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Robust Algorithm Using Delay for Multi-Agent Systems*
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IFAC-PapersOnLine 49-13 (2016) 025–030
1 Robust Robust Algorithm Algorithm Using Using Delay Delay for for Multi-Agent Multi-Agent Systems Systems1111 Robust Using Robust Algorithm Algorithm Using Delay Delay for for Multi-Agent Multi-Agent Systems Systems ,
N. Nekhoroshikh**, Mikhail S. Tarasov** Igor B. **, Artem Artem Mikhail S. Tarasov** Igor B. Furtat* Furtat*,,,**, Artem N. N. Nekhoroshikh**, Nekhoroshikh**, Mikhail S. Tarasov** Igor Igor B. B. Furtat* Furtat*,,**, **, Mikhail S. Tarasov** **, Artem Artem N. N. Nekhoroshikh**, Nekhoroshikh**, Mikhail S. Tarasov** Igor B. Furtat* *Institute for Problems of Mechanical Engineering Russian Academy of Sciences, 61 Bolshoy ave V.O., St.-Petersburg, *Institute for Problems of Mechanical Engineering Russian Academy of Sciences, 61 Bolshoy ave V.O., St.-Petersburg, *Institute for Problems of Mechanical Engineering Russian Academy of Sciences, 61 Bolshoy ave V.O., St.-Petersburg, 199178, Russia (Tel: +7-812-321-47-66; e-mail: [email protected]). *Institute for Problems of Mechanical Engineering Russian Academy of Sciences, 61 Bolshoy ave V.O., St.-Petersburg, *Institute for Problems of Mechanical Engineering Russian Academy of Sciences, 61 Bolshoy ave V.O., St.-Petersburg, 199178, Russia (Tel: +7-812-321-47-66; e-mail: [email protected]). 199178, Russia (Tel: +7-812-321-47-66; e-mail: [email protected]). **ITMO University, 49 Kronverkskiy ave, Saint Petersburg, 197101, Russia. 199178, Russia (Tel: +7-812-321-47-66; e-mail: [email protected]). 199178, Russia (Tel: +7-812-321-47-66; e-mail: [email protected]). **ITMO University, 49 Kronverkskiy ave, Saint Petersburg, 197101, Russia. **ITMO University, 49 Kronverkskiy ave, Saint Petersburg, 197101, Russia. **ITMO **ITMO University, University, 49 49 Kronverkskiy Kronverkskiy ave, ave, Saint Saint Petersburg, Petersburg, 197101, 197101, Russia. Russia. Abstract: The paper describes the robust control algorithm for linear multi-agent systems under Abstract: The The paper paper describes describes the the robust robust control control algorithm algorithm for for linear linear multi-agent multi-agent systems systems under under Abstract: parametric and structural uncertainties and external unmeasured disturbances. The proposed algorithm is Abstract: The paper describes the robust control algorithm for linear multi-agent systems under Abstract: The paper describes the robust control algorithm for linear multi-agent systems under parametric and and structural structural uncertainties uncertainties and and external external unmeasured unmeasured disturbances. disturbances. The The proposed proposed algorithm algorithm is is parametric based on left hand side differences for estimation of the derivatives. The resulting algorithm ensures parametric and structural uncertainties and external unmeasured disturbances. The proposed algorithm is parametric and structural uncertainties and external unmeasured disturbances. The proposed algorithm is based on on left left hand hand side side differences differences for for estimation estimation of of the the derivatives. derivatives. The The resulting resulting algorithm algorithm ensures ensures based required accuracy ofside difference between the plant plant output output and the reference reference signal. Thealgorithm modelingensures results based on left differences for of derivatives. The resulting based onaccuracy left hand handof side differences for estimation estimation of the theand derivatives. The signal. resulting algorithm ensures required accuracy of difference between the and the signal. The modeling results required difference between the plant output the reference The modeling results illustrate the effectiveness of the algorithm. required accuracy of difference between the plant output and the reference signal. The modeling results required of difference the plant output and the reference signal. The modeling results illustrate accuracy the effectiveness effectiveness of the the between algorithm. illustrate the of algorithm. illustrate the effectiveness of the algorithm. illustrate the effectiveness of the algorithm. © 2016, IFAC (International Federation Automatic Control) Hostingfunctional, by Elsevierdescriptor Ltd. All rights reserved. Keywords: Robust control, time time delay, of LMI, Lyapunov-Krasovskii method. Keywords: Robust control, delay, LMI, Lyapunov-Krasovskii functional, descriptor descriptor method. method. Keywords: Robust control, time delay, LMI, Lyapunov-Krasovskii functional, Keywords: Robust control, time delay, LMI, Lyapunov-Krasovskii functional, descriptor method. Keywords: Robust control, time delay, LMI, Lyapunov-Krasovskii functional, descriptor method.
with aa high-gain dynamical observer. The papers (Bazylev, with dynamical observer. The papers (Bazylev, with aa high-gain high-gain dynamical observer. The papers (Bazylev, Pyrkin, 2013, Bazylev, Zimenko, Margun et al., 2014, 1. INTRODUCTION with high-gain dynamical observer. The papers (Bazylev, with a high-gain dynamical observer. The papers (Bazylev, Pyrkin, 2013, Bazylev, Zimenko, Margun et al., 2014, 1. INTRODUCTION Pyrkin, 2013, Bazylev, Zimenko, Margun et al., 2014, 1. INTRODUCTION Bazylev, Margun, Zimenko, 2014, Pyrkin, Bobtsov, Pyrkin, 2013, Bazylev, Zimenko, Margun et al., 2014, 1. Pyrkin, 2013, Bazylev, Zimenko, Margun et al., 2014, 1. INTRODUCTION INTRODUCTION Bazylev, Margun, Zimenko, 2014, Pyrkin, Bobtsov, Bazylev, Margun, Zimenko, 2014, Pyrkin, Bobtsov, Design of simple control systems under parametric Kolyubin et al., 2013, Furtat, 2013, Furtat and Putov, 2013, Bazylev, Margun, Zimenko, 2014, Pyrkin, Bobtsov, Design of simple control systems under parametric Design of simple control systems under parametric Bazylev, Margun, Zimenko, 2014, Pyrkin, Bobtsov, Kolyubin et al., 2013, Furtat, 2013, Furtat and Putov, 2013, Kolyubin et al., 2013, Furtat, 2013, Furtat and Putov, 2013, uncertainties and external when only plant Design of control systems parametric Furtat, 2014, )) are used the adaptive control high gain Design of simple simple controldisturbances systems under under parametric Kolyubin et 2013, Furtat, 2013, and Putov, 2013, uncertainties and external disturbances when only plant uncertainties and external disturbances when only plant Kolyubin et al., al., 2013, Furtat, 2013, Furtat Furtat andwith Putov, 2013, Furtat, 2014, are used the adaptive control with high gain Furtat, 2014, ) are used the adaptive control with high gain output is available for measurement is important problem of uncertainties and external disturbances when only plant observers. Furtat, 2014, ) are used the adaptive control with high gain uncertainties and external disturbances when only plant output is available for measurement is important problem of output is available for measurement is important problem of Furtat, 2014, ) are used the adaptive control with high gain observers. observers. control theory and practice. To construct such control output is available for measurement is important problem of observers. output is available for measurement is important problem of control theory and practice. To construct such control control theory and practice. To construct such control observers. Analysis of the above results shows that the designed schemes many solutions have been this If control theory and To construct such control Analysis of the above results shows that the designed control theory and practice. practice. To proposed constructin suchregard. control schemes many solutions have been proposed in this regard. If Analysis of the above results shows that the designed schemes many solutions have been proposed in this regard. If algorithms use integrators for estimation of the derivatives. the plant relative degree exceeds one then implementation of Analysis of the above results shows that the designed schemes many solutions have been proposed in this regard. If Analysis of the above results shows that the designed algorithms use integrators for estimation of the derivatives. schemes many solutions have been proposed in this regard. If the plant relative degree exceeds one then implementation of algorithms use integrators for estimation of the derivatives. the plant relative degree exceeds one then implementation of Unlike these results we design the observer using left-hand adaptive and robust control systems requires the estimation algorithms use integrators for estimation of the derivatives. the plant relative degree exceeds one then implementation of algorithms use integrators for estimation of the derivatives. Unlike these results we design the observer using left-hand the plant and relative degree exceeds onerequires then implementation of adaptive robust control systems the estimation Unlike these results we design the observer using left-hand adaptive and robust control systems requires the estimation of side differences. This allows designing robust the derivatives of the plant input and output. these we design observer using left-hand adaptive and robust robust control systems requires the estimation estimation of of Unlike Unlike these results results weobserver design the the observer usingthe left-hand adaptive and control systems requires the side differences. This observer allows designing the robust the derivatives of the plant input and output. side differences. This observer allows designing the robust the derivatives of the plant input and output. control system without using integrators but using delays. For side differences. This observer allows designing the the derivatives of the and output. side differences. This observer allows designing the robust robust the derivatives ofobserver the plant plantisinput input andused output. control system without using integrators but using delays. For control system without using integrators but using delays. For The Luenberger widely to estimate the plant algorithm synthesis the results of (Furtat, 2015) will be used. control system without using integrators but using delays. For The Luenberger Luenberger observer observer is is widely widely used used to to estimate estimate the the plant plant control The system without using integrators but using delays. For algorithm synthesis the results of (Furtat, 2015) will be used. algorithm synthesis the results of (Furtat, 2015) will be used. model state vector with known parameters (Luinberger, The Luenberger observer is widely used to estimate the plant The Luenberger observer is widely used to estimate the plant algorithm synthesis the results of (Furtat, 2015) will be used. model state vector with known parameters (Luinberger, model state vector with known parameters (Luinberger, algorithm synthesis the results of (Furtat, 2015) will be used. The paper is organized as follows. The problem statement is 1966). to estimates state vector model state vector with known parameters (Luinberger, The paper is organized as follows. The problem statement is model The stateKalman vector filter withallows known parametersthe (Luinberger, 1966). The Kalman filter allows to estimates the state vector The paper is organized as follows. The problem statement is 1966). The Kalman filter allows to estimates the state vector presented in Section 2. In Section 3 we design the control of a dynamical system under disturbances and noises The paper is organized as follows. The problem statement is 1966). The Kalman filter allows to estimates the state vector The paper is organized as follows. The problem statement is presented in Section 2. In Section 3 we design the control 1966). The Kalman filter allows to estimates the state vector of a dynamical system under disturbances and noises presented in Section 2. In Section 3 we design the control of a dynamical system under disturbances and noises algorithm for multi-agent systems with known relative degree (Kalman, 1960). Under parametric uncertainty and external presented in Section 2. In Section 3 we design the control of a dynamical system under disturbances and noises presented in Section 2. In Section 3 we design the control algorithm for multi-agent systems with known relative degree of a dynamical system under disturbances and noises (Kalman, 1960). Under parametric uncertainty and external algorithm for multi-agent systems with known relative degree (Kalman, 1960). Under parametric uncertainty and external of each agent. In Section 3 we propose the control algorithm disturbances in the paper (Esfandiary and Khalil, soalgorithm for systems with relative degree (Kalman, Under parametric uncertainty and external algorithm for multi-agent multi-agent systems with known known relative degree of each agent. In Section 3 we propose the control algorithm (Kalman, 1960). 1960). Under parametric uncertainty and1992) external disturbances in the paper (Esfandiary and Khalil, 1992) soof each agent. In Section 3 we propose the control algorithm disturbances in the paper (Esfandiary and Khalil, 1992) sofor multi-agent systems with unknown relative degree of each called high-gain observer is proposed. Other form of an of each agent. In Section 3 we propose the control algorithm disturbances in the paper (Esfandiary and Khalil, 1992) soof each agent. In Section 3 we propose the control algorithm for multi-agent systems with unknown relative degree of each disturbances in theobserver paper (Esfandiary andOther Khalil, 1992) socalled high-gain is proposed. form of an for multi-agent systems with unknown relative degree of each called high-gain observer is proposed. Other form of an agent. In Section 5 we consider simulation results and discuss observer with a high gain is considered in (Slotine et al., for multi-agent systems with unknown relative degree of each called high-gain observer is proposed. Other form of an for multi-agent systems with unknown relative degree of each called high-gain observer is proposed. Other form of an agent. In Section 5 we consider simulation results and discuss observer with aa high gain is considered in (Slotine et al., agent. In Section 5 we consider simulation results and discuss observer with high gain is considered in (Slotine et al., efficiency of the proposed scheme. Concluding remarks an 1987). In (Utkin, 1992, Han, 1995) the robust sliding-mode agent. In Section 5 we consider simulation results and discuss observer with a high gain is considered in (Slotine et al., agent. In Section 5 we consider simulation results and discuss observer with a high gain is considered in (Slotine et al., of scheme. Concluding remarks an efficiency 1987). In 1992, Han, 1995) the robust sliding-mode efficiency of the the proposed proposed scheme. Concluding remarks 1987). Inis(Utkin, (Utkin, 1992, Han, 1995) the robust sliding-mode are given in Section 6. Appendix A gives the proof of the efficiency of scheme. Concluding remarks observer proposed. In (Wang and Gao, 2003) the authors an 1987). (Utkin, 1992, Han, 1995) the robust sliding-mode 1987). In Inis (Utkin, 1992, Han, 1995) the robust sliding-mode efficiency of the the proposed proposed scheme. Concluding remarks are given in Section 6. Appendix A gives the proof of an observer proposed. In (Wang and Gao, 2003) the authors are given in Section 6. Appendix A gives the proof of the the observer is proposed. In (Wang and Gao, 2003) the authors control system. are given in Section 6. Appendix A gives the proof of design a nonlinear extended state observer based on a observer is proposed. In (Wang and Gao, 2003) the authors given in Section 6. Appendix A gives the proof of the the control system. observeraa isnonlinear proposed. extended In (Wang state and Gao, 2003)based the authors design observer on aa are control system. design nonlinear extended state observer based on generalization of aa high-gain and aa sliding-mode design aa nonlinear extended state based control system. system. design nonlinear extended observer state observer observer based on on aa control generalization of observer and generalization of aa high-gain high-gain observer and aathesliding-mode sliding-mode observer. In (Veluvolu et al., 2011) observers generalization of high-gain observer and sliding-mode generalization of a high-gain observer and a sliding-mode observer. In (Veluvolu et al., 2011) the observers 2. PROBLEM STATEMENT observer. In (Veluvolu et al., 2011) the observers 2. (Luinberger, 1966, Esfandiary and Khalil, 1992, Utkin, 1992, observer. In (Veluvolu et al., 2011) the observers 2. PROBLEM PROBLEM STATEMENT STATEMENT observer. In (Veluvolu et al., 2011) the observers (Luinberger, 1966, Esfandiary and Khalil, 1992, Utkin, 1992, 2. STATEMENT (Luinberger, 1966, Esfandiary and Khalil, 1992, Utkin, 1992, 2.ofPROBLEM PROBLEM STATEMENT Han, 1995, Wang and Gao, 2003) are investigated for second (Luinberger, 1966, Esfandiary and Khalil, 1992, Utkin, 1992, the multi-agent system S be described by Let each agent S (Luinberger, 1966,and Esfandiary and are Khalil, 1992, Utkin, 1992, Let Han, 1995, Wang Gao, 2003) investigated for second Han, 1995, Wang and Gao, 2003) are investigated for second the multi-agent system S be described by each SSiii of of the multi-agent system S be described by Let following each agent agentequation order dynamical systems. In paper (Veluvolu et al., 2011) a Han, 1995, Wang and Gao, 2003) are investigated for second the of the multi-agent system S be described Let each agent S Han, 1995, Wang and Gao, 2003) are investigated for second ii of the multi-agent system S be described by order dynamical systems. In paper (Veluvolu et al., 2011) aa the order dynamical systems. In paper (Veluvolu et al., 2011) by Let each agent S following equation i the following equation comparative analysis for each observer under different kinds order dynamical dynamical systems. In paper paper (Veluvolu et al., al., 2011) 2011) following equation order systems. In (Veluvolu et aa the comparative analysis for each observer under different kinds comparative analysis for each observer under different kinds the following equation N of parametric uncertainties, external disturbances, and noises comparative analysis for observer under kinds N comparative analysis for each each observer under different different kinds of parametric uncertainties, external disturbances, and noises N of parametric uncertainties, external disturbances, and noises Q ( p ) y ( t ) k R ( p ) u ( t ) )) yy j ((tt )) ff i ((tt ), N are given. Robust observers have found many applications in of parametric uncertainties, external disturbances, and noises i i i i i N Yij (( p Q ( p ) y ( t ) k R ( p ) u ( t ) of parametric uncertainties, external disturbances, and noises are given. Robust observers have found many applications in Q Yij (( p p )) yy jj ((tt )) ff ii ((tt ), ), ii ( p ) y ii (t ) k ii Rii ( p )u ii (t ) N Y are given. Robust observers have found many applications in j 1, Yij Q ( p ) y ( t ) k R ( p ) u ( t ) p the synthesis of control systems under uncertainties. In are given. Robust observers have found many applications in i i i i i ij j Q ( p ) y ( t ) k R ( p ) u ( t ) Y ( p ) y ( t ) f iii (t ), ), (1) j 1 , i i i i i ij j (1) are given. Robust observers have found many applications in the synthesis of control systems under uncertainties. In i i i i i ij j j 1i , (1) the synthesis of control systems under uncertainties. In i jj 1 , (Atassi and Khalil, 1999) the stabilization of nonlinear (1) the synthesis of control systems under uncertainties. In i 1 1,, j (1) the synthesis of control systems under uncertainties. In (Atassi and Khalil, 1999) the stabilization of nonlinear k 1 jj ii (Atassi and Khalil, 1999) the stabilization of nonlinear j i n, i 1, ..., N , p y ( 0 ) y , k 1 , ..., kk 1 dynamical plants is considered by using the control law that (Atassi and Khalil, 1999) the stabilization of nonlinear 0 k 1 p y ( 0 ) y , k 1 , ..., n , i 1 , ..., N , (Atassi and Khalil, 1999) the stabilization of nonlinear dynamical plants is considered by using the control law that p y ( 0 ) y , k 1 , ..., n , i 1 , ..., N , 0 k 1 dynamical plants is considered by using the control obtained law that 0 kk p depends on estimates of plant output dynamical plants is by using the law p kk 11 yy ((0 0)) yy 000kkk ,, kk 1 1,, ..., ..., n n,, ii 1 1,, ..., ..., N N ,, dynamical plants is considered considered by usingderivatives the control control obtained law that that depends on estimates of plant output derivatives depends on estimates of plant output derivatives obtained depends on on estimates estimates of of plant plant output output derivatives derivatives obtained obtained depends 1 is supported solely by the grant from the Russian Science 1 1 The The proof proof of of control control algorithms algorithms was was proposed proposed in in Appendix Appendix A A is supported solely by the grant from the Russian Science The proof of control algorithms was proposed in Appendix A is supported solely by the grant from the Russian Science 1 1Foundation (project No. 14-29-00142) in IPME RAS. The algorithm design in Section 3 and simulation results in Section 1 The proof of control algorithms was proposed in Appendix A is supported solely by the grant from the Russian Science The proof (project of control algorithms was in proposed in Appendix A is supported solely by3 the grant from the Russian Science5 Foundation No. 14-29-00142) IPME RAS. The algorithm design in Section and simulation results in Section 5 Foundation (project No. 14-29-00142) in IPME RAS. The algorithm design in Section 3 and simulation results in Section 5 were supported solely by the Russian Federation President Grant (No. 14.W01.16.6325-MD (MD-6325.2016.8)). The other Foundation (project No. 14-29-00142) in IPME RAS. The algorithm design in Section 3 and simulation results in Section 5 Foundation (project No. 14-29-00142) in IPME RAS. The algorithm design in Section 3 and simulation results in Section 5 were supported solely by the Russian Federation President Grant (No. 14.W01.16.6325-MD (MD-6325.2016.8)). The other were supported solely by the Russian Federation President Grant (No. 14.W01.16.6325-MD (MD-6325.2016.8)). The other researches were partially supported by grants of RFBR (16-08-00282, 16-08-00686, 14-08-01015), Ministry of Education and were supported solely by the Russian Federation President Grant (No. 14.W01.16.6325-MD (MD-6325.2016.8)). The other were supported solely by the Russian Federation President Grant (No. 14.W01.16.6325-MD (MD-6325.2016.8)). The other researches were partially supported by grants of RFBR (16-08-00282, 16-08-00686, 14-08-01015), Ministry of Education and researches were partially supported by grants of RFBR (16-08-00282, 16-08-00686, 14-08-01015), Ministry of Education and Science of Russian Federation (Project and Government of Russian Federation, Grant 074-U01. researches were supported by grants 16-08-00686, 14-08-01015), Ministry of researches were partially partially supported by 14.Z50.31.0031) grants of of RFBR RFBR (16-08-00282, (16-08-00282, 16-08-00686, 14-08-01015), Ministry of Education Education and and Science of Russian Federation (Project 14.Z50.31.0031) and Government of Russian Federation, Grant 074-U01. Science of Russian Federation (Project 14.Z50.31.0031) and Government of Russian Federation, Grant 074-U01. Science of Russian Federation (Project 14.Z50.31.0031) and Government of Russian Federation, Grant 074-U01. Science of Russian Federation (Project 14.Z50.31.0031) and Government of Russian Federation, Grant 074-U01.
Copyright 2016 IFAC 1 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC 1 Copyright 2016 responsibility IFAC 1 Control. Peer review© of International Federation of Automatic Copyright ©under 2016 IFAC IFAC 1 Copyright © 2016 1 10.1016/j.ifacol.2016.07.921
IFAC ALCOSP 2016 26 Igor B. Furtat et al. / IFAC-PapersOnLine 49-13 (2016) 025–030 June 29 - July 1, 2016. Eindhoven, The Netherlands
where yi(t) R are local outputs, ui(t) R are local inputs, fi(t) R are local uncontrollable bounded disturbances, Qi(p), Ri(p) and Yij(p) are linear differential operators with unknown coefficients, deg Qi(p) = ni, deg Ri(p) = mi, deg Yij(p) = nij, ni > nij, ki > 0, i, j = 1, ..., N, y0k, k = 1, ..., n are unknown initial conditions, p = d / dt.
For implementation of control law (4) we use the following algorithm
Assume that the coefficients of Qi(p), Ri(p), Yij(p), ki belong to a known compact set and the polynomials Ri() are Hurwitz, where i, j = 1 ,..., N, is a complex variable.
(6)
ei (t ) ei (t ), ei(1) (t )
( 1) j C vj e(t jh), i 1, ..., N , (7) j 0 v
Rewrite the function i(t) i(t)=ikiRi(p)gi(t)+i(t), where
ij ( p ) y mj (t ) ,
(3)
i
g i (t ) Di ( p)ei (t )
(v ) vi ei (t )
.
Then equation (5) can be written as
i
i 1, ..., N ,
d v 0
Since the derivatives of yi(t) and ui(t) are not available, then we introduce the control law in the form
i (t ) Ai i (t ) i k i B1i g i (t ) B2i i (t )
(4)
v 0
N
where i > 0 are design parameters, the coefficients d0i, d1i, ..., d i ,i are chosen such that the polynomial
Di ( ) d i ,i i d i 1,i i 1 ... d1i d 0i
sum
j 1, j i
f i (t ) Qi ( p) y mi (t ), i 1, ..., N .
(v ) vi ei (t ),
the
Y
ij ( p ) y j (t )
j 1, j i
as
N
i(t) = fi(t) – Qi(p)ymi(t)
N
will
B
3ij j (t ),
ei (t ) Li i (t ), i 1, ..., N ,
(8)
j 1, j i
where Ai, B1i, B2i, B3ij and Li = [1 0 ... 0] are matrices obtained at the transition from (5) to (8).
be
Hurwitz, i = ni – mi 1, i is a relative degree of ith agent, ei(v ) (t ) is an estimate of the vth derivative of the signal ei(t).
Taking into account control law (7) and the structure of the function gi(t), rewrite the first equation of system (8) in the form
Substituting (4) into (3), rewrite (3) as follows
i (t ) Ai i (t ) i k i B1i iT i (t ) i k i B1i
N
Fi ( p)ei (t ) i (t )
d vi v v 0 h i
v! . j! (v j )!
where C vj
First, consider the case where orders of the operators Qi(p), Ri(p) and Yij(p), i, j = 1 ,..., N are known. Taking into account equations (1) and (2) rewrite the tracking error ei(t) = yi(t) – ymi(t) in the form
d
( 1)
ei
(t )
ui (t ) i
3. ALGORITHM FOR STRUCTURALLY CERTAINTY AGENTS
u i (t ) i
( i 1)
(t ) ei i (t h) , i 1, ..., N . h Substituting (6) into (4), rewrite control law (4) in the form ( i )
ei
where i = 1 ,..., N, > 0 is a an accuracy, T > 0 is a transient time.
Y
ei(1) (t ) ei(1) (t h) , h
ei( 2) (t )
The problem is to design the control system such that the following condition holds (2) y i (t ) y mi (t ) for t > T,
Qi ( p)ei (t ) k i Ri ( p)u i (t )
ei (t ) ei (t h) , h
Y
ij ( p) y j (t ),
i 1, ..., N ,
i d vi Li i (t ) v 0 h v
(5)
j 1, j i
i
v
d vi
h v 1 j 1
v
(1) j C vj Li i (t jh) (9)
N
where Fi(p) = Qi(p) + ikiRi(p)Di(p),
i (t ) f i (t ) Qi ( p) y mi (t )
B
3ij j (t )
B2i i (t ), i 1, ..., N ,
j 1, j i
i d vi ei(v ) (t ) . i k i Ri ( p) Di ( p)ei (t ) v 0
where the vectors i are composed of coefficients of the operator Di(p).
Since we know the compact set , then there exist coefficients i and polynomials Di(λ) such that the polynomials Fi(λ) are Hurwitz, i = 1, ..., N.
Let us denote
2
IFAC ALCOSP 2016 June 29 - July 1, 2016. Eindhoven, The Netherlands Igor B. Furtat et al. / IFAC-PapersOnLine 49-13 (2016) 025–030
i d vi ~ Ai Ai i ki B1i iT L , v i v 0 h
27
max i ,
where
As = A + Y +
i 1, ..., N
v
F
vj ,
P > 0,
v 1 j 1
~ ~ ~ A diag A1 , A2 , ..., AN ,
P2 > 0, P3 > 0, Svj > 0, Rvj > 0, v, j = 1, 2, …, are matrices of corresponding dimension, I is an identity matrix.
d vi j j ~ i i k i B1i v (1) C v Li , if v i , Fvj h , else,
Theorem 1. Given a scalar > 0. If there exist constant β > 0 and matrices P > 0, P2 > 0, P3 > 0, Svj > 0, Rvj > 0, v, j = 1, 2, …, such that < 0, then control system (7) ensures goal (2) with an accuracy
~ ~ ~ Fvj diag Fvj1 , Fvj2 , ..., FvjN , i, j = 1, ..., N, B Y 321 B3 N 1
B3 N 2
Proof of Theorem 1 in Appendix A. 4. ALGORITHM FOR STRUCTURALLY UNCERTAINTY AGENTS Now we consider the case when orders of the operators Qi(p), Ri(p) and Yij(p), i, j = 1 ,..., N are unknown. However, we assume that we known the upper bounds i of the relative
T
(t ) 1T (t ), 2T (t ), ..., NT (t ) , (t ) 1 (t ), 2 (t ), ..., N (t )T ,
degrees i, i.e. i i , i = 1 ,..., N. The goal is to ensure condition (2). Taking into account equations (1) and (2), rewrite the equation of the tracking error ei(t) = yi(t) – ymi(t) in the form (3). Introduce the control law as follows
where is a zero matrix. Taking into account notations, rewrite equation (9) as follows
(t ) ( A Y ) (t )
t
B diagB21, B22 , ..., B2 N ,
v
F (t jh) B (t ).
(10)
vj
v 1 j 1
P3
vj
12 11 11 , 11 11 22 11 *
Fv1
2 P2T
v 1
v 1
Fv 2 P2T
2 P3T
v 1
F
v2
v 1
11 12 * 22 * *
v
ei(1) (t )
, T P3 Fv v 1
22 he 2 h diag Rv1 , 2 Rv 2 , ..., v 1 v 1 33 I ,
F v 1
Fv1
ei( 2) (t )
R , v
v 1
... d1,i d 0,i
will
be
Hurwitz. Substituting (9) into (3), we obtain the equation kind of (5). Obviously, there exist i and Di(λ) such that Fi(λ) are Hurwitz, i = 1 ,..., N. For estimation the derivatives of the signal ei(t) we use the following algorithm ei (t ) ei (t ),
,
v 1 j 1
T P2 12 h P3T
(12)
polynomials i 1
Di ( ) d i ,i d i 1,i
v
jhR
i 1, ..., N ,
the i
P3T
(v ) vi ei (t ),
where the coefficients d0i, d1i, …, d i are chosen such that
12 P P2T AsT P3 , S vj , 11
v 1 j 1
22 11
d v 0
v
i
u i (t ) i
Before formulating the main result, let us introduce the following notations: 11 11 P2T As AsT P2
(11)
where 2 sup T (t ) (t ) .
B31N B32N ,
B312
1 0.5 1min ( P) ,
ei (t ) ei (t k1 h) , k1 h
ei(1) (t ) ei(1) (t k 2 h) , k2h
(13)
( i )
ei
P2T B O , I
(t )
( 1) ei i (t )
( 1) ei i (t
k i h
k i h)
, i 1, ..., N .
Here ei(v ) (t ) , v = 0, 1, ..., i are estimates of the vth derivative of the signals ei(t), k1 k 2 ... k i 0 . The delay in (13) is reduced in each subsequent equation because it is necessary to increase the accuracy of the estimate of each subsequent derivative. 3
IFAC ALCOSP 2016 28 Igor B. Furtat et al. / IFAC-PapersOnLine 49-13 (2016) 025–030 June 29 - July 1, 2016. Eindhoven, The Netherlands
Consider the set i = {1, ..., i }, i=1, ..., N. Substituting (13) into (12), we obtain e (t ) ei (t k1 h) ... u i (t ) i d 0i ei (t ) d1i i k1 h
di
... h i
i
e (t ) ei (t h ) ui (t ) i ei (t ) 2 i h e (t ) 2ei (t h ) ei (t 2h ) 0.05 i h2
e (t ) ei (t k i1 h) i i 1 ki
i h 2 [h 2 2h 0.05]ei (t ) [2h 0.1]ei (t h )
Let parameters in (15) be presented as follows
i1 1
...
q11 = 3, q12 = 3, q13 = 3, k1 = 3, r11 = 3, n12 = 3, f1(t) = 1+sin t, y1(0) = 1, y1 (0) 1 , y1 (0) 1 ;
(14)
2
i1 i2
e t hk
(1) i i!
1 ei t h k i1 k i2 2! i , i , 1
i i1 , i2 , ..., i i i , i1 i2 ... i i
0.05ei (t 2h ) , i 1, 2.
i1
k i2 ... k i
i
q21 = 3, q22 = 3, k2 = 3, n21 = 3, f1(t) = 2+3sin 3t, y2(0) = 1, y 2 (0) 1 .
,
Let ym1(t) = ym2(t) = 1 + sin0.7t. Fig. 1 shows simulation results on errors ei(t) = yi(t) – ymi(t) for αi = 0.003 and h = 0.01 (s), i = 1, 2. Fig. 2 shows simulation results of errors ei(t) = yi(t) – ymi(t) for αi =0.03 and h = 0.01 (s), i = 1, 2.
i 1, ..., N . As the result, the equation of the closed-loop system is represented by expression (8) where matrices Ai, B1i, B2i, B3ij and Li are obtained by the transition from (5) to (8) and taking into account, that
e1 (t )
Di ( ) d i i d i 1 i 1 ... d1i d 0i .
i > 0, Theorem 2. There exist coefficients k1 k 2 ... k i 0 , i = 1, ..., N and h > 0 such that control t, s
law (14) ensures goal (2) and boundedness of all signals in the closed-loop system. The proof of Theorem 2 is similar to the proof of Theorem 1, therefore, the proof of Theorem 2 is not in the paper. e2 (t )
5. EXAMPLES 1) Consider a plant model in the form
p
q
3
q11 p 2 q12 p q13 y1 (t )
k1 p r11 u1 (t ) n12 y 2 (t ) f1 (t ), 20 p
2
(15)
t, s
q21 p q22 y1 (t ) k 2 u2 (t ) n21 y 2 (t ) f 2 (t ).
The set is determined by the following inequalities: Fig. 1. The transients on tracking errors ei(t) = yi(t) – ymi(t) for αi = 0.003 and h = 0.01 (s), i = 1, 2.
q20 = 1, 3 q1i 3 , 3 q 2 j 3 , i = 1, 2, 3, j = 1, 2,
1 k1 3 , 1 r11 5 , 1 k 2 5 , 1 n12 12 , 1 n21 12 .
f1 (t ) 10 , equation orders in (15).
We assume that
f 2 (t ) 10 and we know e1 (t )
Define d0 = 1, d1 = 2, d2 = 1. Then, control law (7) is rewritten as follows
e2 (t ) t, s
Fig. 2. The transients on tracking errors ei(t) = yi(t) – ymi(t) for αi = 0.03 and h = 0.01 (s), i = 1, 2.
4
IFAC ALCOSP 2016 June 29 - July 1, 2016. Eindhoven, The Netherlands Igor B. Furtat et al. / IFAC-PapersOnLine 49-13 (2016) 025–030
2) Consider a plant model (15), where q20 = 0 and q21 = 1. Assume that we unknown equation orders in (15).
Bazylev D., Pyrkin A. (2013). Stabilization of biped robot standing on nonstationary plane, Proc. 18th Int. Conf. on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, 459-463. Bazylev D., Zimenko K., Margun A. et al. (2014). Adaptive control system for quadrotor equiped with robotic arm, Proc. 19th Int. Conf. on Methods and Models in Automation and Robotics, MMAR 2014; Miedzyzdroje; Poland, 705-710. Esfandiary F. and H.K. Khalil (1992). Output feedback stabilization of fully linearizable systems, Int. J. Control, 56(5), 1007-1037. Fridman E. (2014). Introduction to Time-Delay Systems. Analysis and Control, Birkhauser. Fridman E. (2014). Tutorial on Lyapunov-based methods for time-delay systems, European Journal of Control, 20, 271–283. Furtat I.B. (2013). Robust synchronization of the structural uncertainty nonlinear network with delay and disturbances, Proc. 11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing, ALCOSP 2013, Caen, France, Vol. 11, Issue PART, 227-232. Furtat I.B. (2014). Adaptive predictor-free control of a plant with delayed input signal, Automation and remote control, 75(1), 139-151. Furtat I.B. (2015). Robust static control algorithm for linear objects, Automation and Remote Control, 76(3), 446457. Furtat I.B., V.V Putov (2013). Suboptimal control of aircraft lateral motion, Proc. 2nd IFAC Workshop on Research, Education and Development of Unmanned Aerial Systems, RED-UAS 2013, Compiegne, France, Vol. 2, Issue PART 1, 276-282. Han J. (1995). A class of extended state observers for uncertain system, Control Decision, 10(1), 85-88. Kalman, R.E. (1960). A new approach to linear filtering and prediction problems, Trans. ASME – J. Basic Engineer, D(82), 35-45. Luinberger, D. (1966). Observers for multivariable systems, IEEE Trans. Automat. Control, AC-11(2), 190-197. Slotine J.J.E., J.K. Hedrick, and E.A. Misawa (1987). On sliding observers for nonlinear systems, J. Dynam. Syst., Measurement, Control, 109, 245-252. Utkin V.I. (1992). Sliding-modes in control optimization. Berlin: Springer-Verlag. Veluvolu K.C., M.Y. Kim, D. Lee (2011). Nonlinear sliding mode high-gain observers for fault estimation, Int. J. Syst. Sci., 42(7), 1065-1074. Wang W. and Z. Gao (2003). A comparison study of advanced state observer design techniques, Proc. Amer. Control Conf., 4754-4759.
Let 1 2 2. Define h = 0.01, k1 = 1, k2 = 0.1, d0 = 1, d1 = 2, d2 = 1. Then, control law (7) is defined as follows
ui (t ) i ei (t ) 2h 1 ei (t ) ei (t h )
10h 2 ei (t ) ei (t h ) ei (t 0.1h ) ei (t 1.1h ) 10 i h
2
(0.1h
2
0.2h 1)ei (t ) (0.2h 1)ei (t h )
ei (t 0.1h ) ei (t 1.1h ) , i 1, 2.
Let parameters in (15) be presented as in previous example, only where q20 = 0, q21 = 1 and y2(0) = 1. Fig. 3 shows simulation results of errors ei(t) = yi(t) – ymi(t) when αi = 0.003 and h = 0.01 (s), i = 1, 2.
e1 (t )
e2 (t )
29
t, s
Fig. 3. The transients on tracking errors ei(t) = yi(t) – ymi(t) for αi = 0.003 and h = 0.01 (s), i = 1, 2. It follows from Fig. 1, 2 that parametric uncertainties and external disturbances are compensated by control system with the required accuracy = 0.01 achieved after 11 s. It follows from Fig. 3 the required accuracy = 0.04 is achieved after 11 s. Simulation results show that the value δ can be reduced by decrease of the value h. 6. CONCLUSIONS The robust algorithm for multi-agent dynamical systems under parametric uncertainties and external unmeasured bounded disturbances is proposed. In this paper we use the observer based on left hand side differences for estimation of the derivatives. The algorithm compensates parametric uncertainties and disturbances with a given accuracy. Since the proposed control algorithm uses time delay, then it practical implementation can be based on digital systems. For example, the algorithm can be implemented by using MatLab. REFERENCES
APPENDIX A.
Atassi A.N. and H.K. Khalil (1999). A separation principle for the stabilization of class of nonlinear systems, IEEE Trans. Automat. Control, 44(9), 1672-1687. Bazylev D, Margun A., Zimenko K., Kremlev A. (2014). UAV equipped with a robotic manipulator, Proc. 22nd Mediterranean Conf. on Control and Automation, Palermo, Italy, 1177-1182.
Proof of Theorem. Consider the following Lemma. Lemma (Furtat, 2015). Let the system be described by the differential equation x(t ) Ax(t ) f (t ) g x(t ), x(t 1 ), ..., x(t n ), , (16) 5
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where x(t)Rn, A R nn is Hurwitz matrix, f(t)Rn is bounded function, i > 0 is a time delay, i = 1, …, n, g(x(t), x(t–1), …, x(t–n), )Rn is a continuous function for all arguments except maybe the case when = 0, moreover, when 0 and g(x(t), x(t–1), …, x(t–n), ) 0 g(0, …, 0, ) = 0. Additionally, the function g(·) satisfies the Lipschitz condition
V T (t ) P (t )
t
v
e
2 ( s t ) T
0
v
e
2 ( s t ) T
( s ) Rvj ( s )dsd .
Differentiating (18) along the trajectories of (17), we have V 2 T (t ) P(t ) 2 T (t ) P T T (t ) P T
g x 2 (t ), x 2 (t 1 ), ..., x 2 (t n ),
(t ) As (t )
L0 ( ) x1 (t ) x 2 (t ) L1 ( ) x1 (t 1 ) x 2 (t 1 ) ... Ln ( ) x1 (t n ) x 2 (t n ) ,
2
where L0() >0, L1() >0, …, Ln() >0 are Lipschitz constants, moreover, lim Li ( ) 0 , i = 0, 1, …, n.
v 1 j 1
v
vj (t )
v 1 j 1
v
e
2 jh T
(t jh) S i (t jh)
v 1 j 1
2
v
0
t
e
2 ( s t ) T
( s ) Rvj ( s )dsd
v 1 j 1 jh t
T (t )
lim x(t , ) x (t ) .
v
jhR (t )
(19)
vj
v 1 j 1
Let us use Lemma for analysis of equation (8). Since fi(t) are bounded functions and ymi(t) are smooth bounded functions together with its derivatives, therefore, the functions i(t) are bounded, i = 1, 2, ..., N. Since ei(t) is a continuous function, then according to (6)
t
v
e
2 ( s t ) T
jh
v 1 j 1
( s ) Rvj ( s )ds.
t jh
Consider the following notations: 1 (t ) col (t ), (t ), t t t 1 1 1 2 (t ) col ( s)ds, ( s)ds, ..., ( s)ds , h 2 t 2h t h t h (t ) col 1 (t ), 2 (t ), (t ). Differentiating (18), we find 2 W V 2 V T .
ei(1) (t ) pei (t ) , ei( 2) (t ) p 2 ei (t ) , … ,
(t ) p i ei (t ) when h0, i = 1, 2, ..., N.
Hence, gi(t)0 when h0. Since the matrix Ai is Hurwitz, then Lemma conditions will be hold for equation (8). Therefore, all the variables in the system (8) will be bounded.
Given > 0. There exist constant β > 0 and matrices P > 0, P2 > 0, P3 > 0, Svj > 0, Rvj > 0, v, j = 1, 2, …, such that < 0, then W < 0. Therefore, given > 0, where 2 sup T (t ) (t ) , the ellipsoid
Now let us show that there exists h > 0 such that algorithm (7) ensures condition (2).
Let us use the descriptor method (Fridman, 2014). Using the relation
t
K R n : T P 0.5 1 2 is an exponentially attractive. From (20) find the upper bound for as follows
t
1 (21) 0.5 1min ( P) . Thereby, the value δ in (2) can be chosen by variation of parameters of (12) and (15). For example, the value in (2) can be reduced by decreasing the value h. Theorem is proved.
t
(t h ) (t ) ( s )ds, t h
rewrite (10) as follows
v 1 j 1
( s ) S vj ( s )ds
S
0
t jh
2 ( s t ) T
≤ µ0 the solution x(t, ) of the original system (16) satisfies |x(t, )| < b. Moreover, the following condition holds
Fvj
( s )ds B (t )
t
v
T (t )
where x (t ) b . Then there exists µ0 > 0 such that for
v
v 1 j 1 t jh
x (t ) Ax (t ) f (t )
3
t
v
e
Let x (t ) be a solution of the equation
(t ) As (t )
2
Fvj
0
( i )
(18)
t
v 1 j 1 jh t
g x1 (t ), x1 (t 1 ), ..., x1 (t n ),
ei
( s ) S vj ( s )ds
v 1 j 1 t jh
(s)ds B (t ).
(17)
t jh
Consider a Lyapunov-Krasovskii functional in the form
6
(20)
ScienceDirect
IFAC-PapersOnLine 49-13 (2016) 025–030
1 Robust Robust Algorithm Algorithm Using Using Delay Delay for for Multi-Agent Multi-Agent Systems Systems1111 Robust Using Robust Algorithm Algorithm Using Delay Delay for for Multi-Agent Multi-Agent Systems Systems ,
N. Nekhoroshikh**, Mikhail S. Tarasov** Igor B. **, Artem Artem Mikhail S. Tarasov** Igor B. Furtat* Furtat*,,,**, Artem N. N. Nekhoroshikh**, Nekhoroshikh**, Mikhail S. Tarasov** Igor Igor B. B. Furtat* Furtat*,,**, **, Mikhail S. Tarasov** **, Artem Artem N. N. Nekhoroshikh**, Nekhoroshikh**, Mikhail S. Tarasov** Igor B. Furtat* *Institute for Problems of Mechanical Engineering Russian Academy of Sciences, 61 Bolshoy ave V.O., St.-Petersburg, *Institute for Problems of Mechanical Engineering Russian Academy of Sciences, 61 Bolshoy ave V.O., St.-Petersburg, *Institute for Problems of Mechanical Engineering Russian Academy of Sciences, 61 Bolshoy ave V.O., St.-Petersburg, 199178, Russia (Tel: +7-812-321-47-66; e-mail: [email protected]). *Institute for Problems of Mechanical Engineering Russian Academy of Sciences, 61 Bolshoy ave V.O., St.-Petersburg, *Institute for Problems of Mechanical Engineering Russian Academy of Sciences, 61 Bolshoy ave V.O., St.-Petersburg, 199178, Russia (Tel: +7-812-321-47-66; e-mail: [email protected]). 199178, Russia (Tel: +7-812-321-47-66; e-mail: [email protected]). **ITMO University, 49 Kronverkskiy ave, Saint Petersburg, 197101, Russia. 199178, Russia (Tel: +7-812-321-47-66; e-mail: [email protected]). 199178, Russia (Tel: +7-812-321-47-66; e-mail: [email protected]). **ITMO University, 49 Kronverkskiy ave, Saint Petersburg, 197101, Russia. **ITMO University, 49 Kronverkskiy ave, Saint Petersburg, 197101, Russia. **ITMO **ITMO University, University, 49 49 Kronverkskiy Kronverkskiy ave, ave, Saint Saint Petersburg, Petersburg, 197101, 197101, Russia. Russia. Abstract: The paper describes the robust control algorithm for linear multi-agent systems under Abstract: The The paper paper describes describes the the robust robust control control algorithm algorithm for for linear linear multi-agent multi-agent systems systems under under Abstract: parametric and structural uncertainties and external unmeasured disturbances. The proposed algorithm is Abstract: The paper describes the robust control algorithm for linear multi-agent systems under Abstract: The paper describes the robust control algorithm for linear multi-agent systems under parametric and and structural structural uncertainties uncertainties and and external external unmeasured unmeasured disturbances. disturbances. The The proposed proposed algorithm algorithm is is parametric based on left hand side differences for estimation of the derivatives. The resulting algorithm ensures parametric and structural uncertainties and external unmeasured disturbances. The proposed algorithm is parametric and structural uncertainties and external unmeasured disturbances. The proposed algorithm is based on on left left hand hand side side differences differences for for estimation estimation of of the the derivatives. derivatives. The The resulting resulting algorithm algorithm ensures ensures based required accuracy ofside difference between the plant plant output output and the reference reference signal. Thealgorithm modelingensures results based on left differences for of derivatives. The resulting based onaccuracy left hand handof side differences for estimation estimation of the theand derivatives. The signal. resulting algorithm ensures required accuracy of difference between the and the signal. The modeling results required difference between the plant output the reference The modeling results illustrate the effectiveness of the algorithm. required accuracy of difference between the plant output and the reference signal. The modeling results required of difference the plant output and the reference signal. The modeling results illustrate accuracy the effectiveness effectiveness of the the between algorithm. illustrate the of algorithm. illustrate the effectiveness of the algorithm. illustrate the effectiveness of the algorithm. © 2016, IFAC (International Federation Automatic Control) Hostingfunctional, by Elsevierdescriptor Ltd. All rights reserved. Keywords: Robust control, time time delay, of LMI, Lyapunov-Krasovskii method. Keywords: Robust control, delay, LMI, Lyapunov-Krasovskii functional, descriptor descriptor method. method. Keywords: Robust control, time delay, LMI, Lyapunov-Krasovskii functional, Keywords: Robust control, time delay, LMI, Lyapunov-Krasovskii functional, descriptor method. Keywords: Robust control, time delay, LMI, Lyapunov-Krasovskii functional, descriptor method.
with aa high-gain dynamical observer. The papers (Bazylev, with dynamical observer. The papers (Bazylev, with aa high-gain high-gain dynamical observer. The papers (Bazylev, Pyrkin, 2013, Bazylev, Zimenko, Margun et al., 2014, 1. INTRODUCTION with high-gain dynamical observer. The papers (Bazylev, with a high-gain dynamical observer. The papers (Bazylev, Pyrkin, 2013, Bazylev, Zimenko, Margun et al., 2014, 1. INTRODUCTION Pyrkin, 2013, Bazylev, Zimenko, Margun et al., 2014, 1. INTRODUCTION Bazylev, Margun, Zimenko, 2014, Pyrkin, Bobtsov, Pyrkin, 2013, Bazylev, Zimenko, Margun et al., 2014, 1. Pyrkin, 2013, Bazylev, Zimenko, Margun et al., 2014, 1. INTRODUCTION INTRODUCTION Bazylev, Margun, Zimenko, 2014, Pyrkin, Bobtsov, Bazylev, Margun, Zimenko, 2014, Pyrkin, Bobtsov, Design of simple control systems under parametric Kolyubin et al., 2013, Furtat, 2013, Furtat and Putov, 2013, Bazylev, Margun, Zimenko, 2014, Pyrkin, Bobtsov, Design of simple control systems under parametric Design of simple control systems under parametric Bazylev, Margun, Zimenko, 2014, Pyrkin, Bobtsov, Kolyubin et al., 2013, Furtat, 2013, Furtat and Putov, 2013, Kolyubin et al., 2013, Furtat, 2013, Furtat and Putov, 2013, uncertainties and external when only plant Design of control systems parametric Furtat, 2014, )) are used the adaptive control high gain Design of simple simple controldisturbances systems under under parametric Kolyubin et 2013, Furtat, 2013, and Putov, 2013, uncertainties and external disturbances when only plant uncertainties and external disturbances when only plant Kolyubin et al., al., 2013, Furtat, 2013, Furtat Furtat andwith Putov, 2013, Furtat, 2014, are used the adaptive control with high gain Furtat, 2014, ) are used the adaptive control with high gain output is available for measurement is important problem of uncertainties and external disturbances when only plant observers. Furtat, 2014, ) are used the adaptive control with high gain uncertainties and external disturbances when only plant output is available for measurement is important problem of output is available for measurement is important problem of Furtat, 2014, ) are used the adaptive control with high gain observers. observers. control theory and practice. To construct such control output is available for measurement is important problem of observers. output is available for measurement is important problem of control theory and practice. To construct such control control theory and practice. To construct such control observers. Analysis of the above results shows that the designed schemes many solutions have been this If control theory and To construct such control Analysis of the above results shows that the designed control theory and practice. practice. To proposed constructin suchregard. control schemes many solutions have been proposed in this regard. If Analysis of the above results shows that the designed schemes many solutions have been proposed in this regard. If algorithms use integrators for estimation of the derivatives. the plant relative degree exceeds one then implementation of Analysis of the above results shows that the designed schemes many solutions have been proposed in this regard. If Analysis of the above results shows that the designed algorithms use integrators for estimation of the derivatives. schemes many solutions have been proposed in this regard. If the plant relative degree exceeds one then implementation of algorithms use integrators for estimation of the derivatives. the plant relative degree exceeds one then implementation of Unlike these results we design the observer using left-hand adaptive and robust control systems requires the estimation algorithms use integrators for estimation of the derivatives. the plant relative degree exceeds one then implementation of algorithms use integrators for estimation of the derivatives. Unlike these results we design the observer using left-hand the plant and relative degree exceeds onerequires then implementation of adaptive robust control systems the estimation Unlike these results we design the observer using left-hand adaptive and robust control systems requires the estimation of side differences. This allows designing robust the derivatives of the plant input and output. these we design observer using left-hand adaptive and robust robust control systems requires the estimation estimation of of Unlike Unlike these results results weobserver design the the observer usingthe left-hand adaptive and control systems requires the side differences. This observer allows designing the robust the derivatives of the plant input and output. side differences. This observer allows designing the robust the derivatives of the plant input and output. control system without using integrators but using delays. For side differences. This observer allows designing the the derivatives of the and output. side differences. This observer allows designing the robust robust the derivatives ofobserver the plant plantisinput input andused output. control system without using integrators but using delays. For control system without using integrators but using delays. For The Luenberger widely to estimate the plant algorithm synthesis the results of (Furtat, 2015) will be used. control system without using integrators but using delays. For The Luenberger Luenberger observer observer is is widely widely used used to to estimate estimate the the plant plant control The system without using integrators but using delays. For algorithm synthesis the results of (Furtat, 2015) will be used. algorithm synthesis the results of (Furtat, 2015) will be used. model state vector with known parameters (Luinberger, The Luenberger observer is widely used to estimate the plant The Luenberger observer is widely used to estimate the plant algorithm synthesis the results of (Furtat, 2015) will be used. model state vector with known parameters (Luinberger, model state vector with known parameters (Luinberger, algorithm synthesis the results of (Furtat, 2015) will be used. The paper is organized as follows. The problem statement is 1966). to estimates state vector model state vector with known parameters (Luinberger, The paper is organized as follows. The problem statement is model The stateKalman vector filter withallows known parametersthe (Luinberger, 1966). The Kalman filter allows to estimates the state vector The paper is organized as follows. The problem statement is 1966). The Kalman filter allows to estimates the state vector presented in Section 2. In Section 3 we design the control of a dynamical system under disturbances and noises The paper is organized as follows. The problem statement is 1966). The Kalman filter allows to estimates the state vector The paper is organized as follows. The problem statement is presented in Section 2. In Section 3 we design the control 1966). The Kalman filter allows to estimates the state vector of a dynamical system under disturbances and noises presented in Section 2. In Section 3 we design the control of a dynamical system under disturbances and noises algorithm for multi-agent systems with known relative degree (Kalman, 1960). Under parametric uncertainty and external presented in Section 2. In Section 3 we design the control of a dynamical system under disturbances and noises presented in Section 2. In Section 3 we design the control algorithm for multi-agent systems with known relative degree of a dynamical system under disturbances and noises (Kalman, 1960). Under parametric uncertainty and external algorithm for multi-agent systems with known relative degree (Kalman, 1960). Under parametric uncertainty and external of each agent. In Section 3 we propose the control algorithm disturbances in the paper (Esfandiary and Khalil, soalgorithm for systems with relative degree (Kalman, Under parametric uncertainty and external algorithm for multi-agent multi-agent systems with known known relative degree of each agent. In Section 3 we propose the control algorithm (Kalman, 1960). 1960). Under parametric uncertainty and1992) external disturbances in the paper (Esfandiary and Khalil, 1992) soof each agent. In Section 3 we propose the control algorithm disturbances in the paper (Esfandiary and Khalil, 1992) sofor multi-agent systems with unknown relative degree of each called high-gain observer is proposed. Other form of an of each agent. In Section 3 we propose the control algorithm disturbances in the paper (Esfandiary and Khalil, 1992) soof each agent. In Section 3 we propose the control algorithm for multi-agent systems with unknown relative degree of each disturbances in theobserver paper (Esfandiary andOther Khalil, 1992) socalled high-gain is proposed. form of an for multi-agent systems with unknown relative degree of each called high-gain observer is proposed. Other form of an agent. In Section 5 we consider simulation results and discuss observer with a high gain is considered in (Slotine et al., for multi-agent systems with unknown relative degree of each called high-gain observer is proposed. Other form of an for multi-agent systems with unknown relative degree of each called high-gain observer is proposed. Other form of an agent. In Section 5 we consider simulation results and discuss observer with aa high gain is considered in (Slotine et al., agent. In Section 5 we consider simulation results and discuss observer with high gain is considered in (Slotine et al., efficiency of the proposed scheme. Concluding remarks an 1987). In (Utkin, 1992, Han, 1995) the robust sliding-mode agent. In Section 5 we consider simulation results and discuss observer with a high gain is considered in (Slotine et al., agent. In Section 5 we consider simulation results and discuss observer with a high gain is considered in (Slotine et al., of scheme. Concluding remarks an efficiency 1987). In 1992, Han, 1995) the robust sliding-mode efficiency of the the proposed proposed scheme. Concluding remarks 1987). Inis(Utkin, (Utkin, 1992, Han, 1995) the robust sliding-mode are given in Section 6. Appendix A gives the proof of the efficiency of scheme. Concluding remarks observer proposed. In (Wang and Gao, 2003) the authors an 1987). (Utkin, 1992, Han, 1995) the robust sliding-mode 1987). In Inis (Utkin, 1992, Han, 1995) the robust sliding-mode efficiency of the the proposed proposed scheme. Concluding remarks are given in Section 6. Appendix A gives the proof of an observer proposed. In (Wang and Gao, 2003) the authors are given in Section 6. Appendix A gives the proof of the the observer is proposed. In (Wang and Gao, 2003) the authors control system. are given in Section 6. Appendix A gives the proof of design a nonlinear extended state observer based on a observer is proposed. In (Wang and Gao, 2003) the authors given in Section 6. Appendix A gives the proof of the the control system. observeraa isnonlinear proposed. extended In (Wang state and Gao, 2003)based the authors design observer on aa are control system. design nonlinear extended state observer based on generalization of aa high-gain and aa sliding-mode design aa nonlinear extended state based control system. system. design nonlinear extended observer state observer observer based on on aa control generalization of observer and generalization of aa high-gain high-gain observer and aathesliding-mode sliding-mode observer. In (Veluvolu et al., 2011) observers generalization of high-gain observer and sliding-mode generalization of a high-gain observer and a sliding-mode observer. In (Veluvolu et al., 2011) the observers 2. PROBLEM STATEMENT observer. In (Veluvolu et al., 2011) the observers 2. (Luinberger, 1966, Esfandiary and Khalil, 1992, Utkin, 1992, observer. In (Veluvolu et al., 2011) the observers 2. PROBLEM PROBLEM STATEMENT STATEMENT observer. In (Veluvolu et al., 2011) the observers (Luinberger, 1966, Esfandiary and Khalil, 1992, Utkin, 1992, 2. STATEMENT (Luinberger, 1966, Esfandiary and Khalil, 1992, Utkin, 1992, 2.ofPROBLEM PROBLEM STATEMENT Han, 1995, Wang and Gao, 2003) are investigated for second (Luinberger, 1966, Esfandiary and Khalil, 1992, Utkin, 1992, the multi-agent system S be described by Let each agent S (Luinberger, 1966,and Esfandiary and are Khalil, 1992, Utkin, 1992, Let Han, 1995, Wang Gao, 2003) investigated for second Han, 1995, Wang and Gao, 2003) are investigated for second the multi-agent system S be described by each SSiii of of the multi-agent system S be described by Let following each agent agentequation order dynamical systems. In paper (Veluvolu et al., 2011) a Han, 1995, Wang and Gao, 2003) are investigated for second the of the multi-agent system S be described Let each agent S Han, 1995, Wang and Gao, 2003) are investigated for second ii of the multi-agent system S be described by order dynamical systems. In paper (Veluvolu et al., 2011) aa the order dynamical systems. In paper (Veluvolu et al., 2011) by Let each agent S following equation i the following equation comparative analysis for each observer under different kinds order dynamical dynamical systems. In paper paper (Veluvolu et al., al., 2011) 2011) following equation order systems. In (Veluvolu et aa the comparative analysis for each observer under different kinds comparative analysis for each observer under different kinds the following equation N of parametric uncertainties, external disturbances, and noises comparative analysis for observer under kinds N comparative analysis for each each observer under different different kinds of parametric uncertainties, external disturbances, and noises N of parametric uncertainties, external disturbances, and noises Q ( p ) y ( t ) k R ( p ) u ( t ) )) yy j ((tt )) ff i ((tt ), N are given. Robust observers have found many applications in of parametric uncertainties, external disturbances, and noises i i i i i N Yij (( p Q ( p ) y ( t ) k R ( p ) u ( t ) of parametric uncertainties, external disturbances, and noises are given. Robust observers have found many applications in Q Yij (( p p )) yy jj ((tt )) ff ii ((tt ), ), ii ( p ) y ii (t ) k ii Rii ( p )u ii (t ) N Y are given. Robust observers have found many applications in j 1, Yij Q ( p ) y ( t ) k R ( p ) u ( t ) p the synthesis of control systems under uncertainties. In are given. Robust observers have found many applications in i i i i i ij j Q ( p ) y ( t ) k R ( p ) u ( t ) Y ( p ) y ( t ) f iii (t ), ), (1) j 1 , i i i i i ij j (1) are given. Robust observers have found many applications in the synthesis of control systems under uncertainties. In i i i i i ij j j 1i , (1) the synthesis of control systems under uncertainties. In i jj 1 , (Atassi and Khalil, 1999) the stabilization of nonlinear (1) the synthesis of control systems under uncertainties. In i 1 1,, j (1) the synthesis of control systems under uncertainties. In (Atassi and Khalil, 1999) the stabilization of nonlinear k 1 jj ii (Atassi and Khalil, 1999) the stabilization of nonlinear j i n, i 1, ..., N , p y ( 0 ) y , k 1 , ..., kk 1 dynamical plants is considered by using the control law that (Atassi and Khalil, 1999) the stabilization of nonlinear 0 k 1 p y ( 0 ) y , k 1 , ..., n , i 1 , ..., N , (Atassi and Khalil, 1999) the stabilization of nonlinear dynamical plants is considered by using the control law that p y ( 0 ) y , k 1 , ..., n , i 1 , ..., N , 0 k 1 dynamical plants is considered by using the control obtained law that 0 kk p depends on estimates of plant output dynamical plants is by using the law p kk 11 yy ((0 0)) yy 000kkk ,, kk 1 1,, ..., ..., n n,, ii 1 1,, ..., ..., N N ,, dynamical plants is considered considered by usingderivatives the control control obtained law that that depends on estimates of plant output derivatives depends on estimates of plant output derivatives obtained depends on on estimates estimates of of plant plant output output derivatives derivatives obtained obtained depends 1 is supported solely by the grant from the Russian Science 1 1 The The proof proof of of control control algorithms algorithms was was proposed proposed in in Appendix Appendix A A is supported solely by the grant from the Russian Science The proof of control algorithms was proposed in Appendix A is supported solely by the grant from the Russian Science 1 1Foundation (project No. 14-29-00142) in IPME RAS. The algorithm design in Section 3 and simulation results in Section 1 The proof of control algorithms was proposed in Appendix A is supported solely by the grant from the Russian Science The proof (project of control algorithms was in proposed in Appendix A is supported solely by3 the grant from the Russian Science5 Foundation No. 14-29-00142) IPME RAS. The algorithm design in Section and simulation results in Section 5 Foundation (project No. 14-29-00142) in IPME RAS. The algorithm design in Section 3 and simulation results in Section 5 were supported solely by the Russian Federation President Grant (No. 14.W01.16.6325-MD (MD-6325.2016.8)). The other Foundation (project No. 14-29-00142) in IPME RAS. The algorithm design in Section 3 and simulation results in Section 5 Foundation (project No. 14-29-00142) in IPME RAS. The algorithm design in Section 3 and simulation results in Section 5 were supported solely by the Russian Federation President Grant (No. 14.W01.16.6325-MD (MD-6325.2016.8)). The other were supported solely by the Russian Federation President Grant (No. 14.W01.16.6325-MD (MD-6325.2016.8)). The other researches were partially supported by grants of RFBR (16-08-00282, 16-08-00686, 14-08-01015), Ministry of Education and were supported solely by the Russian Federation President Grant (No. 14.W01.16.6325-MD (MD-6325.2016.8)). The other were supported solely by the Russian Federation President Grant (No. 14.W01.16.6325-MD (MD-6325.2016.8)). The other researches were partially supported by grants of RFBR (16-08-00282, 16-08-00686, 14-08-01015), Ministry of Education and researches were partially supported by grants of RFBR (16-08-00282, 16-08-00686, 14-08-01015), Ministry of Education and Science of Russian Federation (Project and Government of Russian Federation, Grant 074-U01. researches were supported by grants 16-08-00686, 14-08-01015), Ministry of researches were partially partially supported by 14.Z50.31.0031) grants of of RFBR RFBR (16-08-00282, (16-08-00282, 16-08-00686, 14-08-01015), Ministry of Education Education and and Science of Russian Federation (Project 14.Z50.31.0031) and Government of Russian Federation, Grant 074-U01. Science of Russian Federation (Project 14.Z50.31.0031) and Government of Russian Federation, Grant 074-U01. Science of Russian Federation (Project 14.Z50.31.0031) and Government of Russian Federation, Grant 074-U01. Science of Russian Federation (Project 14.Z50.31.0031) and Government of Russian Federation, Grant 074-U01.
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where yi(t) R are local outputs, ui(t) R are local inputs, fi(t) R are local uncontrollable bounded disturbances, Qi(p), Ri(p) and Yij(p) are linear differential operators with unknown coefficients, deg Qi(p) = ni, deg Ri(p) = mi, deg Yij(p) = nij, ni > nij, ki > 0, i, j = 1, ..., N, y0k, k = 1, ..., n are unknown initial conditions, p = d / dt.
For implementation of control law (4) we use the following algorithm
Assume that the coefficients of Qi(p), Ri(p), Yij(p), ki belong to a known compact set and the polynomials Ri() are Hurwitz, where i, j = 1 ,..., N, is a complex variable.
(6)
ei (t ) ei (t ), ei(1) (t )
( 1) j C vj e(t jh), i 1, ..., N , (7) j 0 v
Rewrite the function i(t) i(t)=ikiRi(p)gi(t)+i(t), where
ij ( p ) y mj (t ) ,
(3)
i
g i (t ) Di ( p)ei (t )
(v ) vi ei (t )
.
Then equation (5) can be written as
i
i 1, ..., N ,
d v 0
Since the derivatives of yi(t) and ui(t) are not available, then we introduce the control law in the form
i (t ) Ai i (t ) i k i B1i g i (t ) B2i i (t )
(4)
v 0
N
where i > 0 are design parameters, the coefficients d0i, d1i, ..., d i ,i are chosen such that the polynomial
Di ( ) d i ,i i d i 1,i i 1 ... d1i d 0i
sum
j 1, j i
f i (t ) Qi ( p) y mi (t ), i 1, ..., N .
(v ) vi ei (t ),
the
Y
ij ( p ) y j (t )
j 1, j i
as
N
i(t) = fi(t) – Qi(p)ymi(t)
N
will
B
3ij j (t ),
ei (t ) Li i (t ), i 1, ..., N ,
(8)
j 1, j i
where Ai, B1i, B2i, B3ij and Li = [1 0 ... 0] are matrices obtained at the transition from (5) to (8).
be
Hurwitz, i = ni – mi 1, i is a relative degree of ith agent, ei(v ) (t ) is an estimate of the vth derivative of the signal ei(t).
Taking into account control law (7) and the structure of the function gi(t), rewrite the first equation of system (8) in the form
Substituting (4) into (3), rewrite (3) as follows
i (t ) Ai i (t ) i k i B1i iT i (t ) i k i B1i
N
Fi ( p)ei (t ) i (t )
d vi v v 0 h i
v! . j! (v j )!
where C vj
First, consider the case where orders of the operators Qi(p), Ri(p) and Yij(p), i, j = 1 ,..., N are known. Taking into account equations (1) and (2) rewrite the tracking error ei(t) = yi(t) – ymi(t) in the form
d
( 1)
ei
(t )
ui (t ) i
3. ALGORITHM FOR STRUCTURALLY CERTAINTY AGENTS
u i (t ) i
( i 1)
(t ) ei i (t h) , i 1, ..., N . h Substituting (6) into (4), rewrite control law (4) in the form ( i )
ei
where i = 1 ,..., N, > 0 is a an accuracy, T > 0 is a transient time.
Y
ei(1) (t ) ei(1) (t h) , h
ei( 2) (t )
The problem is to design the control system such that the following condition holds (2) y i (t ) y mi (t ) for t > T,
Qi ( p)ei (t ) k i Ri ( p)u i (t )
ei (t ) ei (t h) , h
Y
ij ( p) y j (t ),
i 1, ..., N ,
i d vi Li i (t ) v 0 h v
(5)
j 1, j i
i
v
d vi
h v 1 j 1
v
(1) j C vj Li i (t jh) (9)
N
where Fi(p) = Qi(p) + ikiRi(p)Di(p),
i (t ) f i (t ) Qi ( p) y mi (t )
B
3ij j (t )
B2i i (t ), i 1, ..., N ,
j 1, j i
i d vi ei(v ) (t ) . i k i Ri ( p) Di ( p)ei (t ) v 0
where the vectors i are composed of coefficients of the operator Di(p).
Since we know the compact set , then there exist coefficients i and polynomials Di(λ) such that the polynomials Fi(λ) are Hurwitz, i = 1, ..., N.
Let us denote
2
IFAC ALCOSP 2016 June 29 - July 1, 2016. Eindhoven, The Netherlands Igor B. Furtat et al. / IFAC-PapersOnLine 49-13 (2016) 025–030
i d vi ~ Ai Ai i ki B1i iT L , v i v 0 h
27
max i ,
where
As = A + Y +
i 1, ..., N
v
F
vj ,
P > 0,
v 1 j 1
~ ~ ~ A diag A1 , A2 , ..., AN ,
P2 > 0, P3 > 0, Svj > 0, Rvj > 0, v, j = 1, 2, …, are matrices of corresponding dimension, I is an identity matrix.
d vi j j ~ i i k i B1i v (1) C v Li , if v i , Fvj h , else,
Theorem 1. Given a scalar > 0. If there exist constant β > 0 and matrices P > 0, P2 > 0, P3 > 0, Svj > 0, Rvj > 0, v, j = 1, 2, …, such that < 0, then control system (7) ensures goal (2) with an accuracy
~ ~ ~ Fvj diag Fvj1 , Fvj2 , ..., FvjN , i, j = 1, ..., N, B Y 321 B3 N 1
B3 N 2
Proof of Theorem 1 in Appendix A. 4. ALGORITHM FOR STRUCTURALLY UNCERTAINTY AGENTS Now we consider the case when orders of the operators Qi(p), Ri(p) and Yij(p), i, j = 1 ,..., N are unknown. However, we assume that we known the upper bounds i of the relative
T
(t ) 1T (t ), 2T (t ), ..., NT (t ) , (t ) 1 (t ), 2 (t ), ..., N (t )T ,
degrees i, i.e. i i , i = 1 ,..., N. The goal is to ensure condition (2). Taking into account equations (1) and (2), rewrite the equation of the tracking error ei(t) = yi(t) – ymi(t) in the form (3). Introduce the control law as follows
where is a zero matrix. Taking into account notations, rewrite equation (9) as follows
(t ) ( A Y ) (t )
t
B diagB21, B22 , ..., B2 N ,
v
F (t jh) B (t ).
(10)
vj
v 1 j 1
P3
vj
12 11 11 , 11 11 22 11 *
Fv1
2 P2T
v 1
v 1
Fv 2 P2T
2 P3T
v 1
F
v2
v 1
11 12 * 22 * *
v
ei(1) (t )
, T P3 Fv v 1
22 he 2 h diag Rv1 , 2 Rv 2 , ..., v 1 v 1 33 I ,
F v 1
Fv1
ei( 2) (t )
R , v
v 1
... d1,i d 0,i
will
be
Hurwitz. Substituting (9) into (3), we obtain the equation kind of (5). Obviously, there exist i and Di(λ) such that Fi(λ) are Hurwitz, i = 1 ,..., N. For estimation the derivatives of the signal ei(t) we use the following algorithm ei (t ) ei (t ),
,
v 1 j 1
T P2 12 h P3T
(12)
polynomials i 1
Di ( ) d i ,i d i 1,i
v
jhR
i 1, ..., N ,
the i
P3T
(v ) vi ei (t ),
where the coefficients d0i, d1i, …, d i are chosen such that
12 P P2T AsT P3 , S vj , 11
v 1 j 1
22 11
d v 0
v
i
u i (t ) i
Before formulating the main result, let us introduce the following notations: 11 11 P2T As AsT P2
(11)
where 2 sup T (t ) (t ) .
B31N B32N ,
B312
1 0.5 1min ( P) ,
ei (t ) ei (t k1 h) , k1 h
ei(1) (t ) ei(1) (t k 2 h) , k2h
(13)
( i )
ei
P2T B O , I
(t )
( 1) ei i (t )
( 1) ei i (t
k i h
k i h)
, i 1, ..., N .
Here ei(v ) (t ) , v = 0, 1, ..., i are estimates of the vth derivative of the signals ei(t), k1 k 2 ... k i 0 . The delay in (13) is reduced in each subsequent equation because it is necessary to increase the accuracy of the estimate of each subsequent derivative. 3
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Consider the set i = {1, ..., i }, i=1, ..., N. Substituting (13) into (12), we obtain e (t ) ei (t k1 h) ... u i (t ) i d 0i ei (t ) d1i i k1 h
di
... h i
i
e (t ) ei (t h ) ui (t ) i ei (t ) 2 i h e (t ) 2ei (t h ) ei (t 2h ) 0.05 i h2
e (t ) ei (t k i1 h) i i 1 ki
i h 2 [h 2 2h 0.05]ei (t ) [2h 0.1]ei (t h )
Let parameters in (15) be presented as follows
i1 1
...
q11 = 3, q12 = 3, q13 = 3, k1 = 3, r11 = 3, n12 = 3, f1(t) = 1+sin t, y1(0) = 1, y1 (0) 1 , y1 (0) 1 ;
(14)
2
i1 i2
e t hk
(1) i i!
1 ei t h k i1 k i2 2! i , i , 1
i i1 , i2 , ..., i i i , i1 i2 ... i i
0.05ei (t 2h ) , i 1, 2.
i1
k i2 ... k i
i
q21 = 3, q22 = 3, k2 = 3, n21 = 3, f1(t) = 2+3sin 3t, y2(0) = 1, y 2 (0) 1 .
,
Let ym1(t) = ym2(t) = 1 + sin0.7t. Fig. 1 shows simulation results on errors ei(t) = yi(t) – ymi(t) for αi = 0.003 and h = 0.01 (s), i = 1, 2. Fig. 2 shows simulation results of errors ei(t) = yi(t) – ymi(t) for αi =0.03 and h = 0.01 (s), i = 1, 2.
i 1, ..., N . As the result, the equation of the closed-loop system is represented by expression (8) where matrices Ai, B1i, B2i, B3ij and Li are obtained by the transition from (5) to (8) and taking into account, that
e1 (t )
Di ( ) d i i d i 1 i 1 ... d1i d 0i .
i > 0, Theorem 2. There exist coefficients k1 k 2 ... k i 0 , i = 1, ..., N and h > 0 such that control t, s
law (14) ensures goal (2) and boundedness of all signals in the closed-loop system. The proof of Theorem 2 is similar to the proof of Theorem 1, therefore, the proof of Theorem 2 is not in the paper. e2 (t )
5. EXAMPLES 1) Consider a plant model in the form
p
q
3
q11 p 2 q12 p q13 y1 (t )
k1 p r11 u1 (t ) n12 y 2 (t ) f1 (t ), 20 p
2
(15)
t, s
q21 p q22 y1 (t ) k 2 u2 (t ) n21 y 2 (t ) f 2 (t ).
The set is determined by the following inequalities: Fig. 1. The transients on tracking errors ei(t) = yi(t) – ymi(t) for αi = 0.003 and h = 0.01 (s), i = 1, 2.
q20 = 1, 3 q1i 3 , 3 q 2 j 3 , i = 1, 2, 3, j = 1, 2,
1 k1 3 , 1 r11 5 , 1 k 2 5 , 1 n12 12 , 1 n21 12 .
f1 (t ) 10 , equation orders in (15).
We assume that
f 2 (t ) 10 and we know e1 (t )
Define d0 = 1, d1 = 2, d2 = 1. Then, control law (7) is rewritten as follows
e2 (t ) t, s
Fig. 2. The transients on tracking errors ei(t) = yi(t) – ymi(t) for αi = 0.03 and h = 0.01 (s), i = 1, 2.
4
IFAC ALCOSP 2016 June 29 - July 1, 2016. Eindhoven, The Netherlands Igor B. Furtat et al. / IFAC-PapersOnLine 49-13 (2016) 025–030
2) Consider a plant model (15), where q20 = 0 and q21 = 1. Assume that we unknown equation orders in (15).
Bazylev D., Pyrkin A. (2013). Stabilization of biped robot standing on nonstationary plane, Proc. 18th Int. Conf. on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, 459-463. Bazylev D., Zimenko K., Margun A. et al. (2014). Adaptive control system for quadrotor equiped with robotic arm, Proc. 19th Int. Conf. on Methods and Models in Automation and Robotics, MMAR 2014; Miedzyzdroje; Poland, 705-710. Esfandiary F. and H.K. Khalil (1992). Output feedback stabilization of fully linearizable systems, Int. J. Control, 56(5), 1007-1037. Fridman E. (2014). Introduction to Time-Delay Systems. Analysis and Control, Birkhauser. Fridman E. (2014). Tutorial on Lyapunov-based methods for time-delay systems, European Journal of Control, 20, 271–283. Furtat I.B. (2013). Robust synchronization of the structural uncertainty nonlinear network with delay and disturbances, Proc. 11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing, ALCOSP 2013, Caen, France, Vol. 11, Issue PART, 227-232. Furtat I.B. (2014). Adaptive predictor-free control of a plant with delayed input signal, Automation and remote control, 75(1), 139-151. Furtat I.B. (2015). Robust static control algorithm for linear objects, Automation and Remote Control, 76(3), 446457. Furtat I.B., V.V Putov (2013). Suboptimal control of aircraft lateral motion, Proc. 2nd IFAC Workshop on Research, Education and Development of Unmanned Aerial Systems, RED-UAS 2013, Compiegne, France, Vol. 2, Issue PART 1, 276-282. Han J. (1995). A class of extended state observers for uncertain system, Control Decision, 10(1), 85-88. Kalman, R.E. (1960). A new approach to linear filtering and prediction problems, Trans. ASME – J. Basic Engineer, D(82), 35-45. Luinberger, D. (1966). Observers for multivariable systems, IEEE Trans. Automat. Control, AC-11(2), 190-197. Slotine J.J.E., J.K. Hedrick, and E.A. Misawa (1987). On sliding observers for nonlinear systems, J. Dynam. Syst., Measurement, Control, 109, 245-252. Utkin V.I. (1992). Sliding-modes in control optimization. Berlin: Springer-Verlag. Veluvolu K.C., M.Y. Kim, D. Lee (2011). Nonlinear sliding mode high-gain observers for fault estimation, Int. J. Syst. Sci., 42(7), 1065-1074. Wang W. and Z. Gao (2003). A comparison study of advanced state observer design techniques, Proc. Amer. Control Conf., 4754-4759.
Let 1 2 2. Define h = 0.01, k1 = 1, k2 = 0.1, d0 = 1, d1 = 2, d2 = 1. Then, control law (7) is defined as follows
ui (t ) i ei (t ) 2h 1 ei (t ) ei (t h )
10h 2 ei (t ) ei (t h ) ei (t 0.1h ) ei (t 1.1h ) 10 i h
2
(0.1h
2
0.2h 1)ei (t ) (0.2h 1)ei (t h )
ei (t 0.1h ) ei (t 1.1h ) , i 1, 2.
Let parameters in (15) be presented as in previous example, only where q20 = 0, q21 = 1 and y2(0) = 1. Fig. 3 shows simulation results of errors ei(t) = yi(t) – ymi(t) when αi = 0.003 and h = 0.01 (s), i = 1, 2.
e1 (t )
e2 (t )
29
t, s
Fig. 3. The transients on tracking errors ei(t) = yi(t) – ymi(t) for αi = 0.003 and h = 0.01 (s), i = 1, 2. It follows from Fig. 1, 2 that parametric uncertainties and external disturbances are compensated by control system with the required accuracy = 0.01 achieved after 11 s. It follows from Fig. 3 the required accuracy = 0.04 is achieved after 11 s. Simulation results show that the value δ can be reduced by decrease of the value h. 6. CONCLUSIONS The robust algorithm for multi-agent dynamical systems under parametric uncertainties and external unmeasured bounded disturbances is proposed. In this paper we use the observer based on left hand side differences for estimation of the derivatives. The algorithm compensates parametric uncertainties and disturbances with a given accuracy. Since the proposed control algorithm uses time delay, then it practical implementation can be based on digital systems. For example, the algorithm can be implemented by using MatLab. REFERENCES
APPENDIX A.
Atassi A.N. and H.K. Khalil (1999). A separation principle for the stabilization of class of nonlinear systems, IEEE Trans. Automat. Control, 44(9), 1672-1687. Bazylev D, Margun A., Zimenko K., Kremlev A. (2014). UAV equipped with a robotic manipulator, Proc. 22nd Mediterranean Conf. on Control and Automation, Palermo, Italy, 1177-1182.
Proof of Theorem. Consider the following Lemma. Lemma (Furtat, 2015). Let the system be described by the differential equation x(t ) Ax(t ) f (t ) g x(t ), x(t 1 ), ..., x(t n ), , (16) 5
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where x(t)Rn, A R nn is Hurwitz matrix, f(t)Rn is bounded function, i > 0 is a time delay, i = 1, …, n, g(x(t), x(t–1), …, x(t–n), )Rn is a continuous function for all arguments except maybe the case when = 0, moreover, when 0 and g(x(t), x(t–1), …, x(t–n), ) 0 g(0, …, 0, ) = 0. Additionally, the function g(·) satisfies the Lipschitz condition
V T (t ) P (t )
t
v
e
2 ( s t ) T
0
v
e
2 ( s t ) T
( s ) Rvj ( s )dsd .
Differentiating (18) along the trajectories of (17), we have V 2 T (t ) P(t ) 2 T (t ) P T T (t ) P T
g x 2 (t ), x 2 (t 1 ), ..., x 2 (t n ),
(t ) As (t )
L0 ( ) x1 (t ) x 2 (t ) L1 ( ) x1 (t 1 ) x 2 (t 1 ) ... Ln ( ) x1 (t n ) x 2 (t n ) ,
2
where L0() >0, L1() >0, …, Ln() >0 are Lipschitz constants, moreover, lim Li ( ) 0 , i = 0, 1, …, n.
v 1 j 1
v
vj (t )
v 1 j 1
v
e
2 jh T
(t jh) S i (t jh)
v 1 j 1
2
v
0
t
e
2 ( s t ) T
( s ) Rvj ( s )dsd
v 1 j 1 jh t
T (t )
lim x(t , ) x (t ) .
v
jhR (t )
(19)
vj
v 1 j 1
Let us use Lemma for analysis of equation (8). Since fi(t) are bounded functions and ymi(t) are smooth bounded functions together with its derivatives, therefore, the functions i(t) are bounded, i = 1, 2, ..., N. Since ei(t) is a continuous function, then according to (6)
t
v
e
2 ( s t ) T
jh
v 1 j 1
( s ) Rvj ( s )ds.
t jh
Consider the following notations: 1 (t ) col (t ), (t ), t t t 1 1 1 2 (t ) col ( s)ds, ( s)ds, ..., ( s)ds , h 2 t 2h t h t h (t ) col 1 (t ), 2 (t ), (t ). Differentiating (18), we find 2 W V 2 V T .
ei(1) (t ) pei (t ) , ei( 2) (t ) p 2 ei (t ) , … ,
(t ) p i ei (t ) when h0, i = 1, 2, ..., N.
Hence, gi(t)0 when h0. Since the matrix Ai is Hurwitz, then Lemma conditions will be hold for equation (8). Therefore, all the variables in the system (8) will be bounded.
Given > 0. There exist constant β > 0 and matrices P > 0, P2 > 0, P3 > 0, Svj > 0, Rvj > 0, v, j = 1, 2, …, such that < 0, then W < 0. Therefore, given > 0, where 2 sup T (t ) (t ) , the ellipsoid
Now let us show that there exists h > 0 such that algorithm (7) ensures condition (2).
Let us use the descriptor method (Fridman, 2014). Using the relation
t
K R n : T P 0.5 1 2 is an exponentially attractive. From (20) find the upper bound for as follows
t
1 (21) 0.5 1min ( P) . Thereby, the value δ in (2) can be chosen by variation of parameters of (12) and (15). For example, the value in (2) can be reduced by decreasing the value h. Theorem is proved.
t
(t h ) (t ) ( s )ds, t h
rewrite (10) as follows
v 1 j 1
( s ) S vj ( s )ds
S
0
t jh
2 ( s t ) T
≤ µ0 the solution x(t, ) of the original system (16) satisfies |x(t, )| < b. Moreover, the following condition holds
Fvj
( s )ds B (t )
t
v
T (t )
where x (t ) b . Then there exists µ0 > 0 such that for
v
v 1 j 1 t jh
x (t ) Ax (t ) f (t )
3
t
v
e
Let x (t ) be a solution of the equation
(t ) As (t )
2
Fvj
0
( i )
(18)
t
v 1 j 1 jh t
g x1 (t ), x1 (t 1 ), ..., x1 (t n ),
ei
( s ) S vj ( s )ds
v 1 j 1 t jh
(s)ds B (t ).
(17)
t jh
Consider a Lyapunov-Krasovskii functional in the form
6
(20)