GPI controllers andestimated cancelation of the nonlinear input matrix gain:



˙u1u2

ˆ˙x2

(ˆ˙x1)2+ (ˆ˙x2) 2

(x1− x ∗1(t))

ν2 = ¨x2(t) −



k22s2+k21s+k20s(s+k23)

(x2− x ∗2(t))

(33)

and where the estimated velocity variables: ˆ˙x1, ˆ˙x2, are generated, respectively, by the variables

ρ11andρ12in the following single iterated integral injection GPI observers (i.e., with m=1),

˙ˆy10= ˆy1+λ13(y10− ˆy10)

˙ˆy1=ρ11+λ12(y10− ˆy10)

˙ˆy20=ˆy2+λ23(y20 − ˆy20)

˙ˆy2=ρ12+λ22(y20 − ˆy20)

Theorem 7. Given a set of desired reference trajectories, (x∗(t), y ∗(t)), for the desiredposition in the plane of the kinematic model of the car, described by (30); given aset initial conditions, (x(0), y(0)), sufficiently close to the initial value of the desirednominal trajectories,(x∗(0), y ∗(0)), then, the above described GPI observers and the linearmulti-variable dynamical feedback controllers, (32)-(35), forces the closed loop controlledsystem trajectories to asymptotically converge towards a small as desired neighborhood ofthe desired reference trajectories,(x∗

1(t), x ∗2(t)), provided the observer and controller gains

1 Here we have combined, with an abuse of notation, frequency domain and time domain signals.

469Robust Linear Control of Nonlinear Flat Systems

are chosen so that the roots of the corresponding characteristic polynomials describing,respectively, the integral injection estimation error dynamics and the closed loop system, arelocated deep into the left half of the complex plane Moreover, the greater the distance ofthese assigned poles to the imaginary axis of the complex plane, the smaller the neighborhoodthat ultimately bounds the reconstruction errors, the trajectory tracking errors, and their timederivatives.

Proof Since the system is differentially flat, in accordance with the results in Maggiore

& Passino (2005), it is valid to make use of the separation principle, which allows us topropose the above described GPI observers The characteristic polynomials associated withthe perturbed integral injection error dynamics of the above GPI observers, are given by,

P ε1(s) =P ε2(s) = (s+2μ1σ1s+σ2)(s+2μ2σ2s+σ2)

s ∈ C, μ1,μ2,σ1,σ2∈R+

Since the estimated states, ˆ˙x1=ρ11, ˆ˙x2=ρ12, asymptotically exponentially converge towards

a small as desired vicinity of the actual states: ˙x1, ˙x2, substituting (32) into (31), transformsthe control problem into one of controlling two decoupled double chains of integrators Oneobtains the following dominant linear dynamics for the closed loop tracking errors:

e(4)1 +k13e(3)1 +k12¨e1+k11˙e1+k10e1=0 (36)

e(4)2 +k23e(2)2 +k22¨e2+k21˙e2+k20e2=0 (37)The pole placement for such dynamics has to be such that both corresponding associatedcharacteristic equations guarantee a dominant exponentially asymptotic convergence Settingthe roots of these characteristic polynomials to lie deep into the left half of the complex planeone guarantees an asymptotic convergence of the perturbed dynamics to a small as desiredvicinity of the origin of the tracking error phase space

4.3 Experimental results

An experimental implementation of the proposed controller design method was carried out

to illustrate the performance of the proposed linear control approach The used experimentalprototype was a parallax “Boe-Bot" mobile robot (see figure 5) The robot parameters are the

following: The wheels radius is R = 0.7[m]; its axis length is L = 0.125[m] Each wheelradius includes a rubber band to reduce slippage The motion system is constituted by two

servo motors supplied with 6 V dc current The position acquisition system is achieved by

means of a color web cam whose resolution is 352×288 pixels The image processing wascarried out by the MATLAB image acquisition toolbox and the control signal was sent to therobot micro-controller by means of a wireless communication scheme The main function of

the robot micro-controller was to modulate the control signals into a PWM input for the motor.The used micro-controller was a BASIC Stamp 2 with a blue-tooth communication card Figure

4 shows a block diagram of the experimental framework The proposed tracking tasks was asix-leaved “rose" defined as follows:

x ∗1(t) =sin(3ωt+η)sin(2ωt+η)

x ∗2(t) =sin(3ωt+η)cos(2ωt+η)

The design parameters for the observers were set to be,μ1 =1.8,μ2 =2.3,σ1 = 3,σ2 =4and for the corresponding parameters for the controllers,ζ1 = ζ3 = 1.2, ζ2 = ζ4 = 1.5,

ω n1 = ω n3 = 1.8, ω n2 = ω n4 = 1.9 Also, we compared the observer response with that

of a GPI observer without the integral injection (x1_, x2_) Luviano-Juárez et al (2010) Theexperimental implementation results of the control law are depicted in figures, 6 and 7, wherethe control inputs and the tracking task are depicted Notice that in the case of figure 8, there is

a clear difference between the integral injection observer and the usual observer; the filteringeffect of the integral observer helped to reduce the high noisy fluctuations of the control inputdue to measurement noises On the average, the absolute error for the tracking task, for boothschemes, is less than 1 [cm] This is quite a reasonable performance considering the height ofthe camera location and its relatively low resolution

MicroController

BluetoothAntenna

USB Camera

USB Port

Fig 5 Mobile Robot Prototype

5 Conclusions

In this chapter, we have proposed a linear observer-linear controller approach for the robusttrajectory tracking task in nonlinear differentially flat systems The nonlinear inputs-to-flatoutputs representation is viewed as a linear perturbed system in which only the orders ofintegration of the Kronecker subsystems and the control input gain matrix of the system areconsidered to be crucially relevant for the controller design The additive nonlinear terms

in the input output dynamics can be effectively estimated, in an approximate manner, bymeans of a linear, high gain, Luenberger observer including finite degree, self updating,polynomial models of the additive state dependent perturbation vector components Thisperturbation may also include additional unknown external perturbation inputs of uniformlyabsolutely bounded nature A close approximate estimate of the additive nonlinearities isguaranteed to be produced by the linear observers thanks to customary, high gain, poleplacement procedure With this information, the controller simply cancels the disturbancevector and regulates the resulting set of decoupled chain of perturbed integrators after adirect nonlinear input gain matrix cancelation A convincing simulation example has beenpresented dealing with a rather complex nonlinear physical system We have also shownthat the method efficiently results in a rather accurate trajectory tracking output feedbackcontroller in a real laboratory implementation A successful experimental illustration waspresented which considered a non-holonomic mobile robotic system prototype, controlled by

an overhead camera

0 50 100 150 200

−0.1

−0.05

00.05

Aguilar, L E., Hamel, T & Soueres, P (1997) Robust path following control for wheeled robots

via sliding mode techniques, International Conference of Intelligent Robotic Systems,

pp 1389–1395

Bazanella, A S., Kokotovic, P & Silva, A (1999) A dynamic extension for l g v controllers, IEEE

Transactions on Automatic Control 44(3).

Cortés-Romero, J., Luviano-Juárez, A & Sira-Ramírez, H (2009) Robust gpi controller for

trajectory tracking for induction motors, IEEE International Conference on Mechatronics,

Málaga, Spain, pp 1–6

Diop, S & Fliess, M (1991) Nonlinear observability, identifiability and persistent trajectories,

Proceedings of the 36th IEEE Conference on Decision and Control, Brighton, England,

pp 714 – 719

Dixon, W., Dawson, D., Zergeroglu, E & Behal, A (2001) Nonlinear Control of Wheeled Mobile

Robots, Vol 262 of Lecture Notes in Control and Information Sciences, Springer, Great

Britain

Fliess, M & Join, C (2008) Intelligent PID controllers, Control and Automation, 2008 16th

Mediterranean Conference on, pp 326 –331.

Fliess, M., Join, C & Sira-Ramirez, H (2008) Non-linear estimation is easy, International Journal

of Modelling, Identification and Control 4(1): 12–27.

Fliess, M., Lévine, J., Martin, P & Rouchon, P (1995) Flatness and defect of non-linear systems:

introductory theory and applications, International Journal of Control 61: 1327–1361.

Fliess, M., Marquez, R., Delaleau, E & Sira-Ramírez, H (2002) Correcteurs proportionnels

intègraux généralisés, ESAIM: Control, Optimisation and Calculus of Variations

7(2): 23–41

Fliess, M & Rudolph, J (1997) Corps de hardy et observateurs asymptotiques locaux pour

systèmes différentiellement plats, C.R Academie des Sciences de Paris 324, Serie II

b: 513–519

Gao, Z (2006) Active disturbance rejection control: A paradigm shift in feedback

control system design, American Control Conference, Minneapolis, Minnesota, USA,

p 2399 ˝U2405

Gao, Z., Huang, Y & Han, J (2001) An alternative paradigm for control system design, 40th

IEEE Conference on Decision and Control, Vol 5, pp 4578–4585.

Han, J (2009) From PID to active disturbance rejection control, IEEE Transactions on Industrial

Electronics 56(3): 900–906.

Hingorani, N G & Gyugyi, L (2000) Understanding FACTS, IEEE Press, Piscataway, N.J.

Hou, Z., Zou, A., Cheng, L & Tan, M (2009) Adaptive control of an electrically driven

nonholonomic mobile robot via backstepping and fuzzy approach, IEEE Transactions

on Control Systems Technology 17(4): 803–815.

Johnson, C D (1971) Accommodation of external disturbances in linear regulator and

servomechanism problems, IEEE Transactions on Automatic Control AC-16(6).

Johnson, C D (1982) Control and Dynamic Systems: Advances in Theory and applications, Vol 18,

Academic Press, NY, chapter Discrete-Time Disturbance-Accommodating ControlTheory, pp 224–315

Johnson, C D (2008) Real-time disturbance-observers; origin and evolution of the idea part

1: The early years, 40th Southeastern Symposium on System Theory, New Orleans, LA,

USA

Kailath, T (1979) Linear Systems, Information and System Science Series, Prentice-Hall, Upper

Saddle River, N.J

Kim, D & Oh, J (1999) Tracking control of a two-wheeled mobile robot using input-output

linearization, Control Engineering Practice 7(3): 369–373.

Leroquais, W & d’Andrea Novel, B (1999) Modelling and control of wheeled mobile robots

not satisfying ideal velocity constraints: The unicycle case, European Journal of Control

5(2-4): 312–315

Luenberger, D (1971) An introduction to observers, IEEE Transactions on Automatic Control

16: 592–602

Luviano-Juárez, A., Cortés-Romero, J & Sira-Ramírez, H (2010) Synchronization of chaotic

oscillators by means of generalized proportional integral observers, International Journal of Bifurcation and Chaos 20(5): 1509 ˝U–1517

Maggiore, M & Passino, K (2005) Output feedback tracking: A separation principle

approach, IEEE Transactions on Automatic Control 50(1): 111–117.

disturbance-accommodation, 22nd IEEE Conference on Conference on Decision and Control, San Antonio, Texas.

Pai, M A (1989) Energy Function Analysis for Power System Stability, Kluwer Academic

Publishers

475Robust Linear Control of Nonlinear Flat Systems

Parker, G A & Johnson, C D (2009) Decoupling linear dynamical systems using

disturbance accommodation control theory, 41st Southeastern Symposium on System Theory, Tullahoma, TN, USA.

Peng, J., Wang, Y & Yu, H (2007) Advances in Neural Networks, Vol 4491 of Lecture Notes

in Computer Science, Springer Berlin / Heidelberg, chapter Neural Network-Based

Robust Tracking Control for Nonholonomic Mobile Robot, pp 804–812

Pomet, J (1992) Explicit design of time-varying stabilizing control laws for a class of

controllable systems without drift., Systems and Control Letters 1992 18(2): 147–158 Samson, C (1991) Advanced Robot Control, Vol 162 of Lecture Notes in Control and Information

Sciences, Springer, chapter Velocity and torque feedback control of a nonholonomic

cart, pp 125–151

Sira-Ramírez, H & Agrawal, S (2004) Differentially flat systems, Marcel Dekker Inc.

Sira-Ramírez, H & Feliu-Battle, V (2010) Robust σ − δ modulation based sliding mode

observers for linear systems subject to time polynomial inputs, International Journal of Systems Science To appear.

Sira-Ramírez, H & Fliess, M (2004) On the output feedback control of a synchronous

generator, Proceedings of the 43rd IEEE Conference on Decision and Control, Bahamas.

Sugisaka, M & Hazry, D (2007) Development of a proportional control method for a mobile

robot, Applied Mathematics and Computation 186: 74–82.

Sun, B & Gao, Z (2005) A dsp-based active disturbance rejection control design for a

1-kw h-bridge dc ˝Udc power converter, IEEE Transactions on Industrial Electronics

52(5): 1271 ˝U1277

Sun, D (2007) Comments on active disturbance rejection control, IEEE Transactions on

Industrial Electronics 54(6): 3428–3429.

Tian, Y & Cao, K (2007) Time-varying linear controllers for exponential tracking of

non-holonomic systems in chained form, International Journal of Robust and Nonlinear Control 17: 631–647.

Wang, Z., Li, S & Fei, S (2009) Finite-time tracking control of a nonholonomic mobile robot,

Asian Journal of Control 11(3): 344–357.

Wit, C C D & Sordalen, O (1992) Exponential stabilization of mobile robots with

nonholonomic constraints, IEEE Transactions on Automatic Control 37(11): 1791–1797.

Yang, J & Kim, J (1999) Sliding mode control for trajectory tracking of nonholonomic

wheeled mobile robots, IEEE Transactions on Robotics and Automation 15(3): 578–587.

Part 5

Robust Control Applications

Hao Zhang1and Huaicheng Yan2

1Department of Control Science and Engineering, Tongji University, Shanghai 200092

2School of Information Science and Engineering, East China University of Science and

Internet-based control is an interesting and challenging topic One of the major challenges

in Internet-based control systems is how to deal with the Internet transmission delay Theexisting approaches of overcoming network transmission delay mainly focus on designing

a model based time-delay compensator or a state observer to reduce the effect of thetransmission delay Being distinct from the existing approaches, literatures (7–9) have beeninvestigating the overcoming of the Internet time-delay from the control system architectureangle, including introducing a tolerant time to the fixed sampling interval to potentiallymaximize the possibility of succeeding the transmission on time Most recently, a dual-ratecontrol scheme for Internet-based control systems has been proposed in literature (10) Atwo-level hierarchy was used in the dual-rate control scheme At the lower level a localcontroller which is implemented to control the plant at a higher frequency to stabilize theplant and guarantee the plant being under control even the network communication is lostfor a long time At the higher level a remote controller is employed to remotely regulate thedesirable reference at a lower frequency to reduce the communication load and increase thepossibility of receiving data over the Internet on time The local and the remote controller arecomposed of some modes, which mode is enabled due to the time and state of the network

The mode may changes at instant time k, k ∈ { N+}and at each instant time only one mode

of the controller is enabled A typical dual-rate control scheme is demonstrated in a processcontrol rig (7; 8) and has shown a great potential to over Internet time-delay and bring thisnew generation of control systems into industries However, since the time-delay is variableand the uncertainty of the process parameters is unavoidable, a dual-rate Internet-basedcontrol system may be unstable for certain control intervals The interest in the stability of

Passive Robust Control for Internet-Based

Time-Delay Switching Systems

21

networked control systems have grown in recent years due to its theoretical and practicalsignificance [11-21], but to our knowledge there are very few reports dealing with the robustpassive control for such kind of Internet-based control systems The robust passive controlproblem for time-delay systems was dealt with in (24; 25) This motivates the present passivityinvestigation of multi-rate Internet-based switching control systems with time-delay anduncertainties.

In this paper, we study the modelling and robust passive control for Internet-based switchingcontrol systems with multi-rate scheme, time-delay, and uncertainties The controller isswitching between some modes due to the time and state of the network, either different time

or the state changing may cause the controller changes its mode and the mode may changes ateach instant time Based on remote control and local control strategy, a new class of multi-rateswitching control model with time-delay is formulated Some new robust passive properties

of such systems under arbitrary switching are investigated An example is given to illustratethe effectiveness of the theoretical results

Notation: Through the paper I denotes identity matrix of appropriate order, and ∗representsthe elements below the main diagonal of a symmetric block matrix The superscript 

represents the transpose L2[0,∞) refers to the space of square summable infinite vector

sequences The notation X > 0(≥,<,≤ 0)denotes a symmetric positive definite (positive

semi-definite, negative, negative semi-definite) matrix X Matrices, if not explicitly stated, are assumed to have compatible dimensions Let N = {1, 2,· · · } and N+ = {0, 1, 2,· · · }denotethe sets of positive integer and nonnegative integer, respectively

2 Problem formulation

A typical multi-rate control structure with remote controller and local controller can be shown

as Fig 1 The control architrave gives a discrete dynamical system, where plant is in circle

with broken line, x(k) ∈ R n is the system state, z(k) ∈ R qis the output, andω(k) ∈ R pis

the exogenous input, which is assumed to belong to L2[0,∞), r(k)is the input and for the

passivity analysis one can let r(k) = 0, u1(k) and u2(k) are the output of remote control

and local control, respectively A1, B1, B2 and C are parameter matrices of the model with appropriate dimensions, K 2i and K 1jare control gain switching matrices where the switching

rules are given by i(k) = s(x(k) , k) and j(k) = σ(x(k) , k), and i ∈ {1, 2,· · · , N1} , j ∈ {1, 2,· · · , N2} , N1, N2∈ N, which imply that the switching controllers have N1and N2modes,respectively.τ1andτ2are time-delays caused by communication delay in systems

For the system given by Fig 1, it is assumed that, the sampling interval of remote controller is

the m multiple of local controller with m being positive integer, and the switching device SW1 closes only at the instant time k = nm, n ∈ N+, and otherwise, it switches off

Correspondingly, remote controller u1(k) updates its state at k = nm, n ∈ N+ only, andotherwise, it keeps invariable Also, it is assumed that the benchmark of discrete systems isthe same as local controller In this case, the system can be described by the following discretesystem with time-delay

Fig 1 Multi-rate network control loop with time-delays

where remote controller u1(k− τ2)is given by

x(k+1)=(A1− B2K 2i) x(k) − B2B1K 1j x(k − τ) + Eω(k) , k=nm,

x(k+1)=(A1− B2K 2i) x(k) − B2B1K 1j x(nm − τ) + Eω(k) , k ∈ { nm+1,· · · , nm+m −1},

z(k) =C x(k) + Dω(k),

(5)whereτ=τ1+τ2> 0, k ∈ N+ , n ∈ N+ , m >0 is a positive integer

Obviously, if define A i=A1 − B2K2i , B j = −B2 B1K1j, then the controlled system (5) becomes

where A1, B1, B2, C, D, E are matrices with appropriate dimensions, K 1j and K 2iare mode gain

matrices of the remote controller and local controller At each instant time k, there is only

one mode of each controller is enabled.τ = τ1+τ2 > 0 and m > 0 are integers, k ∈ N+,

with 0≤ τ ≤ h ≤ τ+m −1, andΔA(k),ΔB(k)andΔE being structured uncertainties, and

are assumed to have the form of

ΔA(k) =D1F(k)E a, ΔB(k) =D1F(k)E b, ΔE(k) =D1F(k)E e, (9)

where D1, E a , E b and E eare known constant real matrices with appropriate dimensions It isassumed that

For the case of structured uncertainties, it can be described by

where A i(k) = A i+ΔA(k), B j(k) = B j+ΔB(k), E(k) = E+ΔE(k), and it is assumed that(9) and (10) are satisfied Our problem is to test whether system (11) and (12) are passive withthe switching controllers To this end, we introduce the following fact and related definition

whereβ is some constant which depends on the initial condition of system.

In the sequel, we provide condition under which a class of discrete-time switching dynamicalsystems with time-delay and uncertainties can be guaranteed to be passive

System (11) can be recast as

It is noted that (11) is completely equivalent to (16)

Theorem 1. System (11) is passive under arbitrary switching rules s and σ, if there exist matrices P1> 0, P2, P3, W1, W2, W3, M1, M2, S1> 0, S2>0 such that the following LMIs hold

Proof Construct Lyapunov function as

B j

x(k) +

0

=η(k)hWη(k) +2η (k)(M− P

0

B j

)(x(k) −x(k − h)) + k−1∑

From (19)-(22) we can get

0

Eω(k)

−2(x(k)C ω(k) + ω (k)D ω(k)).Letξ (k) = [x(k), y (k), x (k− h),ω (k)], thenΔV(k) −2z (k)ω(k) ≤ξ(k)  υξ(k), where



0 P1+hS1

If v <0, then V(k) − 2z (k)ω(k) <0, which gives

where Q1, Q2, Q3, W, M are defined in Theorem 1 and E a , E b , E eare given by (9) and (10).

Proof.Replacing A i , B j and E in (17) with A i+D1F(k)E a , B j+D1F(k)E b and E+D1F(k)E e,respectively, we find that (17) for (12) is equivalent to the following condition

⎤

⎥

⎦ F(k)E a 0 E b E e

+

ReplacingλP, λS1,λS2,λM and λW with P, S1 , S2, M and W respectively, and applying the

Schur complement shows that (26) is equivalent to (24) This completes the proof

4 A numerical example

In this section, we shall present an example to demonstrate the effectiveness and applicability

of the proposed method Consider system (12) with parameters as follows:

B3=



−1 0

0 −1

C=0.1−0.2

, E=

0.20.1

, E a=

0.5 00.1 0.2

, E b=

0.6 0

0 0.3

, D1=

0.1 0

0 1

,

D=0.1, h=5

Applying Theorem 2, with i ∈ {1, 2} , j ∈ {1, 2, 3} It has been found by using software LMIlabthat the switching discrete time-delay system (12) is the passive and we obtain the solution asfollows:

P1=10−3 ×

0.1586 0.0154

∗ 0.2660

, P2=

0.5577 0.3725

−1.6808 1.0583

, P3=

0.1689 −0.0786

−0.0281 0.1000

,

S1=10−4 ×

0.4207 0.0405

∗ 0.6941

, S2=

2.6250 0.8397

∗ 2.0706

, W1=

0.2173−0.0929

,

,

M2=10−5 ×

0.0985−0.43040.1231−0.9483