The use of differential geometry methods for linearization of nonlinear electrical circuits with multiple inputs and multiple outputs (MIMOs)

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1 Electrical Engineering (08) 00: ORIGINAL PAPER The use o dierential geometry methods or linearization o nonlinear electrical circuits with multiple inputs and multiple outputs (MIMOs) Sebastian Różowicz Andrzej Zawadzki Received: 6 June 07 / Accepted: 8 September 08 / Published online: 3 October 08 The Author(s) 08 Abstract The paper presents a transormation o nonlinear MIMO electrical circuit into linear one by a change in coordinates (local dieomorphism) with the use o a closed eedback loop The necessary conditions that must be ulilled by a nonlinear system to make linearizing procedures possible are presented Numerical solutions o state equations or the nonlinear system and equivalent linearized system are included Keywords Nonlinear MIMO circuit Linearization State space transormation Feedback linearization Local dieomorphism Lie algebras Introduction Electrical circuits with concentrated parameters or electromechanical systems can be described by a inite number o mutually coupled ordinary dierential equations and algebraic relations Most mathematical models o physical systems are complex They are characterized by high-order and numerous nonlinearities A well-known nonlinear model with multiple inputs and multiple outputs which describe its dynamics in the state space can be represented by the ollowing equations: dx(t) dt (x(t)) + m i g i (x)u i (t), x(t) R n, u(t) R m, y(t) h(x(t) [h (x(t)),,h m (x(t))] T, y(t) R p, () where: and g i are smooth vector ields determined on a maniold M R n, called state space, h is a smooth mapping speciied or the state space M in p-dimensional output space, R p, h : M R p, u(t) input vector, y(t) output vector B Sebastian Różowicz srozowicz@tukielcepl B Andrzej Zawadzki azawadzki@tukielcepl Faculty o Electrical Engineering, Automatics and Computer Science, Kielce University o Technology, al Tysiąclecia Państwa Polskiego 7, 5-34 Kielce, Poland The analysis o nonlinear systems (), especially in dynamic states, is a very diicult task o circuit theory In most cases, there is no analytical solution o the problem (which is requently sought) and the inormation about the current low and voltage distribution can be obtained using the methods o numerical integration As shown in Gear [], Butcher [], Najm [3], although they are universal and applicable to any number o dierential equations, they generate a numerical solution (compiled in the orm o tables, graphs, etc) Thereore, in the search or analytical solutions the transormation o nonlinear description into a linear one by linearization (ensuring local balance o the system dynamics) is very useul in solving practical problems relating to the behaviour o nonlinear circuits The methods o dierential geometry provide a tool or linearization and decoupling, and decomposition o a system o Eq () to a linear orm The application o geometric approach to solve nonlinear problems initiated by Brockett [4] was used in control theory with observability and controllability o the systems taken into account [5, 6] The methods o dierential geometry allowed the development o eicient techniques or the analysis and synthesis o such systems The series o publications by Byrnes and Isidori [7], Celikovsky and Nijmeijer [8], Jakubczyk and Respondek [9] which considered the problem o nonlinear transormation o a linear system by changing the coordinates (local dieomorphism) and using eedback [, 0, ] considerably contributed to the development o the techniques mentioned Furthermore, the problems o transormation o nonlinear systems to lin- 3

2 86 Electrical Engineering (08) 00:85 84 ear orms have been widely studied in the literature by Isidori [, 3], Nijmeijer and van der Shat [4] Bodson [5], Su [6], and Hunt et al [7] As Eq () describes, a nonlinear system in local coordinates x in R n [where x (x, x,,x n ) is a local coordinate system on a smooth maniold M X s x], one normally considers a transormation o state variables [ 4, 8 0] whose operation can be presented as ollows: S : x(t) z(t) () As a result o the transormation, the state vector in new coordinates assumes the ollowing orm: z S(x), (3) where: S(x) [S (x), S (x),, S n (x)] T is the mapping deined on the open set o R n space with R n values The transormation o this kind is called dieomorphism [9, 3] The necessary and suicient conditions or the existence o S(x) transormation o nonlinear system into linear one are given in [4] Transormation S(x) is diicult to ind- especially or multidimensional systems so the systems which cannot be globally linearized and are transormed only into quasi-linear systems can be urther linearized by the eedback In this case, a combination o linearization by transormation o state variables and input transormation u(t) using a eedback is applied [, 5] Ater transormation involving the change in coordinates and the introduction o the eedback, the state vector in a new coordinate system can be illustrated as ollows: z S(x) + eedback (4) The cited publications indicate that although geometric methods are mainly applied in control theory, they can be also used in other areas, eg in the theory o electrical circuits Such attempts have already been made (or example [6 30]), but they were too ew and not exhaustive enough It appears that geometrical methods can easily ind other applications Since the presented issues are still valid, we attempt to use geometric methods in the theory o electrical circuits It seems that the results presented can easily ind other applications As the considered issues are still valid, we attempt to use geometrical methods in theory o electrical circuits Some signiicant developments have been made in robust eedback linearization in [3 33], but most o them are applicable or single-input nonlinear systems or when parametric uncertainties exist The act that the there are many more publications analysing SISO systems than those dealing with MIMO systems prompted the authors to use the methods o dierential geometry or linearization o a class o nonlinear electrical circuits with multiple inputs (power sources) and multiple outputs The paper is organized as ollows: Section presents the elements o Lie algebra, used in the construction o a new base o state space The basic deinitions and theorem concerning the conditions to be met by nonlinear system to perorm the linearizing procedures are included Section 3 deals with the construction o linearizing transormation The eective transormation and digital simulation o mathematical model o nonlinear electrical circuit MIMO showing that linearization is correct are given in Sects 4 and 5 Conclusions and comments are presented in Sect 6 The paper ends with a list o reerences Preliminaries Some basic deinitions and concepts o dierential geometry are presented Attempts have been made to discuss them in a simpliied and compact orm More detailed inormation can be ound in the reerences [34 36] In the analysis o nonlinear systems, the Lie derivative deserves particular attention It is the operation involving real-valued unction h and vector ield deined on maniold M o R n space The result o the operation is a smooth realvalued unction deined or each x rom M set Deinition Let h mapping: R n R n be a smooth scalar unction o n variables, x (x, x,,x n ) T R n and : R n R n a vector ield deined on maniold M R n, then the Lie derivative o scalar unction h(x) h(x, x,x n ) along the ield is a scalar unction given by the ormula: L h h x n i h x i i ( h), (5) where: stands or gradient; stands or scalar product For example: h h [ h x, h, h ] T x x x n The Lie derivative is a directional derivative o a scalar unction along the vector ield I g is another vector ield deined on the same maniold M R n then a derivative o scalar unction along the vector ield g is deined as ollows: L g (L h) L g L h x (L h) g ( ) h x x g ((L h) g) (6) 3

3 Electrical Engineering (08) 00: We consider a general case o nonlinear system described by Eq () with m power sources (inputs) and m outputs The output vector o y(t) system is related to u(t) vector o power sources through state variables and nonlinear equation o state The task o linearization consists in inding a new vector v(t) o power source () such that each m output block depends only on one power source, that is, has the outputs decoupled rom the inputs The terms o decoupling outputs rom inputs are given by the deinition: Deinition In nonlinear system (), outputs are decoupled rom inputs i the ollowing conditions are satisied: or each i {,, m}, y i outputs are invariant with respect to u j input or j i; y i output is not invariant with respect to u i input or i {,, m} The necessary condition or the invariance o outputs is given by the ollowing theorem: Theorem Let us consider a nonlinear system () with y output invariant with respect to u input Then, or any k 0 and or any vector ields τ, τ,, τ k selected rom {, g,,g m }weget:l u L τ L τk h(x) 0 or each x Hence, the suicient condition or decoupling outputs rom inputs: L g j L τ L τk h(x) 0, (7) or each k 0 and τ, τ,,τ k {, g,,g m } The starting point to determine dieomorphism S(x) transorming the system () into linear system is the deinition o a relative degree o the system, sometimes called the characteristic number [ 4] Deinition 3 A relative degree r (x),, r p (x) oasmooth nonlinear system includes such small natural numbers that or each j {,,p}: ( ) L g L k h j (x) L g L k h j (x),,l gm L k h j (x) 0, k 0,,r j, x U, (8) L g L r j (x) h j (x) ( L g L r j (x) 0, or certain x U h j (x),,l gm L r j (x) h j (x) I: L g L k h j(x) ( L g L k h j(x),,l gm L k h j(x) ) 0 k 0 and x U, it is assumed that r j It should be emphasized that each number r i is associated with the ith output o the h i system It is also worth noting ) (9) that by dierentiating the y i output with respect to time r i, we obtain: y r i i L r i h i(x)+l gi L r i h i(x)u i, i {,,m}, (0) ie r i is the number that tells us how many times the system output y i (t) should be dierentiated to obtain the supply u i in an explicit orm, that is, the linear relation between supply u i and output y i Deinition 4 Nonlinear system () has outputs decoupled rom inputs around x o point i there exists such a neighbourhood V o x o or which (7) holds true or each x V and i a relative degree o the system r,,r m is inite and constant on V It should be pointed out that i the system has outputs decoupled rom inputs then or x R n the set L gi L r i h i (x) 0, i m contains V I a nonlinear system does not have outputs decoupled rom inputs, a regular static eedback can be used to change it into a system with outputs decoupled rom inputs As a result, we obtain Eq (7) Deinition 5 Regular static eedback or a nonlinear system () is deined by the relation: u α(x)+β(x)v, () where: u (u,,u m ) T and α: M R m and β: M R mxm are smooth mappings such that matrix β(x) is non-singular or any x, and v =(v,,v m ) is a new input vector Thereore, in order to solve the problem o linearization o the system with multiple inputs and multiple outputs with initial state x o one should ind a regular static eedback deined in such neighbourhood V o x o point that each y i output is aected by only one v i input, i m The eedback in this case has the orm: u v b (x) u v E (x) b (x), () b m (x) u m v m where: E(x) decoupling matrix and vector b(x) aregiven by the relations: L g L r h,,l gm L r h E(x) L g L r p h p,,l gm L r p b(x) L r h (x) L r p h p (x) h p ; (3) 3

4 88 Electrical Engineering (08) 00:85 84 Theorem Let us consider a given nonlinear system () and the point x o The necessary and suicient condition or the solution o input output linearization problem is the existence o a non-singular matrix E(x) at x x o, that is, the rank o matrix dim E(x o ) m The proo o the theorem is presented in [4] The method used to determine the transormation S(x) linearizing a nonlinear system () and coordinates z(t) othe linearized system is presented in the next section 3 Determination o transormation linearizing nonlinear MIMO system Let us consider the ollowing nonlinear system () with m inputs and m outputs: ẋ (x)+ m g i (x)u i (x), k y h (x), y j h j (x), y m h m (x) (4) Let outputs be divided into m separable blocks Let the system have a relative degree r,,r m at x o, and let the rank o decoupling matrix E(x o ) equal m Then r + + r m n and or i m the transormation o coordinates is written as: S i (x) h i(x), S i (x) L h i (x), S i r i (x) L r i h i (x) (5) I r r + + r m is less than n it is always possible to ind such n r o unctions S r+ (x),, S n (x) that the mapping: S(x) [ S (x),,s r (x),,s m (x),,sm r m (x), S r+ (x),s n (x) ] T (6) has a Jacobi matrix which is non-singular at x o and thereore determines local coordinates o transormation in the neighbourhood o x o To determine a relative degree o a multidimensional system, we dierentiate output j according to the relation: ẏ j L h j (x)+ m ( Lgi h j (x) ) u i (7) i I L gi h j (x) 0 dierentiation should be continued or each i until or some natural r in the ith step, we obtain: L gi L r j h j (x) 0 We dierentiate outputs according to a general relation: y (i) j L r j h j(x)+ m i L g L r j h j (x)u i (8) Thus i L gi L r j h j (x) 0, the output equation takes the ollowing orm: y r j L r h u y r j L r h + E(x) u, (9) y r m j L r m h m u m where: E(x) is a decoupling matrix o m m dimension described by the relation (3) As a result, we obtain a relative degree o the system r r + + r p,or p m I det E(x) 0, then E(x) matrix is non-singular and we can determine the eedback linearizing the system The eedback is described by the ollowing relation (): u L r h (x) u L r h (x) (E(x)) + (E(x)) u m L r m h m (x) v v (E(x)) b(x)+(e(x)) v (0) v m To determine the transormation o state variables S(x), we use Eq (5), which can be written in the ollowing general relation Sn i (x) Lk h i (x) Hence, the state variables are given by the system o equations: [ ] [ ] z, z,,z ri S i, Si,,Si n [ ] h (x), L h (x),l r i h p (x) () Their derivatives are determined as ollows: ż d dt (h (x)) dh dx (x)ẋ dh dx (x)(x) L h (x) z () 3

5 Electrical Engineering (08) 00: Fig Diagram o nonlinear electrical circuit o the ourth order with two power sources Thus, new dynamics o a linearized system are described by an m set o the ollowing equations: ż i zi ; żi zi 3,,żi r i zi r i ; ż i r i v i ; ż i r+ q r+,,ż i n q n (3) The outputs o the system are given by relation (9) Equations żr+ i q r+,, żn i q n in (3) are determined according to the ollowing equations: ) q i (z) L S i (S (z), r + i n (4) The solution o the considered problem is deined locally in the state space or x close to x o, in which the decoupling matrix E(x) is non-singular It should be noted that nonsingularity o the matrix is also a necessary condition or a solution to exist 4 Example We consider an indeinite state in an electrical circuit, comprising two power sources and two nonlinear elements, as shown in Fig We assume that the initial state o the circuit is zero (zero initial conditions), and that at t=0 the switches W and W are closed simultaneously Nonlinear current voltage characteristics o the nonlinear resistive element are described by the second-degree polynomial o the ollowing orm: u(i) b i, (5) where: b is a coeicient o ( A ) dimension A nonlinear coil (without losses) is described by the ollowing relation: u(t) L(i) di dt, (6) where: L(i) (a i) and a is a coeicient o ( A ) dimension We assume that i x, i x, i 4 x 3, u C x 4 are state variables and the considered system has the orm: e x 4 + L(i )ẋ 0, x 4 + R (x x 3 )+L ẋ + b x e e 0, L ẋ 3 R (x x 3 ) 0, Cẋ 4 x x (7) We order variables, adopt the notations: /L k; R/L c; b/l d; R/L w, /C l, and employ the expression or inductance o a nonlinear coil L(i ) L(x ) (a x ) to obtain a model system o equations: ẋ a x x 4 a x e ẋ k x 4 c (x x 3 ) d x + k e + k e ẋ 3 w(x x 3 ) ẋ 4 l(x x ) (8) 3

6 80 Electrical Engineering (08) 00:85 84 Fig Time characteristics o nonlinear system outputs y (t) andy (t) Fig 3 Numerical simulation o the solution o the system ater transormation onto the original space: x(t) S (z(t)) or which the corresponding vectors (x), g (x), g (x), (, g R 4, n 4 ) have the orm: (x) a x x 4 k x 4 c (x x 3 ) d x w x w x 3 l x l x g (x) a x k 0 0 ; ; 0 g (x) k 0 (9) 0 The equations o response (output) are as ollows: y x 3 y x 4 (30) Hence, the output unctions have the orm:h (x) x 3 and h (x) x 4 In the considered circuit, the outputs y(t) are related to the power supply vector e(t) through state variables and nonlinear equation o state To solve the problem, we have to ind a new power source deined by a regular, static eedbackthe irst step o linearization is to determine a relative degree o the system according to deinition The considered system has two inputs and two outputs From Eq (30), we can directly determine dierentials o unctions h (x) and h (x): h (x) (x) h (x) (x) [ h (x) x [ h (x) x h (x) x h (x) x h (x) x 3 h (x) x 3 ] h (x) x 4 [ 000 ] ] h (x) x 4 [ 000 ] Calculating the Lie derivatives o unction h(x) along g(x) ield or h (x) unction, we get: L g h (x) [ 000 ][ ax k 00 ] T 0 L g h (x) [ 000 ][ 0 k 00 ] T 0 Similarly, or unction h (x) we obtain: 3

7 Electrical Engineering (08) 00: L g h (x) [ 000 ][ ax k 00 ] T 0 L g h (x) [ 000 ][ 0 k 00 ] T 0 As the irst derivatives equal zero, we calculate the derivatives o higher orders For h (x) unction along g (x) ield, we obtain: a x x 4 L g L h (x) L g [ ] 000 kx 4 c(x x 3 ) dx w x w x 3 l x l x L g (w x w x 3 ) a x [ 0 w w 0 ] k 0 k w 0 0 Similarly, or g (x) we get: L g L h (x) L g (w x w x 3 ) k w 0 Thus, the relative degree o the subsystem related to unction h (x) isr For the h (x) unction, we have: L g L h (x) L g (l x l x ) alx + kl 0, L g L h (x) L g (l x l x ) kl 0 This means that or the h (x) unction, a relative degree o the subsystem is also r Finally, we obtain that the relative (vector) degree o the considered system is {r, r } {, } or each point x (x, x, x 3, x 4 ) at which x 0 Next, we deine a decoupling matrix E(x)(3) which has the ollowing orm: E(x) [ ] k w k w (3) a l x k l The rank o E(x) equals Since det E(x) k w l a x 0, it ollows that matrix E(x) is non-singular and it is possible to determine eedback linearizing input output system: [ e e ] E (x) [( v v ) ( )] b (x), (3) b (x) where: v [v, v ] T is a new power source vector We can now determine new coordinates o the considered system according to (): z (x) h (x) x 3, z (x) L h (x) w x w x 3, z 3 (x) h (x) x 4, z 4 (x) L h (x) l x l x (33) To ind a normal orm o the system, we have to determine b(x) vector and E (x) matrix rom Eq (3) [ ] E kl kw (x) det E(x) alx kw [ ] [ ] kl kw wax lax kwlax alx kw kw lax (34) Since: b(x) [ ] b (x) b (x) [ ] L h (x) L h, (35) (x) we have to calculate corresponding Lie derivatives o h(x) unction along the vector ield (x): L h (x) L (w x w x 3 ) [0 w w 0] (x) kwx 4 cw(x x 3 ) dwx and w x + w x 3, L h (x) L (l x l x ) [ l l 0 0] (x) alx x 4 klx 4 cl(x x 3 ) dlx Now, the normal orm o the linearized system rom () is written as: ż (x) d dt h (x) d dt (x 3) ẋ 3 w x w x 3 z (x), ż (x) d dt (L h (x)) d dt (w x w x 3 ) w ẋ w ẋ 3 wkx 4 cw(x x 3 ) dwx + kwe + kwe w x + w x 3, ż 3 (x) d dt h (x) d dt (x 4) ẋ 4 l x l x z 4 (x), ż 4 (x) d dt (L h (x)) d dt (l x l x ) l ẋ l ẋ lkx 4 cl(x x 3 ) 3

8 8 Electrical Engineering (08) 00:85 84 dlx + kle + kle alx x 4 + alx e The expression or linearizing eedback (3) can be written as: e det E(x) (a ( b (x)+v ) a ( b (x)+v )), e det E(x) ( a ( b (x)+v ) + a ( b (x)+v )) Thus, ater transormations we get: v b (x)+a (x) e + a (x) e wkx 4 cw(x x 3 ) dwx (36) w x + w x 3 + kwe + kwe, v b (x)+a (x) e + a (x) e lkx 4 cl(x x 3 ) dlx alx x 4 +(alx + kl)e + kle (37) For the assumed eedback, the system deined by (3) is described by the ollowing equations: ż (x) z (x), ż (x) v, ż 3 (x) z 4 (x), ż 4 (x) v (38) 5 Veriication o the model and numerical experiments In order to veriy the results obtained a numerical solution o nonlinear equation o state, given by Eqs (8) and (30), and its linearized model (38, 39) was conducted Numerical solution o nonlinear equations o state was carried out or zero initial conditions x(0) 0, parameters a b k, c 5, d 0, w, and or inputs e (t) e (t) (t) Figure presents time characteristics o nonlinear system outputs y (t) and y (t) To illustrate correct perormance o the transormation linearizing a nonlinear system, we compare the obtained simulation results with the results o simulations conducted or Eqs (38) and (39) linearized by transormation onto original space x(t) S (z(t)) For the purpose o comparison, we conduct the ollowing calculations: Since: z (x) x 3 ; z (x) w x w x 3 ; z 3 (x) x 4 ; z 4 (x) l x l x, then determining the variables o x state we get: x (t) z (t)+/w z (t) /l z 4 (t); x (t) z (t)+/w z (t); x 3 (t) z (t); x 4 (t) z 3 (t) New power sources v i v are determined rom the relation: v wkx 4 cw(x x 3 ) dwx + kwe + kwe w x + w x 3 Also rom the structure o power source, it ollows that: y z, y z 3 (39) In matrix notation, the linearized system has the orm: ż 000 z 0 0 ż ż z 000 z v v (40) ż z 4 0 As a result o linearization, we obtain a decomposed and decoupled linear model o the coordinate system z(t) Each state variable is deined by the dierential equation with the right side, represented by the irst-degree polynomial o the simplest orm To sum up, we have determined linear orm o state equations or the considered system in which the ith output y i depends only on the ith power input v i,ori, v lkx 4 cl(x x 3 ) dlx + kle + kle alx x 4 + alx e Substituting the above relations into linear state equation o a linearized system, we obtain variable characteristics o x state o the original system Initial conditions were calculated as ollows: z (0) x 3 (0); z (0) w x (0) w x 3 (0); z 3 (0) x 4 (0); z 4 (0) l x (0) l x (0) Figure 3 presents time characteristics o outputs y (t) and y (t) as shown or a nonlinear system As S(x) is an algebraic transormation o numerical solutions (time characteristics) o nonlinear systems state equations (Fig ), and the corresponding solutions o linearized systems (Fig 3) overlap This conirms that the derived mathematical models o linear systems subjected to linearization are correct 3

9 Electrical Engineering (08) 00: Conclusion The use o geometrical methods or nonlinear mathematical models makes it possible to obtain simple models o linearized systems Thanks to that we can analyse linearized models using methods known rom the theory o linear systems and then transer the results to a nonlinear system () by means o inverse transormation S (z) The transormation o nonlinear description into linear one by linearization (ensuring local balance o the system dynamics) is very useul in solving practical problems relating to the behaviour o nonlinear objects It not only allows a simpler analysis o such systems but also makes it possible to avoid problems associated with the nonlinearity o the system Furthermore, it is worth noting that the numbers r + + r p are called the characteristic numbers and each characteristic number r i is associated with jth output o h i system and tells us how many times y i must be dierentiated to receive power supply u i explicitly I a relative degree o the system r < n then such n r o S r+ (x),, S n (x) coordinates must be deined that the Jacobian o the mapping (6) isoull rank at x o point Then, making an additional assumption that L g S i (x) 0 we obtain additional equation o the system dynamics ż i L S i (x(t)) Nonlinear system o multiple inputs and multiple outputs can be decomposed by a static eedback only i E(x) matrix is non-singular Non-singularity o the matrix is thus a necessary condition or a solution to the problem However, when choosing dierent output unctions o the considered system, it is a requent case that when we look or a regular eedback, a decoupling matrix is singular In this case, dynamic eedback [ 4] is used to solve the problem Open Access This article is distributed under the terms o the Creative Commons Attribution 40 International License ( onsorg/licenses/by/40/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original 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