Optimal Compensation of Bounded External Disturbances and Measurement Noises for Nonlinear Systems

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Optimal Compensation of Bounded Optimal Compensation of Bounded Optimal Compensation of Bounded External Disturbances and Measurement Optimal Compensation of Bounded External Disturbances and Measurement External Disturbances and Measurement Noises for Nonlinear External Disturbances and Systems Measurement Noises for Nonlinear Systems Noises for Nonlinear Systems Noises for Nonlinear Systems ∗ ∗∗ ∗ ∗∗

Alexey Alexey Peregudin Peregudin ∗ Igor Igor B. B. Furtat Furtat ∗∗ Alexey Peregudin ∗∗ Igor B. Furtat ∗∗ ∗∗ ∗ Alexey Peregudin Igor B. Furtat ∗ ITMO University, Saint Petersburg, Russia ITMO University, Saint Petersburg, Russia ∗ Saint Petersburg, Russia (e-mail: ∗ ITMO University, (e-mail: [email protected]) [email protected]) ITMO University, Saint Petersburg, Russia ∗∗ (e-mail: [email protected]) of Mechanical Engineering, ∗∗ Institute for Problems for Problems of Mechanical Engineering, Russian Russian Academy Academy (e-mail: [email protected]) ∗∗ Institute Institute for Problems of Mechanical Engineering, Russian Academy of Sciences (IPME RAS), Russia, ITMO University, Saint ∗∗ of Institute Sciences for (IPME RAS),ofRussia, ITMO University, Russian Saint Petersburg, Petersburg, Problems Mechanical Engineering, Academy of Sciences (IPME RAS), Russia, ITMO University, Saint Petersburg, Russia (e-mail: [email protected]) Russia (e-mail: [email protected]) of Sciences (IPME RAS), Russia, ITMO University, Saint Petersburg, Russia (e-mail: [email protected]) Russia (e-mail: [email protected]) Abstract: Abstract: This This paper paper is is dedicated dedicated to to the the problem problem of of compensation compensation of of bounded bounded external external Abstract: This paper is dedicated to the problem of compensation of bounded external disturbances in the presence of bounded measurement noises for nonlinear systems. The disturbances in the presence of bounded measurement noises for nonlinear systems. The Abstract: This paper is dedicated to minimizing themeasurement problem ofsize compensation of bounded external disturbances in the presence of bounded noises for nonlinear systems. The synthesis of a linear feedback controller the of invariant ellipsoids of a closedsynthesis of a linear feedback controller minimizing the size of invariant ellipsoids of a closeddisturbances in the feedback presence of bounded measurement for nonlinear systems. The synthesis of ais linear controller minimizing the sizenoises of invariant ellipsoids of a closedloop system considered. Based on the method of Lyapunov functions, S-procedure and loop system is considered. Based on the method of Lyapunov functions, S-procedure and synthesis of aislinear feedbackBased controller minimizing the size of invariant ellipsoids of controller a closedloop system considered. on the method of Lyapunov functions, S-procedure and the alternating optimization method, an iterative algorithm for finding optimal the alternating optimization method, an iterative algorithm for finding optimal controller loop systemis isproposed. considered. Based on of the method of Lyapunov functions, S-procedure and the alternating optimization method, an iterative algorithm for finding optimal controller parameters The efficiency the obtained algorithm is demonstrated. parameters is proposed. The efficiency ofantheiterative obtainedalgorithm algorithmfor is demonstrated. the alternating optimization method, finding optimal controller parameters is proposed. The efficiency of the obtained algorithm is demonstrated. parameters proposed. The efficiency of the obtained demonstrated. © 2018, IFACis (International Federation of Automatic Control)algorithm Hosting byisElsevier Ltd. All rights reserved. Keywords: Keywords: Optimal Optimal control, control, robust robust control, control, nonlinear nonlinear systems, systems, optimization, optimization, iterative iterative Keywords: Optimal control, robust control, nonlinear systems, optimization, iterative algorithms, invariant ellipsoids, S-procedure algorithms, invariant ellipsoids, S-procedure Keywords: Optimal robust control, nonlinear systems, optimization, iterative algorithms, invariantcontrol, ellipsoids, S-procedure algorithms, invariant ellipsoids, S-procedure 1. INTRODUCTION INTRODUCTION systems 1. systems with with measurement measurement noises. noises. To To solve solve the the resulting resulting 1. INTRODUCTION systems with measurement noises. To solve the resulting optimization problem, we propose to apply the alternating optimization problem, we propose to apply the alternating 1. INTRODUCTION systems with measurement noises. To solvethe the resulting optimization problem, we propose to apply alternating method, which was previously successfully used for method, which was previously successfully used for solvsolvoptimization problem, we propose to apply the alternating The problem of compensation of external disturbances is method, which was previously successfully used for solvautomatic control problems in Kiriakidis (2001); El The problem of compensation of external disturbances is ing ing automatic control problems in Kiriakidis (2001); El method, which was previously successfully used for solvThe problem of compensation of external disturbances is one of of the the main main problems problems in in the the control control theory. theory. DisturDistur- ing automatic controlTheoretical problems in Kiriakidis (2001); El Ghaoui et al. (1994). grounds of this method one The problem of compensation of external disturbances is Ghaoui et al. (1994). Theoretical grounds of this method automatic control problems in Kiriakidis (2001); El one of the in the control Disturbances are main almostproblems always present present in real real theory. technical pro- ing Ghaoui et al. (1994). Theoretical grounds of this method and its applicability to the problems of biconvex optimizabances are almost always in technical proone of and the main problems in the control theory. Disturand its applicability toTheoretical the problems of biconvex optimizaGhaoui et al. (1994). grounds of this method bances are almost always present in real technical processes can lead to unpredictable deviations from the and its applicability to the problems of biconvex optimizaare studied Gorski et The cesses can lead always to unpredictable fromprothe tion bancesand are almost present Nevertheless, in deviations real technical tion its areapplicability studied in in to Gorski et al. al. (2007). (2007). The advantage advantage the problems of biconvex optimizacesses and can leadof unpredictable deviations from desired behaviour oftothe the system. therethe is and tion are studied in Gorski et al. (2007). The advantage of the resulting algorithm is that external disturbances desired behaviour system. Nevertheless, there is of the resulting algorithm is that external disturbances cesses and can lead to unpredictable deviations from the tion are studied in Gorski et al. (2007). The advantage desired behaviour of the system. Nevertheless, there is wide range range of of algorithms algorithms which which make make it it possible possible to to of the resulting algorithm that external disturbances measurement noises not assumed to sinusoidal aadesired wide of the system. Nevertheless, there to is and and measurement noises are areis assumed to be be sinusoidal of the resulting algorithm isnot that external disturbances a wide behaviour rangeaa negative of algorithms which make itdisturbances. possible compensate negative impact of external external disturbances. and measurement noises are not assumed to be sinusoidal as in Gerasimov et al. (2016); Fedele et al. (2013) compensate impact of a wide range of algorithms which make it possible to as in Gerasimov et al. (2016); Fedele et al. (2013) or or and are not assumed be differently sinusoidal compensate a negative impact of external disturbances. Among widely widely used compensation compensation methods we should should as inmeasurement Gerasimov asnoises et inal.Tsykunov (2016); Fedele etto al. (2013) or even continuous (2012) and, Among used methods we compensate a negative impact of external disturbances. even continuous as in Tsykunov (2012) and, differently as in Gerasimov et al. (2016); Fedele et al. (2013) or Among used of methods we should even mention widely the method method ofcompensation the internal internal model, model, see Gerasimov Gerasimov continuous as in Tsykunov (2012) and, differently from Furtat (2017), no restrictions on derivatives of the mention the the see from Furtat (2017), no restrictions on derivatives of the Among widely used compensation methods we should even continuous as in Tsykunov (2012) and, differently mention the method of the internal model, see Gerasimov et al. al. (2016), (2016), various various methods methods based based on on the the identification identification from Furtat (2017), no imposed. restrictions on derivatives of the measurement noises et mention the method the internal model, Gerasimov noises are are from Furtat (2017), no imposed. restrictions on derivatives of the et al. various methods based on the identification of the(2016), parameters ofof sinusoidal signals, seesee Fedele et al. al. measurement measurement noises are imposed. of the parameters of sinusoidal signals, see Fedele et et al. (2016), various methods based on the identification measurement 2. noises are imposed. of the parameters of sinusoidal signals, see Fedele et al. (2013); Aranovskiy et al. (2017), the method of universal PROBLEM (2013); Aranovskiy ofet sinusoidal al. (2017),signals, the method of universal 2. PROBLEM STATEMENT STATEMENT of the parameters see Fedele et al. (2013); Aranovskiy et al. (2017), the method of universal controllers, see Proskurnikov (2015), the method of invari2. PROBLEM STATEMENT controllers, see Proskurnikov (2015), the method of invari(2013); Aranovskiy et al. (2017), the method of universal 2. PROBLEM STATEMENT controllers, seesee Proskurnikov (2015), the method of invari- Consider aa plant model ant ellipsoids, ellipsoids, see Polyak et et al. al. (2007). plant {︂ model in in the the form form ant Polyak (2007). controllers, seesee Proskurnikov (2015), the method of invari- Consider ant ellipsoids, Polyak et al. (2007). Consider a plant {︂ model in the form 𝑥𝑥 ˙ = Φ(𝑥𝑥) + 𝐵𝐵𝐵𝐵 + {︂ However, the see problem ofetcompensation compensation of external external disdis- Consider a plant model 𝑥𝑥˙ = Φ(𝑥𝑥) + 𝐵𝐵𝐵𝐵 + 𝐷𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷𝐷 in the form ant ellipsoids, Polyakof al. (2007). of (1) However, the problem {︂ 𝑥𝑥 ˙ = Φ(𝑥𝑥) + 𝐵𝐵𝐵𝐵 + 𝐷𝐷𝐷𝐷𝐷 (1) 𝑦𝑦 = 𝐶𝐶𝐶𝐶 + 𝐸𝐸𝐸𝐸𝐸 However, the problem of compensation of external disturbances appears to be much more complicated if the 𝑦𝑦 𝐶𝐶𝐶𝐶 + 𝐸𝐸𝐸𝐸𝐸 (1) 𝑥𝑥 ˙ = Φ(𝑥𝑥) + 𝐵𝐵𝐵𝐵 + 𝐷𝐷𝐷𝐷𝐷 turbances appears to be much more complicated if the However, the problem of compensation of external dis𝑦𝑦 = 𝐶𝐶𝐶𝐶 + 𝐸𝐸𝐸𝐸𝐸 (1)k turbances appears of be much if the where 𝑥𝑥 ∈ Rnn is a state, 𝑢𝑢 ∈ Rm measured signals signals ofto the the plant more carry complicated additional noises. noises. m is a control, 𝑓𝑓 ∈ Rk measured plant carry additional 𝑦𝑦 = 𝐶𝐶𝐶𝐶 + 𝐸𝐸𝐸𝐸𝐸 control, 𝑓𝑓 ∈ Rk turbances appears to the be much more complicated ifmenthe where 𝑥𝑥 ∈ Rn is a state, 𝑢𝑢 ∈ Rm is p a measured signals of plant carry additional noises. Among the solutions proposed to date, we should where 𝑥𝑥 ∈ R is a state, 𝑢𝑢 ∈ R is a control, 𝑓𝑓 ∈ Rk p is an external disturbance, 𝜉𝜉 ∈ R is a measurement Among thesignals solutions proposed date,additional we shouldnoises. men- is an external disturbance, 𝜉𝜉 R ∈ mRis is control, a measurement measured of 𝐻𝐻the plant to carry where 𝑥𝑥 ∈ ∈R Rlln isis a measurable state, 𝑢𝑢 ∈ p a n 𝑓𝑓 ∈ nR Among the solutions proposed to date, we should mention the method of -optimization, see Sanchez-Pena is an external disturbance, 𝜉𝜉 ∈ R is a measurement ∞ noise, 𝑦𝑦 a output, Φ : 𝑅𝑅 → 𝑅𝑅 is n n tion thethe method of 𝐻𝐻proposed -optimization, Sanchez-Pena p 𝑦𝑦 ∈ Rl isdisturbance, a measurable 𝑅𝑅n → 𝑅𝑅n×k Among solutions to date,see weFurtat should(2017); men- noise, is continuous an external 𝜉𝜉 ,output, ∈𝐶𝐶R∈ isΦ a:: ,measurement n is n×m l×n tion the method of 𝐻𝐻∞ see Sanchez-Pena et al. (1998), the auxiliary loop method, ∞ -optimization, 𝑦𝑦 ∈ R is a measurable output, Φ 𝑅𝑅 → 𝑅𝑅 is,, aanoise, map, 𝐵𝐵 ∈ R R 𝐷𝐷 ∈ R n×m l×n n×k et al. (1998), the auxiliary loop method, see Furtat (2017); n ∈ Rn×k continuous 𝐵𝐵 ∈ Rn×m ,output, 𝐶𝐶 ∈ RΦl×n: ,𝑅𝑅𝐷𝐷 -optimization, see Sanchez-Pena tion method ofthe𝐻𝐻∞ 𝑦𝑦 ∈ Rl map, is a measurable → 𝑅𝑅𝜉𝜉 nare is, l×p et al. the (1998), the auxiliary loop method, see Furtat (2017); anoise, Tsykunov (2012), methods of zooming in quantizer, continuous map, 𝐵𝐵matrices. ∈ Rn×mThe , 𝐶𝐶 signals ∈ Rl×n𝑥𝑥, , 𝐷𝐷 ∈ R 𝐸𝐸 ∈ R known 𝑓𝑓𝑓𝑓 and l×p are Tsykunov (2012), the methods of zooming in quantizer, n×k 𝐸𝐸 ∈ R are known matrices. The signals 𝑥𝑥, and 𝜉𝜉 are et al.Baillieul (1998), the auxiliary loop method, seeHowever, Furtat (2017); a𝐸𝐸 continuous map, 𝐵𝐵matrices. ∈ R , 𝐶𝐶 signals ∈ Rthat , 𝐷𝐷 ∈ R𝜉𝜉 are, l×p Tsykunov (2012), the methods of zooming in quantizer, see (2002); Furtat et al. (2015). all of ∈ R are known The 𝑥𝑥, 𝑓𝑓 and not available for measurement. Suppose see Baillieul (2002); et al.of (2015). However, all of not measurement. Suppose that Tsykunov (2012), theFurtat methods zooming inrestrictions, quantizer, 𝐸𝐸 ∈available Rl×p are for known matrices. The signals 𝑥𝑥, 𝑓𝑓 and 𝜉𝜉 are see Baillieul (2002); Furtat et al. have (2015). However, all of not the above algorithms some available for measurement. Suppose that Φ(𝑥𝑥) = 𝐴𝐴𝐴𝐴 + 𝜑𝜑(𝑥𝑥), the Baillieul above mentioned mentioned algorithms have some restrictions, see (2002); Furtat et al. (2015). However, all of Φ(𝑥𝑥) = 𝐴𝐴𝐴𝐴 +Suppose 𝜑𝜑(𝑥𝑥), that not available for measurement. the above mentioned algorithms have some restrictions, for example, in Furtat (2017); Tsykunov (2012) external T= 𝐴𝐴𝐴𝐴 +T𝜑𝜑(𝑥𝑥), Φ(𝑥𝑥) for example, in Furtatalgorithms (2017); Tsykunov (2012) external the above mentioned have some restrictions, 𝜑𝜑 𝑥𝑥𝑥 (2) 𝜑𝜑 for example, in Furtat (2017); Tsykunov (2012) external Φ(𝑥𝑥) 𝐴𝐴𝐴𝐴𝛽𝛽𝛽𝛽 +T disturbances are required to be smooth, and in Baillieul 𝜑𝜑 ≤ ≤ 𝛽𝛽𝛽𝛽 𝑥𝑥𝑥 (2) 𝜑𝜑T T= T𝜑𝜑(𝑥𝑥), disturbances are required to be smooth, and in Baillieul n×n for example, in Furtat (2017); Tsykunov (2012) external 𝜑𝜑 ≤ 𝛽𝛽𝛽𝛽 𝑥𝑥𝑥 (2) 𝜑𝜑 is a known matrix, 𝛽𝛽 ∈ R is a known where 𝐴𝐴 ∈ R n×n T T disturbances are required to be smooth, and in Baillieul (2002); Furtat et al. (2015) the dimension of measurement is a known matrix, 𝛽𝛽 ∈ R is a known where 𝐴𝐴 ∈ R (2) 𝜑𝜑 𝜑𝜑 ≤ 𝛽𝛽𝛽𝛽 𝑥𝑥𝑥 (2002); Furtatare et al. (2015)to thebedimension of measurement n×n disturbances required smooth, and in Baillieul is a known matrix, 𝛽𝛽 ∈ R is a known where 𝐴𝐴 ∈ R constant. The matrix 𝐴𝐴 is not assumed to be stable, (2002); Furtat et al. (2015) the dimension of measurement noises is is assumed assumed to to be be smaller smaller than than the the one one of of the the state state where n×n constant. The matrix 𝐴𝐴 is not assumed to isbea stable, noises is a known matrix, 𝛽𝛽 ∈ R known 𝐴𝐴 ∈ R (2002); Furtat et al. (2015) the dimension of measurement The(𝐴𝐴𝐴𝐴𝐴) matrix 𝐴𝐴 is not assumed to be(𝐴𝐴𝐴𝐴𝐴) stable, but is and is noises isIn smaller observer than theis one of in theFurtat state constant. vector. addition, abe high-gain used but the the pair pair is stabilizable stabilizable and the the pair pair is vector. Inassumed addition,to high-gain observer used in The(𝐴𝐴𝐴𝐴𝐴) matrix 𝐴𝐴 is not assumed to be(𝐴𝐴𝐴𝐴𝐴) stable, noises is assumed toaabe smaller than theis one of for theFurtat state constant. but the pair (𝐴𝐴𝐴𝐴𝐴) is stabilizable and the pair (𝐴𝐴𝐴𝐴𝐴) is detectable. vector. In addition, high-gain observer is used in Furtat et al. (2015), which can lead to large errors highdetectable. et al. (2015), which can lead to large errors for highbut the pair (𝐴𝐴𝐴𝐴𝐴) is stabilizable and the pair (𝐴𝐴𝐴𝐴𝐴) is vector. In addition, a high-gain observer is used in Furtat detectable. et al. (2015), which can lead to large errors for high- detectable. frequency noises. It is assumed that the external disturbance and measurefrequency noises. et al. (2015), which can lead to large errors for high- It is assumed that the external disturbance and measurefrequency noises. It is assumed the external disturbance and measurement noise bounded and the The of frequency noises. noise are arethat bounded and satisfy satisfy the condition condition The purpose purpose of this this paper paper is is to to generalize generalize the the method method ment It is assumed that the and measureT external T disturbance ment noise are bounded and satisfy the condition The purposeellipsoids, of this paper is et to al. generalize thenonlinear method ment noise are bounded of Polyak (2007), (3) 𝑓𝑓𝑓𝑓 T 𝐻𝐻 T 𝐻𝐻2 𝜉𝜉 ≤ 1, 1 𝑓𝑓 + of invariant invariant ellipsoids, Polyak et al. (2007), to to nonlinear + 𝜉𝜉𝜉𝜉 satisfy 1, condition (3) T 𝐻𝐻1 𝑓𝑓and T 𝐻𝐻2 𝜉𝜉 ≤the The purpose of this paper is to generalize the method of invariant ellipsoids, Polyak et al. (2007), to nonlinear (3) 𝑓𝑓 T 𝐻𝐻1 𝑓𝑓 + 𝜉𝜉 T 𝐻𝐻2 𝜉𝜉 ≤ 1, of invariant ellipsoids, Polyak et al. (2007), to nonlinear 𝐻𝐻 𝑓𝑓 + 𝜉𝜉 𝐻𝐻 𝜉𝜉 ≤ 1, (3) 𝑓𝑓 2 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All 1rights reserved.

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IFAC CHAOS 2018 8 Alexey Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018Peregudin et al. / IFAC PapersOnLine 51-33 (2018) 7–11

1 Λ1 = 𝐴𝐴𝐴𝐴 +𝑃𝑃 𝑃𝑃T +𝛼𝛼𝛼𝛼 +𝑌𝑌 T 𝐵𝐵 T +𝐵𝐵𝐵𝐵 + 𝐷𝐷𝐷𝐷1−1 𝐷𝐷T +𝛾𝛾𝛾𝛾𝛾 𝛼𝛼 1 Λ2 = 𝐵𝐵𝐵𝐵 − 𝐷𝐷𝐷𝐷1−1 𝐷𝐷T − 𝛾𝛾𝛾𝛾𝛾 𝛼𝛼 1 Λ3 = (𝐴𝐴+𝐿𝐿i 𝐶𝐶)𝑃𝑃 +𝑃𝑃 (𝐴𝐴+𝐿𝐿i 𝐶𝐶)T +𝛼𝛼𝛼𝛼 + 𝐷𝐷𝐷𝐷1−1 𝐷𝐷T +𝛾𝛾𝛾𝛾𝛾 𝛼𝛼 𝛾𝛾 Λ4 = 𝐿𝐿i 𝐸𝐸𝐸 Λ5 = −𝛼𝛼𝛼𝛼2 , Λ6 = 𝑃𝑃𝑃 Λ7 = − 𝐼𝐼𝐼 𝛽𝛽 determines the controller matrix ̂︀ = 𝑌𝑌̂︀ 𝑃𝑃̂︀−1 𝐾𝐾 ̂︀ and ensures that for system (1)-(5) with matrices (𝐿𝐿i , 𝐾𝐾) the consequence 𝑥𝑥T (0)𝑃𝑃̂︀−1 𝑥𝑥(0) ≤ 1 ⇒ 𝑥𝑥T (𝑡𝑡)𝑃𝑃̂︀−1 𝑥𝑥(𝑡𝑡) ≤ 1

where 𝐻𝐻1 ∈ Rk×k and 𝐻𝐻2 ∈ Rp×p are known positivedefinite matrices. Introduce the observer (4) 𝑥𝑥 ̂︀˙ = 𝐴𝐴̂︀ 𝑥𝑥 + 𝐵𝐵𝐵𝐵 + 𝐿𝐿(𝐶𝐶 𝑥𝑥 ̂︀ − 𝑦𝑦), 𝐿𝐿 ∈ Rn×l , constructed on the basis of measurable signals. The error of the observer is determined by the value of 𝑒𝑒 = 𝑥𝑥 ̂︀ − 𝑥𝑥. Introduce the control law in the form 𝑢𝑢 = 𝐾𝐾 𝑥𝑥𝑥𝑥𝑥 ̂︀ ∈ Rm×n .

(5)

Equations (4), (5) define the structure of a linear dynamic controller, determined by the matrices 𝐿𝐿 and 𝐾𝐾. Thus, the specific control law is uniquely determined by the choice these matrices.

holds for all 𝑡𝑡 ≥ 0.

Suppose that conditions (2), (3) hold and the system (1) is equipped with controller (4), (5) with some given 𝐿𝐿𝐿𝐿𝐿. Let then 𝑃𝑃̂︀ (𝐿𝐿𝐿𝐿𝐿) ∈ Rn×n denote the solution of the optimization problem minimize trace(𝑃𝑃 )

Proof. Closed-loop system (1), (4), (5) with conditions (2),(3) can be presented as 𝑠𝑠˙ = 𝐹𝐹 𝐹𝐹 + 𝐺𝐺𝐺𝐺 + 𝑇𝑇 𝑇𝑇𝑇 (9) 𝑧𝑧 T 𝑅𝑅𝑅𝑅 ≤ 0,

P

subject to

𝑃𝑃 ≻ 0,

𝑥𝑥T (0)𝑃𝑃 −1 𝑥𝑥(0) ≤ 1 ⇒

(6)

where

⇒ 𝑥𝑥T (𝑡𝑡)𝑃𝑃 −1 𝑥𝑥(𝑡𝑡) ≤ 1, ∀𝑡𝑡 ≥ 0. We then define an optimal controller as the one with ˜ 𝐾𝐾) ˜ = argmin trace(𝑃𝑃̂︀(𝐿𝐿𝐿𝐿𝐿)). (𝐿𝐿𝐿 (7)

[︂ ]︂ [︂ ]︂ 𝑥𝑥 𝑓𝑓 , 𝑤𝑤 = , 𝜑𝜑 𝜉𝜉 [︂ ]︂ [︂ ]︂ 𝐴𝐴 + 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 𝐷𝐷 0 𝐹𝐹 = , 𝐺𝐺 = , 0 𝐴𝐴 + 𝐿𝐿𝐿𝐿 −𝐷𝐷 −𝐿𝐿𝐿𝐿 [︂ ]︂ [︂ ]︂ [︂ ]︂ −𝛽𝛽𝛽𝛽 0 𝐻𝐻1 0 𝐼𝐼 𝑅𝑅 = , 𝐻𝐻 = , 𝑇𝑇 = . 0 𝐼𝐼 −𝐼𝐼 0 𝐻𝐻2 Then the validity of the consequence 𝑥𝑥T (0)𝑃𝑃 −1 𝑥𝑥(0) ≤ 1 ⇒ 𝑥𝑥T (𝑡𝑡)𝑃𝑃 −1 𝑥𝑥(𝑡𝑡) ≤ 1, ∀𝑡𝑡 ≥ 0 follows from the validity of the consequence 𝑠𝑠T (0)𝑄𝑄𝑄𝑄(0) ≤ 1 ⇒ 𝑠𝑠T (𝑡𝑡)𝑄𝑄𝑄𝑄(𝑡𝑡) ≤ 1, ∀𝑡𝑡 ≥ 0, where [︂ −1 ]︂ 𝑃𝑃 0 𝑄𝑄 = 0 𝑃𝑃 −1 . For the latter consequence to be true it is sufficient that the consequence ⎧ ⎪ ⎨𝑉𝑉 (𝑠𝑠) ≥ 1 𝑧𝑧 T 𝑅𝑅𝑅𝑅 ≤ 0 ⇒ 𝑉𝑉˙ (𝑠𝑠) ≤ 0 (10) ⎪ ⎩ T 𝑤𝑤 𝐻𝐻𝐻𝐻 ≤ 1 𝑠𝑠 =

L, K

˜ 𝐾𝐾) ˜ is The set {𝑥𝑥 ∈ R : 𝑥𝑥 𝑃𝑃˜ −1 𝑥𝑥 ≤ 1}, where 𝑃𝑃˜ = 𝑃𝑃̂︀ (𝐿𝐿𝐿 called the optimal invariant ellipsoid of the system (1). n

T

Geometrically it means that the optimal controller ensures that the trajectories of closed-loop system (1), (4), (5), started within the ellipsoid 𝑥𝑥T 𝑃𝑃˜ −1 𝑥𝑥 ≤ 1, never leave it in the future as long as conditions (2), (3) hold and that this ellipsoid is the smallest one we can possibly obtain by choosing the matrices 𝐿𝐿 and 𝐾𝐾. By ”smallest” we mean that the sum of squares of its semiaxes (i.e. trace(𝑃𝑃˜ )) is minimal. 3. SOLUTION The second condition of the optimization problem (6) is not an equality or inequality, thus it is not a kind of problem that can be solved directly. In this section we shall give an iterating algorithm for solving this problem based on the following two theorems. Theorem 1. Let an observer matrix 𝐿𝐿 = 𝐿𝐿i be given such that the matrix 𝐴𝐴 + 𝐿𝐿i 𝐶𝐶 is stable. Then the solution 𝑃𝑃̂︀ , 𝑌𝑌̂︀ of the optimization problem minimize

trace(𝑃𝑃 )

subject to

𝑃𝑃 ≻ 0, Λ ⪯ 0,

α, γ, P, Y

where 𝛼𝛼 ∈ R,

𝑤𝑤T 𝐻𝐻𝐻𝐻 ≤ 1,

[︂ ]︂ 𝑥𝑥 , 𝑒𝑒

𝑧𝑧 =

holds for all 𝑠𝑠 ∈ R2n , 𝑤𝑤 ∈ Rk+p , 𝑧𝑧 ∈ R2n , where 𝑉𝑉 (𝑠𝑠) = 𝑠𝑠T 𝑄𝑄𝑄𝑄 is the Lyapunov function, defined over the trajectories of closed-loop system (9). T

Let 𝜓𝜓 = [𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥] ∈ R3n+k+p , then (10) is equivalent to the consequence ⎧ T ⎪ ⎨𝜓𝜓 𝑊𝑊1 𝜓𝜓 ≤ −1 𝜓𝜓 T 𝑊𝑊2 𝜓𝜓 ≤ 0 ⇒ 𝜓𝜓 T 𝑊𝑊4 𝜓𝜓 ≤ 0, ⎪ ⎩ T 𝜓𝜓 𝑊𝑊3 𝜓𝜓 ≤ 1 where ⎤ ⎤ ⎡ ⎡ 0 0 0 −𝛽𝛽𝛽𝛽 0 0 0 0 −𝑄𝑄 ⎢ 0 0 0 0 0⎥ ⎢ 0 0 0⎥ ⎥ ⎥ ⎢ ⎢ 𝑊𝑊1 = ⎢ 0 0 0 0 0⎥ , 𝑊𝑊2 = ⎢ 0 0 𝐼𝐼 0 0⎥, ⎣ 0 0 0 0 0⎦ ⎣ 0 0 0 0 0⎦ 0 0 0 0 0 0 0 0 0 0

(8)

𝛾𝛾 ∈ R, 𝑃𝑃 ∈ Rn×n , 𝑌𝑌 ∈ Rm×n , ⎡ ⎤ Λ1 Λ2 0 Λ6 ⎢ ΛT ⎥ ⎢ 2 ΛT3 Λ4 0 ⎥ Λ=⎣ 0 Λ4 Λ5 0 ⎦ ΛT 0 Λ7 , 6 0 8

IFAC CHAOS 2018 Alexey Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018Peregudin et al. / IFAC PapersOnLine 51-33 (2018) 7–11

⎡

0 ⎢0 ⎢ 𝑊𝑊3 =⎢0 ⎣0 0

0 0 0 0 0

0 0 0 0 0

⎤ ⎡ T ⎤ 0 𝐹𝐹 𝑄𝑄 + 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 0⎥ T ⎥ ⎢ 0 0 0 ⎥. 0⎥, 𝑊𝑊4 =⎣ 𝑇𝑇 𝑄𝑄 ⎦ 0 00 ⎦ T 𝑄𝑄 𝐺𝐺 𝐻𝐻 0 00

9

We note that for fixed 𝛼𝛼𝛼𝛼𝛼 𝛼 0 the optimization problem (14) is convex due to the fact that the minimizing function trace(𝑃𝑃 ) is linear whilst the domain defined by linear matrix inequalities is convex. There are numerous software packages for numerical solution of convex optimization problems (e.g. SeDuMi Toolbox and YALMIP Toolbox for MATLAB).

0 0 0

According to S-lemma, see Polyak (1998), the latter consequence holds if there exist real numbers 𝜏𝜏1 , 𝜏𝜏2 , 𝜏𝜏3 ≥ 0, such that {︂ 𝜏𝜏1 · (−1) + 𝜏𝜏2 · 0 + 𝜏𝜏3 · 1 ≤ 0, 𝜏𝜏1 𝑊𝑊1 + 𝜏𝜏2 𝑊𝑊2 + 𝜏𝜏3 𝑊𝑊3 ⪰ 𝑊𝑊4 .

We introduce the notation 𝐽𝐽 = 𝑃𝑃 −1 . It is obvious that maximization of trace(𝐽𝐽) is equivalent to minimization of trace(𝑃𝑃 ) and in the same way corresponds to the optimal ˜ ≤ 1. invariant ellipsoid 𝑥𝑥T 𝐽𝐽𝐽𝐽 Theorem 2. Let a controller matrix 𝐾𝐾 = 𝐾𝐾i be given such ̂︀ that the matrix 𝐴𝐴 + 𝐵𝐵𝐵𝐵i is stable. Then the solution 𝐽𝐽, 𝑍𝑍̂︀ of the optimization problem

To minimize the desired ellipsoid, following Polyak et al. (2007), take 𝜏𝜏1 = 𝜏𝜏3 = 𝛼𝛼 𝛼 0 and denote 𝜏𝜏2 = γ1 . Then the last matrix inequality presents as [︂ ]︂ ⎡ ⎤ 1 𝛽𝛽𝛽𝛽 0 T 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝐹𝐹 𝑄𝑄 + 𝑄𝑄𝑄𝑄 + 𝛼𝛼𝛼𝛼 + ⎢ ⎥ 𝛾𝛾 0 0 ⎢ ⎥ 1 ⎢ ⎥ ⪯ 0. 𝑇𝑇 T 𝑄𝑄 − 𝐼𝐼 0 ⎦ ⎣ 𝛾𝛾 0 −𝛼𝛼𝛼𝛼 𝐺𝐺T 𝑄𝑄

maximize

trace(𝐽𝐽)

subject to

𝐽𝐽 ≻ 0, Δ ⪰ 0, Γ ⪯ 0,

α, γ, J, Z, R

Due to the property of Schur complement, it can be rewritten as [︂ ]︂ 1 𝛽𝛽𝛽𝛽 0 + 𝐹𝐹 T 𝑄𝑄 + 𝑄𝑄𝑄𝑄 + 𝛼𝛼𝛼𝛼 + 𝛾𝛾 0 0 )︂ (︂ 1 (11) +𝑄𝑄 𝛾𝛾𝛾𝛾𝛾𝛾 T + 𝐺𝐺𝐺𝐺 −1 𝐺𝐺T 𝑄𝑄 ⪯ 0. 𝛼𝛼 Multiplying from the left and right by 𝑄𝑄−1 we obtain [︂ ]︂ [︂ ]︂T 𝑃𝑃 0 𝐴𝐴 + 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 + 0 𝑃𝑃 0 𝐴𝐴 + 𝐿𝐿𝐿𝐿 [︂ ]︂ [︂ ]︂ [︂ ]︂ 𝐴𝐴 + 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 𝑃𝑃 0 𝑃𝑃 0 + + 𝛼𝛼 + (12) 0 𝐴𝐴 + 𝐿𝐿𝐿𝐿 0 𝑃𝑃 0 𝑃𝑃 ⎤ ⎡ [︂ ]︂ [︂ ]︂ [︂ ]︂ 𝛽𝛽 𝐼𝐼 −𝐼𝐼 𝑃𝑃 0 ⎣ 𝐼𝐼 0⎦ 𝑃𝑃 0 + 𝛾𝛾 + + 𝛾𝛾 0 𝑃𝑃 −𝐼𝐼𝐼𝐼 0 𝑃𝑃 0 0 [︂ ]︂ [︂ −1 ]︂ [︂ ]︂T 1 𝐷𝐷 0 𝐻𝐻1 𝐷𝐷 0 0 + ⪯ 0. 0 𝐻𝐻2−1 −𝐷𝐷 −𝐿𝐿𝐿𝐿 𝛼𝛼 −𝐷𝐷 −𝐿𝐿𝐿𝐿

where

(14)

𝐽𝐽 ∈ Rn×n , 𝑍𝑍 ∈ Rn×l , 𝑅𝑅 ∈ Rn×n , ⎤ ⎡ ⎤ 𝑅𝑅 𝑅𝑅 Γ1 Γ4 0 )︂ (︂ −1 ⎦ ⎦ ⎣ , Γ = ⎣ΓT 1 4 Γ2 Γ5 , Δ = −1 T 𝐽𝐽 𝐷𝐷𝐷𝐷 𝐷𝐷 + 𝛾𝛾𝛾𝛾 T 1 0 Γ5 Γ3 𝛼𝛼 𝛽𝛽 Γ1 = 𝐽𝐽(𝐴𝐴 + 𝐵𝐵𝐵𝐵i ) + (𝐴𝐴 + 𝐵𝐵𝐵𝐵i )T 𝐽𝐽 + 𝛼𝛼𝛼𝛼 + 𝐼𝐼 + 𝑅𝑅𝑅 𝛾𝛾 T T T Γ2 = 𝐴𝐴 𝐽𝐽 + 𝐽𝐽𝐽𝐽 + 𝛼𝛼𝛼𝛼 + 𝐶𝐶 𝑍𝑍 + 𝑍𝑍𝑍𝑍 + 𝑅𝑅𝑅 Γ3 = −𝛼𝛼𝛼𝛼2 , Γ4 = 𝐽𝐽𝐽𝐽𝐽𝐽i − 𝑅𝑅𝑅 Γ5 = 𝑍𝑍𝑍𝑍𝑍 determines the observer matrix ̂︀ = 𝐽𝐽̂︀−1 𝑍𝑍̂︀ 𝐿𝐿 𝛼𝛼 ∈ R, ⎡

𝛾𝛾 ∈ R,

̂︀ i ) and ensures that for system (1)-(5) with matrices (𝐿𝐿𝐿𝐿𝐿 the consequence ̂︀ ̂︀ 𝑥𝑥T (0)𝐽𝐽𝐽𝐽(0) ≤ 1 ⇒ 𝑥𝑥T (𝑡𝑡)𝐽𝐽𝐽𝐽(𝑡𝑡) ≤1 holds for all 𝑡𝑡 ≥ 0.

Proof. Taking into account [︂ ]︂ 𝐽𝐽 0 𝑄𝑄 = 0 𝐽𝐽 ,

Setting 𝐿𝐿 = 𝐿𝐿i , introducing the notation 𝑌𝑌 = 𝐾𝐾𝐾𝐾 and Λj from the statement of Theorem 1 and performing elementary operations, rewrite (12) as [︂ ]︂ T Λ1 − Λ6 Λ−1 Λ2 7 Λ6 T ⪯ 0. ΛT Λ3 − Λ4 Λ−1 2 5 Λ4

and introducing the notation 𝑍𝑍 = 𝐽𝐽𝐽𝐽, 𝛽𝛽 𝐼𝐼𝐼 𝛾𝛾 Φ2 = 𝐽𝐽𝐽𝐽𝐽𝐽𝐽 Φ3 = 𝐴𝐴T 𝐽𝐽 + 𝐽𝐽𝐽𝐽 + 𝛼𝛼𝛼𝛼 + 𝐶𝐶 T 𝑍𝑍 T + 𝑍𝑍𝑍𝑍𝑍 )︂ (︂ 1 Γ3 = −𝛼𝛼𝛼𝛼2 , Γ5 = 𝑍𝑍𝑍𝑍𝑍 Ψ = 𝐽𝐽 𝐷𝐷𝐷𝐷1−1 𝐷𝐷T + 𝛾𝛾𝛾𝛾 𝐽𝐽𝐽 𝛼𝛼 rewrite (12) as [︂ ]︂ Φ1 + Ψ Φ2 − Ψ (15) −1 T ⪯ 0. ΦT 2 − Ψ Φ3 + Ψ − Γ5 Γ3 Γ5 Φ1 = 𝐽𝐽(𝐴𝐴 + 𝐵𝐵𝐵𝐵) + (𝐴𝐴 + 𝐵𝐵𝐵𝐵)T 𝐽𝐽 + 𝛼𝛼𝛼𝛼 +

Applying Schur complement property, we arrive at the linear matrix inequality ⎤ ⎡ Λ1 Λ2 0 Λ6 ⎥ ⎢ΛT ⎢ 2 ΛT3 Λ4 0 ⎥ ⪯ 0. (13) ⎣ 0 Λ4 Λ5 0 ⎦ ΛT 0 Λ7 6 0

It is shown in Polyak et al. (2007) that if the matrix 𝐹𝐹 is stable, then the set defined by (11) is nonempty. In our case the matrix 𝐹𝐹 is block-triangular, and its stability follows from the stability of the matrices 𝐴𝐴 + 𝐵𝐵𝐵𝐵 and 𝐴𝐴 + 𝐿𝐿i 𝐶𝐶. Since 𝐾𝐾 is a variable, for the set defined by (13) to be nonempty it is sufficient that the matrix 𝐴𝐴 + 𝐿𝐿i 𝐶𝐶 is stable, and the pair (𝐴𝐴𝐴𝐴𝐴) is stabilizable. 

If there exists some matrix 𝑅𝑅 ∈ Rn×n such that Ψ − 𝑅𝑅 ⪯ 0 then (15) is implied by ]︂ [︂ Φ1 + 𝑅𝑅 Φ2 − 𝑅𝑅 (16) −1 T ⪯ 0. ΦT 2 − 𝑅𝑅 Φ3 + 𝑅𝑅 − Γ5 Γ3 Γ5 9

IFAC CHAOS 2018 10 Alexey Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018Peregudin et al. / IFAC PapersOnLine 51-33 (2018) 7–11

4. EXAMPLE

Indeed, ]︂ [︂ Φ1 + Ψ Φ2 − Ψ −1 T − ΦT 2 − Ψ Φ3 + Ψ − Γ5 Γ3 Γ5 [︂ ]︂ Φ + 𝑅𝑅 Φ2 − 𝑅𝑅 − T1 T = Φ2 − 𝑅𝑅 Φ3 + 𝑅𝑅 − Γ5 Γ−1 3 Γ5

Consider a model of a single inverted pendulum with external disturbance and measurement noises, described as ⎧ ⎨𝑥𝑥˙ 1 = 𝑥𝑥2 , 𝑥𝑥˙ = 𝜔𝜔 2 sin𝑥𝑥1 + 𝑢𝑢 + 𝑓𝑓𝑓 ⎩ 2 𝑦𝑦i = 𝑥𝑥i + 𝜉𝜉i , 𝑖𝑖 = 1, 2 where 𝑥𝑥1 is an angle, 𝑥𝑥2 is an angular velocity, 𝜔𝜔 is a natural frequency of the pendulum, 𝑢𝑢 is a control, 𝑓𝑓 is an external disturbance, 𝜉𝜉i is a measurement noise, 𝑦𝑦i is a measurable output. The origin (0, 0) of a state space 𝑥𝑥 ∈ R2 corresponds to the upright position of the pendulum.

[︂

]︂ [︂ ]︂ Ψ − 𝑅𝑅 𝑅𝑅 − Ψ 1 = = (Ψ − 𝑅𝑅) [1 −1] ⪯ 0. 𝑅𝑅 − Ψ Ψ − 𝑅𝑅 −1 Due to the property of Schur complement, Ψ − 𝑅𝑅 ⪯ 0 can be presented as a linear matrix inequality ⎤ ⎡ 𝑅𝑅 𝑅𝑅 )︂−1 (︂ ⎦ ⪰ 0. ⎣ (17) 1 𝐽𝐽 𝐷𝐷𝐷𝐷1−1 𝐷𝐷T + 𝛾𝛾𝛾𝛾 𝛼𝛼

Let 𝜔𝜔 = 0.707, then this model can be presented in the form (1) with matrices [︂ ]︂ [︂ ]︂ 0 1 0 𝐴𝐴 = , 𝜑𝜑(𝑥𝑥) = , 0 0 0.5 sin𝑥𝑥1 [︂ ]︂ [︂ ]︂ 0 1 0 𝐵𝐵 = 𝐷𝐷 = , 𝐶𝐶 = 𝐸𝐸 = . 1 0 1 Let 𝛽𝛽 = 0.25, then the condition (2) takes the form

Applying Schur complement property again, setting 𝐾𝐾 = 𝐾𝐾i and using the definition of Γ1 , Γ2 , Γ4 from the statement of Theorem 2, rewrite (16) as ⎡ ⎤ Γ1 Γ4 0 ⎣ΓT ⎦ (18) 4 Γ2 Γ5 ⪯ 0. T 0 Γ5 Γ3

0.25 sin2 𝑥𝑥1 ≤ 0.25(𝑥𝑥21 + 𝑥𝑥22 ) and holds for all 𝑥𝑥 ∈ R2 .

Since 𝐿𝐿 is a variable, for the set defined by (17), (18) to be nonempty it is sufficient that the matrix 𝐴𝐴 + 𝐵𝐵𝐵𝐵i is stable, and the pair (𝐴𝐴𝐴 𝐴𝐴) is detectable. 

Let it be known that 1 1 1 |𝑓𝑓 | ≤ , |𝜉𝜉1 | ≤ , |𝜉𝜉2 | ≤ , 3 5 4 and thus the external disturbance and measurement noises at any instant of time satisfy the condition (3), where ⎤ ⎡ 25 0 ⎥ ⎢ 𝐻𝐻1 = 3, 𝐻𝐻2 = ⎣ 3 16 ⎦ . 0 3

We note that for fixed 𝛼𝛼𝛼𝛼𝛼 𝛼 0 the problem (9) is also a convex optimization problem and thus can be solved quickly and accurately using existing numerical methods. Theorems 1 and 2 give us the convenient tool for finding the partial optimum of the main optimization problem (6), (7). To solve the problem of global minimization with respect to all variables (𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃), the following iterative algorithm based on the alternating method (Gorski et al., 2007, section 4.2.1) is proposed.

Let us choose the initial matrix of the observer (4) as [︂ ]︂ −10 0 . 𝐿𝐿1 = 0 −10

The Algorithm: Step 1. Choose an arbitrary observer matrix 𝐿𝐿1 such that the matrix 𝐴𝐴 + 𝐿𝐿1 𝐶𝐶 is stable. It is always possible since the pair (𝐴𝐴𝐴 𝐴𝐴) is detectable. Choose an arbitrarily small number 𝜀𝜀 𝜀 0, which characterizes the accuracy of the search and set 𝑖𝑖 = 1.

As mentioned above, it is sufficient to choose such 𝐿𝐿1 that the matrix 𝐴𝐴 + 𝐿𝐿1 𝐶𝐶 is stable. Using the proposed algorithm we obtain the optimal result 𝛼𝛼 = 0.7, 𝛾𝛾 = 1.1, [︂ ]︂ ˜ = lim 𝐿𝐿i = −11.02 8.64 , 𝐿𝐿 13.46 −39.36 i→∞ ˜ 𝐾𝐾 = lim 𝐾𝐾i = [−10.86 −6.60] , i→∞ [︂ ]︂ 1.89 −2.31 ˜ , 𝑃𝑃 = lim 𝑃𝑃i = −2.31 10.54 i→∞ trace(𝑃𝑃˜ ) = lim trace(𝑃𝑃i ) = 12.43.

Step 2. Solve for fixed 𝐿𝐿i optimization problem (7) and ̂︀ and 𝑃𝑃̂︀ . Set obtain the optimal (for this 𝐿𝐿i ) matrices 𝐾𝐾 ̂︀ ̂︀ 𝐾𝐾i = 𝐾𝐾 and 𝑃𝑃i = 𝑃𝑃 . Step 3. Solve for fixed 𝐾𝐾i optimization problem (8) and ̂︀ and 𝐽𝐽. ̂︀ Set obtain the optimal (for this 𝐾𝐾i ) matrices 𝐿𝐿 −1 ̂︀ ̂︀ 𝐿𝐿i+1 = 𝐿𝐿 and 𝑃𝑃i+1 = 𝐽𝐽 .

i→∞

˜ = Step 4. If trace(𝑃𝑃i ) − trace(𝑃𝑃i+1 ) < 𝜀𝜀 then set 𝐾𝐾 ˜ = 𝐿𝐿i+1 , 𝑃𝑃˜ = 𝑃𝑃i+1 and stop, otherwise augment 𝑖𝑖 𝐾𝐾i , 𝐿𝐿 by 1 and go to Step 2.

For demonstration purposes, we define the external disturbance and measurement noises in square wave form as 1 𝑓𝑓 (𝑡𝑡) = sgn (sin2t) , 3 (︂ )︂ (︁ 𝜋𝜋t )︁ 1 1 2𝜋𝜋t 𝜉𝜉1 (𝑡𝑡) = sgn sin , 𝜉𝜉2 (t) = sgn sin . 5 4 5 5 Fig. 1 shows the plots of 𝑥𝑥(𝑡𝑡) and 𝑢𝑢(𝑡𝑡) of the closed-loop system under such disturbance and noise.

During the execution of the algorithm, the matrix sequence {𝑃𝑃i }i∈N and the corresponding numerical sequence {trace(𝑃𝑃i )}i∈N are formed. The latter sequence decreases due to steps 2 and 3 of the algorithm and also bounded from below, e.g. by zero, and thus converges. Therefore the stopping condition of the algorithm will once be met. 10

IFAC CHAOS 2018 Alexey Eindhoven, The Netherlands, Oct 30 - Nov 1, 2018Peregudin et al. / IFAC PapersOnLine 51-33 (2018) 7–11

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noises are imposed. Efficiency of the proposed algorithm is demonstrated by the numerical example of an inverted pendulum. The obtained results can be generalized to discrete systems, as well as to different robust variants of the problem. REFERENCES Gerasimov D.N., Paramonov A.V., Nikiforov V.O. Adaptive Disturbance Compensation in Linear Systems with Input Arbitrary Delay: Internal Model Approach. 8th International Congress on Ultra Modern Telecom. and Control Systems and Workshops. 2016. pp. 304-309. Fedele G., Ferrise A. Biased Sinusoidal Disturbance Compensation With Unknown Frequency, IEEE Trans. on Automatic Control, vol. 58, no. 12, pp. 3207-3212, 2013. Wang J., Aranovskiy S.V., Bobtsov A.A., Pyrkin A.A. Compensating for a multisinusoidal disturbance based on YoulaKucera parametrization. Automation and Remote Control. 2017. Vol. 78. No. 9. pp. 1559-1571. Proskurnikov A.V. Universal controllers of V.A. Yakubovich: a systematic approach to LQR problems with uncertain external signals, IFAC-PapersOnLine, Volume 48, Issue 11, 2015, pp 557-562. Nazin S.A., Polyak B.T., Topunov, M.V. Rejection of bounded exogenous disturbances by the method of invariant ellipsoids, Automation and Remote Control March 2007, Volume 68, pp 467-486 Sanchez-Pena R. and Sznaier M. Robust systems: theory and applications, New York, Wiley, 1998. Furtat I. B. Control of Linear Time-Invariant Plants with Compensation of Measurement Noises and Disturbances, Proc. of the 56th IEEE CDC, December 12-15, 2017, Melbourne, Australia. Tsykunov A. M. Compensation of perturbations and disturbances in systems with a measured state vector, Vestn. Astrakhan State Technical Univ. Ser. Management, CSI, 2012, no. 2, 67-76 Baillieul J. Feedback Coding for Information-Based Control: Operating Near the Data Rate Limit, Proc. 41st IEEE Conf. Decision Control, Las Vegas, Nevada, USA, pp. 3229-3236, 2002. Furtat I. B., Fradkov A. L., and Liberzon D. Compensation of disturbances for MIMO systems with quantized output, Automatica, vol. 60, pp. 239-244, 2015. Vasiljevic L., Khalil H. Error bounds in differentiation of noisy signals by high-gain observers, Syst. Contr. Lett., vol. 57, pp. 856-862, 2008. Kiriakidis K. Robust stabilization of the TakagiSugeno fuzzy model via bilinear matrix inequalities, IEEE Trans. Fuzzy Syst., vol. 9, no. 2, pp. 269-277, Apr. 2001. El Ghaoui L. and Balakrishnan V. Synthesis of fixedstructure controllers via numerical optimization, IEEE Conference on Decision and Control, Lake Buena Vista, FL, 1994, pp. 2678-2683. Gorski J., Pfeuffer F., Klamroth K. Biconvex sets and optimization with biconvex functions: A survey and extensions, Mathematical Methods of Operations Research, 2007, 66, 373-407. Polyak B.T. Convexity of quadratic transformations and its use in control and optimization, J. Optim. Theory and Appl. 1998. V. 99, 553-583.

Fig. 1. The plots of 𝑥𝑥(𝑡𝑡) and 𝑢𝑢(𝑡𝑡) under disturbance and noise.

Fig. 2. Optimal invariant ellipsoid 𝑥𝑥T 𝑃𝑃˜ −1 𝑥𝑥 = 1 and a trajectory 𝑥𝑥(𝑡𝑡). Fig. 2 shows the optimal invariant ellipsoid 𝑥𝑥T 𝑃𝑃˜ −1 𝑥𝑥 = 1 of the closed-loop system and the trajectory 𝑥𝑥(𝑡𝑡) with initial T state 𝑥𝑥(0) = [1.15 0] , which corresponds to a pendulum movement, started 65.89◦ away from the upright position with zero initial speed. As one can see, the trajectory 𝑥𝑥(𝑡𝑡), started within the ellipsoid, never leaves it and thus the condition (6) holds. 5. ACKNOWLEDGEMENTS Results of Section 3 were supported by Russian Science Foundation (project no. 18-79-10104) in IPMERAS. The other results were partially financially supported by Goverment of Russian Federation (Grant 08-08). 6. CONCLUSIONS The algorithm for finding the optimal parameters of the dynamic linear feedback controller for compensation of external disturbances and measurement noises for nonlinear systems is proposed. Unlike Gerasimov et al. (2016); Fedele et al. (2013) we do not assume that external disturbances are generated by a linear model. Differently from Furtat (2017), no restrictions on derivatives of the measurement 11