Proceedings o the International MultiConerence o Engineers and Computer Scientists 4 Vol I,, March - 4, 4, Hong Kong Designs o Minimal-Order State Observer and Servo Controller or a Robot Arm Using Piecewise Bilinear Models Tadanari Taniguchi, Luka Eciolaza, and Michio Sugeno Abstract This paper proposes a servo control system based on a minimal-order state observer or nonlinear systems approximated by piecewise bilinear (PB) models. The design method is capable o designing the state observer and the servo controller o nonlinear systems separately. The approximated system is ound to be ully parametric. The input-output (I/O) eedback linearization is applied to stabilize PB control systems. Although the controller is simpler than the conventional I/O eedback linearization controller, the control perormance based on PB model is the same as the conventional one. The PB models with eedback linearization are a very powerul tool or the analysis and synthesis o nonlinear control systems. We apply the control method to a robot arm model. An example conirms the easibility o the our proposals. Index Terms nonlinear control, piecewise bilinear model, input-output linearization, robot arm, minimal-order state observer I. INTRODUCTION PIECEWISE linear (PL) systems which are ully parametric have been intensively studied in connection with nonlinear systems [], [], [3], [4]. We are interested in the parametric piecewise approximation o nonlinear control systems based on the original idea o PL approximation. The PL approximation has general approximation capability or nonlinear unctions with a given precision. One o the authors suggested to use the piecewise bilinear (PB) approximation [5]. PB approximation also has general approximation capability or nonlinear unctions with a given precision. We note that a bilinear unction as a basis o PB approximation is, as a nonlinear unction, the second simplest one ater a linear unction. The PB model has the ollowing eatures. ) The PB model is derived rom uzzy i-then rules with singleton consequents. ) It is built on piecewise hyper-cubes partitioned in the state space. 3) It has general approximation capability or nonlinear systems. 4) It is a piecewise nonlinear model, the second simplest ater a PL model. 5) It is continuous and ully parametric. So ar we have shown the necessary and suicient conditions or the stability o PB systems with respect to Lyapunov unctions in the two dimensional case [6], [7] membership unctions are ully taken into account. We derived the stabilizing Manuscript received December 3, 3; revised January 3, 4. This work was supported by a URP grant rom Ford Motor Company which the authors thankully acknowledge. In addition, this work was supported by Grant-in-Aid or Young Scientists (B: 3776) o The Ministry o Education, Culture, Sports, Science and Technology in Japan. T. Taniguchi is with IT Education Center, Tokai University, Hiratsuka, Kanagawa, 599 JAPAN, email: taniguchi@tokai-u.jp. L. Eciolaza and M. Sugeno are with European Centre or Sot Computing, 336 Mieres, Asturias, SPAIN, emails: luka.eciolaza@sotcomputing.es, michio.sugeno@gmail.com conditions [8], [9] based on the eedback linearization, [8] applies the input-output linearization and [9] applies the ull-state linearization. Although the controllers are simpler than the conventional I/O eedback linearization controller, the control perormance based on PB model is the same as the conventional one. This paper proposes a servo control system based on a minimal-order state observer o nonlinear control systems approximated by PB models. The design method is capable o designing the state observer and the servo controller o nonlinear systems separately. Although the controller is simpler than the conventional I/O eedback linearization controller, the control perormance based on PB model is the same as the conventional one. In addition, the perormance o the observer-based PB controller is equivalent to the PB controller without the state observer. This paper is organized as ollows. Section II presents the canonical orm o PB models. Section III presents PB controllers or nonlinear plants with PB modeling and I/O linearization. Section IV presents a servo control o PB models. Section V proposes a minimal-order state observer or PB models. Section VI applies the proposed method to a robot arm model and shows the easibility o the proposed methods. Section VII gives conclusions. II. CANONICAL FORM OF PIECEWISE BILINEAR MODELS A. Open-loop systems In this section, we introduce the PB models suggested in [5]. We deal with the two dimensional case without loss o generality. Deine a vector d(σ,τ) and a rectangle R στ in the two-dimensional space as, respectively, d(σ,τ) (d (σ),d (τ)) T, R στ [d (σ),d (σ )][d (τ),d (τ )]. σ and τ are integers: < σ,τ < d (σ) < d (σ),d (τ) < d (τ ) and d(,) (d (),d ()) T. The superscript T denotes transpose operation. For x R στ, the PB system is expressed as σ τ ẋ = ω (x i )ω j (x )(i,j), () σ τ x = ω(x i )ω j (x )d(i,j),
Proceedings o the International MultiConerence o Engineers and Computer Scientists 4 Vol I,, March - 4, 4, Hong Kong ω(x σ ) = (d (σ ) x )/(d (σ ) d (σ)), ω σ (x ) = (x d (σ))/(d (σ ) d (σ)), ω(x τ () ) = (d (τ ) x )/(d (τ ) d (τ)), ω τ (x ) = (x d (τ))/(d (τ ) d (τ)), and ω,ω i j [, ]. In the above, we assume (,) = and d(,) = to guarantee ẋ = or x =. A key point in the system is that the state variable x is also expressed by a convex combination o d(i,j) with respect to ω i and ω j just as in the case o ẋ. As is seen in Eq. (), x is located inside R στ which is a rectangle: a hypercube in general. That is, the expression o x is polytopic with our verticesd(i,j). The model oẋ = (x) is built on a rectangle including x in the state space and it is also polytopic with our vertices(i,j). We call this orm o the canonical model () parametric expression. Representing ẋ with x in Eqs. () and (), we can obtain the state space expression o the model which is ound to be bilinear (bi-aine) [5]. Thereore, the derived PB model has simple nonlinearity. In the case o the PL approximation, a PL model is built on simplexes partitioned in the state space, triangles in the two dimensional case. Note that any three points in the three dimensional space are spanned with an aine plane: y = abx cx. A PL model is continuous. It is, however, diicult to handle simplexes in the rectangular coordinate system. B. Closed-loop systems We consider a two-dimensional nonlinear control system. {ẋ =o (x)g o (x)u(x), (3) y =h o (x). The PB model (4) can be constructed rom the nonlinear system (3). {ẋ =(x)g(x)u(x), (4) y =h(x), σ τ (x) = ω(x i )ω j (x )(i,j), σ τ g(x) = ω (x i )ω j (x )g(i,j), σ τ h(x) = ω(x i )ω j (x )h(i,j), στ x = ω(x i )ω j (x )d(i,j). The modeling procedure in the region R στ is as ollows. Algorithm.: Piecewise bilinear modeling procedure ) Assign vertices d(i,j) or x = d (σ),d (σ), x = d (τ),d (τ ) o the state vector x, then the state space is partitioned into piecewise regions, see also Fig.. ) Compute the vertices (i,j), g(i,j) and h(i,j) in Eqs. (5), by substituting the values o x = d (σ), d (σ) (5) d (τ ) (σ,τ ) d (τ) ω σ ω τ (σ,τ) d (σ) (x) Fig.. Piecewise region ( (x), x R στ ) (σ,τ ) ω σ ω τ (σ,τ) d (σ ) and x = d (τ), d (τ) into original nonlinear unctions o, g o and h o in the system (3). Fig. illustrates the expression o (x), (x) = ( (x), (x)) T and x R στ. The overall PB model can be obtained automatically when all the vertices are assigned. Note that (x), g(x) and h(x) in the PB model coincide with those in the original system at the vertices o all the regions. III. DESIGN OF PB CONTROLLERS FOR NONLINEAR SYSTEMS WITH PB MODELING AND I/O LINEARIZATION This section deals with the I/O linearization o nonlinear control systems approximated with PB models. We consider, in particular, nonlinear systems and show their I/O linearization based on PB models in detail. First we give a brie introduction to the I/O linearization o PB models [9], []. A. I/O linearization Consider the PB model (4) in the previous section. The derivative ẏ is given by L h(x) = ẏ = h x ((x)g(x)u) = L h(x)l g h(x)u. L g h(x) = n σ h(δ,i,...,i n ) n i =σ ω i n σ h(i,i,...,δ n ) n, h(δ,i,...,i n ) n g i =σ ω i = d (σ ) d (σ ), d (δ ) = d (σ ) d (σ ), = d n (σ n ) d n (σ n ), h(i,i,...,δ n ) g n,
Proceedings o the International MultiConerence o Engineers and Computer Scientists 4 Vol I,, March - 4, 4, Hong Kong h(δ,i,...,i n ) =h(σ,i,...,i n ) h(σ,i,...,i n ), h(i,δ,...,i n ) =h(i,σ,...,i n ) h(i,σ,...,i n ), h(i,i,...,δ n ) =h(i,i,...,σ n ) h(i,i,...,σ n ). I L g h(x) =, then ẏ = L h(x) is independent o u. We continue to calculate the second derivative o y, denoted by y () and then we obtain y () = L h x ((x)g(x)u) = L h(x)l g L h(x)u, L h(x) = L h x L h x n n, L g L h(x) = L h x g L h x n g n, L h x = 3 i 3=σ 3 ω i3 3 n σ i =σ ω i n n h(δ,δ,...,i n ) n d (δ ) h(δ,i,...,δ n ) n h(δ,i,...,i n ) n σ i =σ ω i n (δ,i,...,i n ) n h(i,i,...,δ n ) n (δ,i,...,i n ) n, σ L h σ h(δ = x ω i,i,...,δ n ) n d i (δ ) n=σ σ n n i n =σ n h(i,...,δ,δ n ) d (δ ) n h(δ,i,...,i n ) n σ i =σ ω i σ i =σ ω i σ i =σ ω i (i,i,...,δ n ) h(i,i,...,δ n ) n (i,i,...,δ n ) Once again, i L g L h(x) =, then y () = L h(x) is independent o u. Repeating this process, we see that i h(x) satisies L g L i h(x) =,i =,,...,ρ, L g L ρ h(x) then u does not appear in the equations o y, ẏ,..., y (ρ ) and appears in the equation o y (ρ) with a nonzero coeicient: y (ρ) =L ρ h(x)l gl ρ h(x)u. The oregoing equation shows clearly that the system is input-output linearizable, since the state eedback control u =( L ρ h(x)v)/l gl ρ h(x) reduces the input-output map to y (ρ) = v, which is a chain o ρ integrators. In this case, the integer ρ is called the relative degree o the system. I L g L ρ h(x t ) =, the relative degree cannot be deined at x = x t. In some cases the relative degree can be deined at the point because we can adjust a partition o the state space or PB modeling so that L g L ρ h(x t ). Deinition 3.: The nonlinear system is said to have relative degree ρ, ρ n, in a region D D i L g L i h(x) =, i =,,,ρ L g L ρ h(x), or all x D. The input-output linearized system can be ormulated as { ξ = Aξ Bv, (6) y = Cξ, ξ R ρ, C = (,,,, ) T,...... A =........, B =.. Note that all the eedback linearizable PB systems (4) are transormed into the linear system (6). Thereore it is easy to design the stabilizing controller and analyze stability o the PB systems. According to the relative degree, three cases o linearized systems (6) must be considered. Relative degree: ρ = n In this case, the state vector o the inputoutput linearized system is z = ξ = (h(x), L h(x),, L ρ h(x)) T. The state vector z is necessary to be a dieomorphism. Relative degree: ρ < n There is unobservable state (n ρ dimensions). It is necessary to consider the zero dynamics o the unobservable state µ. The state vector z is necessary to be a dieomorphism. z = ( ξ, µ ) T, ξ R ρ, µ R n ρ, µ(ξ,µ) = ζ (ξ,µ)ζ (ξ,µ)v. µ(,µ) is characterized by zero dynamics. In the case ol g L i h(x) =, i, the proposed approach cannot be applied. When the relative degree ρ n, the input-output linearizing controller is u = α(x)β(x)v, α(x) = L ρ h(x)/l gl ρ h(x), β(x) = /L g L ρ h(x). In the ollowing, we assume the relative degree is n (ull). The stabilizing linear controller v = Fξ o the linearized
Proceedings o the International MultiConerence o Engineers and Computer Scientists 4 Vol I,, March - 4, 4, Hong Kong system (6) can be obtained so that the transer unction G = C(sI A) B is Hurwitz. The linearizing controller is also characterized as the LUT (Look-Up-Table) controller, the LUT-controller is widely used or industrial applications, in particular, or vehicle control because o simplicity and also visibility as a nonlinear controller. In the case o the LUT-controller, control inputs are calculated by interpolation based on the table. When bilinear piecewise interpolation is adopted, the LUT-controller is ound to be exactly the PB system. r Fig.. G E K T - - Linearized system based on PB models F Two-degree o reedom servo control system C y IV. SERVO CONTROL We apply a servo control [] to nonlinear systems with the PB models based on I/O linearization. This is a twodegree o reedom servo control to nonlinear systems as shown in Fig.. The controller is designed to deal with disturbances and robustness. In this igure, T and show the coordinate transormation and the integrator. F, K, and G are the eedback gains. r (ṙ = ) is the setpoint signal and d ist means a disturbance. Due to lack o space, we only discuss the nonlinear system with the relative degree ρ = n. The ollowing approach can be also applied to the nonlinear systems with ρ < n. We consider the linearized system using PB models. {ż =Az Bv (7) y =Cz z R n, A R nn, B R nm, v R m, C R ln and y R l. The control system is o a ollowing orm. {ż =Az Bv, (8) η =r y, the controller v = Fz Kη Gr is designed to make r y as t. We rewrite the system equations (8) as (ż ) ( )( ( ) A BF BK z BG = r (9) η C η) The gains o F and K are calculated such that the system (9) is stable. The gain G can be obtained such that ż( ) = and y( ) = r. Thereore the gain o G can be chosen G = (C(A BF) B). Finally, the two-degree reedom servo controller is designed as u =α(x)β(x)v = Lρ h(x) Fz Kη Gr L g L ρ h(x) V. MINIMAL-ORDER STATE OBSERVER BASED ON PB MODELS () We proposed a ull-order state observer o PB control system in []. In this paper, we propose an observer-based PB controller to estimate the minimal-order state z R n l by using the output y R l. Fig. 3 shows the minimal-order state observer or PB control system. The ollowing system () is a minimal-order state observer, known as Gopinath observer [3]. {ẇ = Âw ˆBv Hy, () ẑ =Ĉw ˆDy w R n l,  R n ln l, ˆB R n lm, H R n ll, Ĉ R nn l, ˆD R nl and ẑ R n l. The observer is designed by using the ollowing steps [3], ) Set a transormation T = [C T M T ] T R nn satisying dett, M R n ln is an arbitrary matrix. ) Ā and B are divided as ollows. ) ) (Ā Ā =T AT Ā = ( B, B = T B = Ā Ā B Ā R l, Ā R n ll, Ā R n ln l, B R l and B R n l. 3) Derive L R n ll so that  = Ā LĀ is Hurwitz. 4) Calculate the ollowing parameters by using L. ˆB = L B B, H = ÂLĀ LĀ, ( ) ( ) Ĉ =T, ˆD = T Il I. n l L The estimation ẑ o () is substituted into the servo controller (), then the observer-based PB controller is designed as u = Lρ h(x) Fẑ Kη Gr L g L ρ, h(x) ẇ =Âw ˆBv Hy, ẑ =Ĉw ˆDy. Note that we can design the state observer or all the PB control systems. Since the linearized systems o all the PB models are the same as the linear system (7) and the system (7) is observable. In addition, the design method is capable o designing the state observer and the servo controller o nonlinear systems separately. VI. ROBOT ARM MODEL We consider a simple one-link robot arm [4]. The rotary motion is controlled by an elastically coupled actuator. The system is represented as {ẋ =o (x)g o (x)u(x), () y =h o (x)
Proceedings o the International MultiConerence o Engineers and Computer Scientists 4 Vol I,, March - 4, 4, Hong Kong r - /S G K - E T C y Servo controller y F Linearized system based on PB models Minimal-order state observer H /S Fig. 3. Minimal-order state observer and servo control systems based on PB models The vectors x, o and g o are given by x = (x,x,x 3,x 4 ) T, x 3 x 4 o (x) = K J N x K J N x F J x 3 K N x K x mgd cosx F x 4 g o (x) = (,, /J, ) T,, x is the angle o the torsional spring on the gear box and x is the angle o the robot arm. x 3 and x 4 are the time derivatives o x and x respectively. J and represent inertia, F and F are viscous riction constraints, K represents the elastic coupling with the joint and N is the transmission gear ratio. m is the mass and d is the position o the center o gravity o the link. g is the acceleration due to gravity. We choose the angle o the arm x as the output, i.e., y =h o (x) = x. Now we divide the state-space o the robot arm model () as x { 5,5,5}, x { 5,5,5}, x 4 { 5,5,5}, x 3 { 4π, 39π/, 38π/,...,4π}, then the PB model is constructed as = = 3 = 4 = 3 ẋ =(x)gu, y = h(x) = x i 3=σ 3 ω i3 3 (x 3 ) (,,i 3, ), 4 i 4=σ 4 ω i4 4 (x 4 ) (,,,i 4 ), 3 i 3=σ 3 (x ) (x )ω i3 3 (x 3) 3 (i,i,i 3, ), 4 i 4=σ 4 (x ) (x )ω i4 4 (x 4) 4 (i,i,,i 4 ), g = (,,,/J ) T, (,,i 3, ) = (j,j,i 3,j 4 ), j = σ,σ,j = σ,σ,j 4 = σ 4,σ 4, (,,,i 4 ) = (j,j,j 3,i 4 ), j = σ,σ,j = σ,σ,j 3 = σ 3,σ 3, 3 (i,i,i 3, ) = 3 (i,i,i 3,j 4 ),j 4 = σ 4,σ 4, 4 (i,i,,i 4 ) = 4 (i,i,j 3,i 4 ),j 3 = σ 3,σ 3. Here (,,i 3, ) is independent o i, i and i 4. It is also the same or (,,,i 3 ), 3 (i,i,i 3, ) and 4 (i,i,,i 4 ). In this case, the system has many local PB models. Note that the linearized systems o the all PB models are the same as the linear system (7). We design the servo controller u u = L 4 h/l g L 3 hν/l g L 3 h, (3) L h = (x), L h = 4 (x), L g L 3 h = K J N, L 3 h = K N (x) (,σ,, ) (,σ,, ) (x) d (σ ) d (σ ) F 4 (x), L 4 h = K N 3(x) (,σ,, ) (,σ,, ) 4 (x) d (σ ) d (σ ) F ( K N (x) F 4 (x) ) (,σ,, ) (,σ,, ) (x), d (σ ) d (σ ) ν = Fz Kη Gr = (5.75,8.,8.57,.57)z.6η 5.75r. Note that the controller (3) based on PB model is simpler than the conventional one since the nonlinear terms o controller (3) are not the original nonlinear terms (e.g., sinx, cosx ) but the PB approximation models.
Proceedings o the International MultiConerence o Engineers and Computer Scientists 4 Vol I,, March - 4, 4, Hong Kong The initial condition is x() = (,,,) T and the setpoint signal is r = π/. The disturbance signal d ist =.5 is added to the state x ater 5 seconds. Fig. 4 shows the state response o x using the PB servo controller with the disturbance d ist. We construct a minimal-order state observer o this robot arm model. In this example we set a transormation T = diag(,,,) and eigenvalues ρ =, ρ =, ρ 3 = o Â. The parameters o the observer () are calculated as L = ( 753,.89 5,.58 7) T, Â = 753.89 5, ˆB =,.58 7 Ĉ =, ˆD = 753.89 5..58 7 The estimation ẑ o () is substituted into the servo controller (), then the observer-based PB controller is obtain as u = Lρ h L g L ρ h (5.75,8.,8.57,.57)ẑ.6η 5.75r L g L ρ, h ẇ =Âw ˆBv Hy, ẑ =Ĉw ˆDy. In Fig. 5, the upper graph shows the responses o the state x (t) without the disturbance and the lower one shows the response with the disturbance. The results conirm the easibility o the servo control and state observer. In addition, the results show that the controller has disturbance rejection eature. The perormance o the observer-based PB controller is equivalent to the PB controller without the state observer. x 3 Fig. 4. 3 5 5 5 3 35 4 State response x using the PB servo controller with the disturbance VII. CONCLUSIONS This paper has proposed a servo control system based on a minimal-order state observer or nonlinear systems approximated by piecewise bilinear (PB) models. The design method is capable o designing the state observer and the servo controller o nonlinear systems separately. The approximated system is ound to be ully parametric. The inputoutput (I/O) eedback linearization is applied to stabilize PB control systems. The PB models with eedback linearization are a very powerul tool or the analysis and synthesis o x x 3 5 5 5 3 35 4 3 5 5 5 3 35 4 time Fig. 5. State responses x using the observer-based PB servo controller without/with the disturbance nonlinear control systems. Although the controller is simpler than the conventional I/O eedback linearization controller, the control perormance based on PB model is the same as the conventional one. We have applied the control method to a robot arm system. An example has conirmed the easibility o our proposal. ACKNOWLEDGMENT The authors would like to thank Dr. Dimitar Filev and Dr. Yan Wang o Ford Motor Company or his valuable comments and discussions. REFERENCES [] E. D. Sontag, Nonlinear regulation: the piecewise linear approach, IEEE Trans. Autom. Control, vol. 6, pp. 346 357, 98. [] M. Johansson and A. Rantzer, Computation o piecewise quadratic lyapunov unctions o hybrid systems, IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 555 559, 998. [3] J. Imura and A. van der Schat, Characterization o well-posedness o piecewise-linear systems, IEEE Trans. Autom. Control, vol. 45, pp. 6 69,. [4] G. Feng, G. P. Lu, and S. S. Zhou, An approach to hininity controller synthesis o piecewise linear systems, Communications in Inormation and Systems, vol., no. 3, pp. 45 54,. [5] M. Sugeno, On stability o uzzy systems expressed by uzzy rules with singleton consequents, IEEE Trans. Fuzzy Syst., vol. 7, no., pp. 4, 999. [6] M. Sugeno and T. Taniguchi, On improvement o stability conditions or continuous mamdani-like uzzy systems, IEEE Tran. Systems, Man, and Cybernetics, Part B, vol. 34, no., pp. 3, 4. [7] T. Taniguchi and M. Sugeno, Stabilization o nonlinear systems based on piecewise lyapunov unctions, in FUZZ-IEEE 4, 4, pp. 67 6. [8], Piecewise bilinear system control based on ull-state eedback linearization, in SCIS & ISIS,, pp. 59 596. [9], Stabilization o nonlinear systems with piecewise bilinear models derived rom uzzy i-then rules with singletons, in FUZZ- IEEE,, pp. 96 93. [] T. Taniguchi, L. Eciolaza, and M. Sugeno, LUT controller design with piecewise bilinear systems using estimation o bounds or approximation errors, Journal o Advanced Computational Intelligence and Intelligent Inormatics, vol. 7, no. 6, 3. [], Look-up-table controller design or nonlinear servo systems with piecewise bilinear models, in FUZZ-IEEE 3, 3. [], Full-order state observer design or nonlinear systems based on piecewise bilinear models, in 4 nd International Conerence on System Modeling and Optimization, 4, (accepted). [3] B. Gopinath, On the control o linear multiple input-output systems, The Bell Journal, vol. 5, pp. 63 8, 97. [4] A. Isidori, Nonlinear Control Systems 3rd ed. Springer, 995.