DEPTH CONTROL OF THE INFANTE AUV USING GAIN-SCHEDULED REDUCED-ORDER OUTPUT FEEDBACK 1. C. Silvestre A. Pascoal

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1 DEPTH CONTROL OF THE INFANTE AUV USING GAIN-SCHEDULED REDUCED-ORDER OUTPUT FEEDBACK 1 C. Silvestre A. Pascoal Institto Sperior Técnico Institte for Systems and Robotics Torre Norte - Piso 8, Av. Rovisco Pais Lisboa, Portgal {cjs,antonio}@isr.ist.tl.pt Abstract. The paper addresses the problem of atonomos nderwater vehicle (AUV) control in the absence of fll state information. An application is made to the control of a prototype AUV in the vertical plane. The methodology adopted for controller design is nonlinear gain schedling control, whereby a set of linear, dynamic redced order otpt feedback controllers are designed and schedled on the vehicle s forward speed. The paper smmarizes the basic controller design steps, describes a techniqe for practical implementation of the nonlinear control systems derived, and presents experimental reslts obtained with the INFANTE AUV dring tests at sea. Copyright c 25 IFAC Keywords. Marine Systems, Atonomos Underwater Vehicles, Gain-Schedled Control, Otpt Feedback. 1. INTRODUCTION This paper describes a soltion to the problem of atonomos nderwater vehicle (AUV) control in the vertical plane, in the absence of fll state information. An application is made to the control of the prototype IN- FANTE AUV, bilt and operated by the Institto Sperior Técnico of Lisbon, Portgal. The paper starts by introdcing a nonlinear dynamic model of the INFANTE AUV shown in Fig. 1. This is followed by control system design for precise manevering in the vertical plane. The techniqe elected for controller design is gain schedling (Rgh et al., 2). Using this approach, a set of linear controllers is first derived for a finite nmber of linearized models of the plant at selected operating points. The reslting con- 1 This work was spported by the Portgese FCT POSI programme nder framework QCA III, projects MAYA of the AdI and MAROV of the PDCTM, and by the EC nder the FREESUB Network. trollers are then interpolated on the vehicle s forward speed. For linear control systems design, the paper exploits the se of redced order feedback (ROF) techniqes, which lead natrally to dynamic otpt feedback control laws with a very simple strctre. In fact, the reslting controllers exhibit only the dynamics introdced by appended integrators (aimed at redcing steady state tracking errors to zero) as well as extra dynamics that act as shaping filters to limit the actation bandwidth. The importance of otpt feedback control strategies cannot be overemphasized: in practice, it is often impossible, difficlt, or too expensive to measre the fll state vector of a given plant. This motivates the development of controllers that rely on otpt variables only, effectively increasing the simplicity and ths the reliability of the control laws adopted. In the case of the INFANTE AUV, for example, it is difficlt to measre the angle of sideslip and the angle of attack in the horizontal and vertical planes, respectively. However, it is crcial to achieve stabilization and high vehicle perfor-

2 mance in both planes. Ths the se of otpt feedback control to meet tight stability and performance criteria. From a theoretical point of view, the redced order otpt feedback control can be converted into a static otpt feedback problem for a related agmented system (Mäkilä, 1985). However, in spite of the availability of necessary and sfficient conditions for plant stabilizability by static otpt feedback, no algorithm is crrently available which garantees to compte a stabilizing gain or determine if sch a gain exists (Iawasaki et al., 1994). Mch of the work in this area is well rooted in the theory of Linear Matrix Ineqalities (LMIs), which are steadily becoming the tool par excellence for advanced control system design. In fact, many control problems can be cast as LMI problems that can be solved efficiently sing convex programming techniqes. However, difficlties arise when designing (sb-optimal) static otpt feedback controllers becase the latter problem can only be cast in terms of an eqivalent one that involves Bilinear Matrix Ineqalities (BMIs) (Grigoriadis et al., 1996). The reslting problem is no longer convex, and no efficient nmerical procedres exist for its soltion as in the case of LMIs. However, the bilinear characteristics of the problem can still be exploited to yield an iterative procedre whereby two sets of LMIs are solved seqentially. This is the approach prsed in this paper, where the reslts described in (El Ghaoi et al., 1997) are sed to develop a simple algorithm to iteratively search for the soltion to (sb-optimal) static otpt feedback (SOF) control problems. Indirectly, the algorithm yields also a comptational procedre to solve the ROF problem stdied in this paper by exploiting the relationship between ROF and SOF control design techniqes described in (Mäkilä, 1985). Althogh the algorithm performs a local search with no garantees of global convergence, it exhibits excellent performance in the crrent application. The interested reader is referred to (Leibfritz, 21) for a lcid discssion of different techniqes for comptational design of static otpt feedback controllers. In the work reported here, a finite nmber of ROF controllers were developed for linearized plant models obtained at different operating conditions determined by the vehicle s forward speed. The controller parameters were then interpolated and schedled on speed. The final implementation of the reslting non-linear gain schedled controller was done sing the D-methodology described in (Kaminer et al., 1995) that garantees a fndamental linearization property and avoids the need to feedforward the vales of the state variables and inpts at trimming. The paper is organized as follows. Section 2 introdces a nonlinear model for the vertical plane dynamics of the INFANTE AUV. Section 3 details the techniqes that were sed for depth control system design and implementation. Finally, Section 4 contains experimental reslts obtained dring sea trials of the vehicle in the Azores, Portgal. 2. VEHICLE DYNAMICS This section describes the dynamic model of the IN- FANTE AUV in the vertical plane. See (Silvestre, 2) for a complete stdy of the AUV dynamics. The vehicle is 4.5 m long, 1.1 m wide and.6 m high. It is eqipped with two main thrsters (propellers and nozzles) for crising and flly moving srfaces (rdders, bow planes and stern planes) for vehicle steering and diving in the horizontal and vertical planes, respectively. Fig. 1. The INFANTE Vehicle The notation sed and the strctre of the vehicle model are standard (Silvestre, 2; Fossen, 1994). The variables and w denote srge and heave speeds, while θ, q, and z denote pitch, pitch rate, and depth, respectively. The symbols δ b, and δ s represent the bow and stern plane deflections, respectively. With this notation, and neglecting the roll stable motion, the dynamics of the AUV in the vertical plane can be written in compact form as m (ẇ q) = (W B) cos(θ) + ρ 2 L2 Z w w + (1) ρ 2 L3 Z q q + ρ 2 L2 2 [Z δb δ b + Z δs δ s ] + ρ 2 L3 Zẇẇ + ρ 2 L4 Z q q, ż = sin(θ) + w cos(θ), (2) I y q = z CB B sin(θ) + ρ 2 L3 M w w + ρ 2 L4 M q q + (3) ρ 2 L3 2 [M δb δ b + M δs δ s ] + ρ 2 L4 Mẇẇ + ρ 2 L5 M q q, θ = q, (4)

3 where eqations (1) and (3) describe the heave and pitch motion respectively Z (.) and M (.) are hydrodynamic derivative terms and z CB represents the metacentric distance. See (Silvestre, 2) for nmerical vales of the hydrodynamic parameters. The variables m, L, W, B, and I y are the vehicle s mass, length, weight, boyancy, and moment of inertia abot the y axis, respectively and ρ is the density of the water. 3. CONTROL SYSTEM DESIGN AND IMPLEMENTATION This section focses on the design of a depth control system for the AUV INFANTE, based on the dynamic model presented in Section 2. The methodology adopted for controller design is nonlinear gain-schedled control, whereby the design of a controller to achieve stabilization and adeqate performance of a given nonlinear plant (system to be controlled) involves the following steps (Rgh et al., 2): i) Linearizing the plant abot a finite nmber of representative operating points, ii) Designing linear controllers for the plant linearizations at each operating point, iii) Interpolating the parameters of the linear controllers of Step ii) to achieve adeqate performance of the linearized closed-loop systems at all points where the plant is expected to operate. The interpolation is performed according to an external schedling variable (vehicle s forward speed), and the reslting family of linear controllers is referred to as a gain schedled controller, iv) Implementing the gain schedled controller on the original nonlinear plant. In what follows a brief smmary is given of the work carried ot at each of the design steps, leading to the development of a controller for the vehicle that is schedled on forward speed. For the sake of brevity, the linear design methodology is illstrated for the case of a single operating condition that corresponds to a forward speed of 2 m/s. Linearization. Open-Loop System Analysis The model for the vertical plane was linearized abot the eqilibrim point determined by (w, q, z, θ ) T = (,,, ) T and = (δ b, δ s ) T =(, ) T. The reslting linearized model eigenvales are presented in Fig. 2. The model exhibits an eigenvale at zero (corresponding to a pre integrator in the depth coordinate z) and three stable eigenvales that link together the variables w, q, and θ. Notice the overall trend in the plot, where the two complex eigenvales at low speed Imaginary Axis m/s 1. m/s 1.5 m/s 2. m/s 2.5 m/s 2.8 m/s 3. m/s 3.5 m/s Real Axis Fig. 2. Linearized model eigenvales as fnctions of the forward speed degenerate into two real eigenvales at higher speed. In the inpt matrix, the bow and stern plane deflections δ b and δ s affect directly the state variables w and q. 3.1 Design Specifications The linear depth controllers were reqired to meet the following design specifications: Zero Steady State Error. Achieve zero steady state vales for the error variable in response to the inpt commands z cmd. Bandwidth Reqirements. The inpt-otpt command response bandwidth for the depth command channel shold be on the order of.1 rad/s; the control loop bandwidth for the bow and stern planes channels shold not exceed 5 rad/s; these figres were selected to ensre that the actators wold not be driven beyond their normal actation bandwidth. Closed Loop Damping and Stability Margins. The closed loop eigenvales shold have a damping ratio of a least.7. It was also reqired that the steady state deflection of the bow planes in response to a step inpt command in depth be δ b =. 3.2 Linear Control System Design The methodology selected for linear control system design was redced order otpt feedback with an H criterion (Grigoriadis et al., 1996). This method rests on a firm theoretical basis and leads natrally to an interpretation of control design specifications in the freqency domain. Frthermore, it provides clear gidelines for the design of controllers so as to achieve robst performance in the presence of plant ncertainty. The Redced Order Otpt Feedback (ROF) control problem can solved by converting it into a Static Otpt Feedback (SOF) control problem, sing a well-known system agmentation techniqe. To that effect, consider the original plant dynamics Σ m = {A m, B m, C m } and

4 the appended dynamics Σ k = {A k = k, B k = I k, C k = I k } of order k. Let k R k, and x k R k, be the control inpt and state of the appended dynamics. It can be shown (Mäkilä, 1985) that the ROF stabilization problem has a soltion of order k if and only if the agmented system [ ] [ ] [ ] Ak Bk Ck A =, B =, C = A m B m C m admits a static otpt-feedback stabilizing soltion. The remainder of this section focses on the SOF problem. w G K Fig. 3. Feedback interconnection z In what follows, the standard set-p and nomenclatre in (Zho et al, 1995) is adopted, leading to the feedback system represented in Fig. 3 with realization ẋ(t) = Ax(t) + B w w(t) + B(t) z(t) = C z x(t) + Dw(t) + E(t), (t) = Ky(t), (5) y(t) = Cx(t) + Fw(t) where x is the state vector. The symbol w denotes the inpt vector of exogenos signals (inclding commands and distrbances), z is the otpt vector of errors to be redced, y is the vector of measrements that are available for feedback, and is the vector of actator signals. The generalized plant G consists of the agmented system described before together with weights that shape the exogenos and internal signals, see Section 3.3. Sppose that the feedback system is well-posed, and let T zw denote the closed loop operator from w to z. The (sboptimal) H SOF synthesis problem consists of finding (if it exists) a static controller K that stabilizes the closed loop system and makes the infinity norm T zw of the operator T zw smaller than a desired bond γ >. The techniqe sed for controller design is based on two following standard reslts. Reslt 1: The closed loop system with realization (4) has all the eigenvales in the semi-plane λ C : Re(λ) < α if a real symmetric matrix X > and a real matrix K exist sch that the closed loop Lyapnov ineqality X(A αi) + XC K B + (A αi)x + BKCX < (6) is satisfied. Reslt 2: The H norm of the operator T zw is less than a positive nmber γ, that is, T zw < γ, if a real y symmetric matrix X > and a real matrix K exist sch that the LMI. [ X(A + C K B ) + (A + BKC)X ] B w γi < (7) C zx + EKC D γi holds. In the case of a sqare fll rank matrix C the standard transformation W = KCX converts the above nonlinear LMIs into convex LMIs. However, in the case of a noninvertible C matrix, the problem of determining a sb-optimal SOF controller involves Bilinear Matrix Ineqalities (BLMIs). In this sitation, the problem at hand is no longer convex, ths making the task of finding nmerical soltions hard. It is important to point ot that given an arbitrary dynamic system, there are no garantees that a SOF controller exists that will stabilize the system. Frthermore, even if the existence of a stabilizing controller can be established, the nonconvex characteristics of the optimization problem are sch that no assrances can be given as to whether a nmerical procedre will converge to a soltion. Therefore, the following algorithms for the comptation of a sboptimal H SOF controller shold only be adopted if sond jdging is applied to establish if a soltion to the (sb-optimal) H SOF synthesis problem can indeed be fond. The algorithms proposed can be briefly explained as follows: i) compte (if it exists) a SOF stabilizing controller for the generalized plant G, and ii) se this controller as a starting point to find an H sb-optimal controller. The first algorithm finds a stabilizing controller sing Reslt 1, starting with an arbitrary K of appropriate dimensions. Algorithm 1: SOF stabilizing controller (1) For an arbitrary K find α sch that Re(λ i (A αi)) <, i = 1,..., n. (2) Fix K and solve LMI (6) (feasibility problem) with respect to variable X. (3) Fix X and solve the optimization problem of minimizing α sbject to the LMI constraint (6) in the variables α, and K. (4) If α go to step 1, else end. The second algorithm comptes a SOF H sb-optimal controller sing Reslt 2, adopting the stabilizing static controller K obtained before as a starting point. Algorithm 2: SOF H sb-optimal controller (1) Fix K and solve the optimization problem of minimizing γ sbject to the LMI constraint (7) in the variables γ and X. Set γ 1 = γ fond.

5 (2) Fix X and solve the optimization problem of minimizing γ sbject to the LMI constraint (7) in the variables γ and K. Set γ 2 = γ fond. (3) If γ 1 γ 2 > ζ go to step 1, else end. In this design exercise ζ was set to.1. Finally, the ROF controller is compted from the agmented system and the gain K. 3.3 Synthesis Model and Controller Design The first step in the controller design procedre is the development of a synthesis model that can serve as an interface between the designer and the H controller synthesis algorithm. Consider the feedback system shown w w 1 w 2 W 4 Fig. 4. Synthesis model P K + e x 1 x 2 S W 1 W 2 W 3 I s in Fig. 4, where P is the agmented linearized model of the AUV in the vertical plane, and K is a SOF controller to be designed. The block G within the dashed line is the synthesis model, which is derived from the linear agmented model of the plant by appending the depicted weights. In practice, the weights serve as tning knobs which the designer can adjst to meet the desired performance specifications. G z y In the figre, w 1 represents the depth command z cmd that mst be tracked. The vector w 2 incldes the inpt noise to each of the sensors that provide measrements of depth, pitch, and pitch rate as well as distrbance inpts to the states w and q of the plant. The signal represents the agmented system control inpts that consist of k and the bow and stern plane deflections δ b, and δ s, respectively, whereas e = w 1 x 1 is the respective depth tracking error. The signal x 2 contains the remaining state variables that mst be penalized in the design process, that is, w, q, and θ. The matrices W i ; i = 1,..., 4 correspond to dynamic weights that penalize inpt, state, and tracking variables. Finally, the signal y consists of the variables x k, q, θ, z, e/s and δ b /s that are available for feedback. To meet the depth step command response reqirement the weighting fnction W 1 was chosen as W 1 =.1. The bow and stern planes deflection bandwidth constraint was achieved with W 2 = diag(.5,.5, 2(s/6 + 1)/(s/3+1), 2(s/6+1)/(s/3+1))]. The weight W 3 was set to diag(22.,.1,.1,.5,.5) to meet the command bandwidth reqirements, and W 4 =.1I 4. Notice the existence of a block of integrators I/s that operates on the tracking errors e and on the entries of the control inpt vector that are selected by the matrix S. Integral action on the errors is reqired to ensre zero steady state in response to step commands in w 1. Integral action on the entries of introdces a washot on the particlar control inpts selected. In the present case the washot ensres zero bow plane deflection at trimming conditions. After several iterations the controller order was set to k = 2 to accommodate the reqired actators bandwidth constraints. 3.4 Non-linear Controller Implementation A set of controllers was designed for a finite nmber of operating points, and their parameters interpolated according to the vehicle s forward speed (schedling variable). The implementation of the reslting non-linear gain schedled controller was done sing the D-methodology described in (Kaminer et al., 1995). This leads to the general strctre for the implementation of discretetime gain schedled controllers depicted in Fig. 5, where F () denotes the block that interpolates the redced order otpt feedback controllers obtained from the discretization of the linear controller designs in Section 3.2. In the present case a sampling freqency of 1 Hz was selected. e x z 1 z F () + z z 1 + Fig. 5. D Controller implementation with anti-windp mechanism 4. TESTS AT SEA To assess the performance of the controller developed, a series of tests were carried ot at sea. The vehicle was operated at constant heading nder the inflence of strong wave action. Figs. 6 throgh 1 show some of the practical reslts obtained dring depth changing manevers, together with the reslts of simlations obtained with a fll nonlinear model of the vehicle. At the beginning of this manever INFANTE was at srface;

6 2 seconds into the manever the depth controller was switched on and a command to dive to 8 meters depth was applied; this was followed by a command to dive to 1 meters at t = 15 seconds. In the figres, the dashed and solid lines represent the experimental and the simlation reslts, respectively. The vehicle s forward speed was kept approximately constant at 1.8 m/s. Depth [m] Fig. 6. Commanded and measred depth - simlated and measred vales Pitch angle [deg] Fig. 7. Pitch angle Pitch rate [deg/sec] Fig. 8. Pitch rate Bow plane [deg] Fig. 9. Bow plane deflection nder strong wave action Figs. 6, 7 and 8 show commanded and measred depth, pitch, and pitch rate activity, respectively. Figs. 9 and 1 Stern plane [deg] Fig. 1. Stern plane deflection nder strong wave action display the activity of the bow and stern planes respectively. Notice the strong copling between wave action and control planes deflection near the srface, mainly indced by pitch and pitch rate. Leaving aside the inflence of the waves (which was not addressed explicitly in the controller design phase), the figres reveal close agreement between predicted and actal manevers. 5. ACKNOWLEDGEMENTS The athors wold like to thank to João Alves, Lis Sebastião, Manel Rfino, and Rdolfo Oliveira from ISR/IST for their valable contribtion to the development and testing of the INFANTE vehicle. 6. REFERENCES El Ghaoi, L., Ostry, F., AitRami, M. (1997). A Cone Complementary Linearization Algorithm for Static Otpt- Feedback and Related Problems, IEEE Trans. Atomatic Control, Vol. 42, No. 8, pp Fossen, T. (1994). Gidance and Control of Ocean Vehicles, John Wiley and Sons. Grigoriadis, K. M. and Skelton, R. E. (1996). Low Order Control Design for LMI problems Using Alternating Projection Methods, Atomatica, Vol. 32, pp Iwasaki, T., Skelton, R., Geromel, J. (1994). Linear Qadratic Sboptimal Control with Static Otpt Feedback, Systems and Control Letters, Vol. 32, No. 6, pp Kaminer, I., Pascoal, A., Khargonekar, P. and Coleman, E. (1995). A Velocity Algorithm for the Implementation of Gain-Schedled Controllers, Atomatica, Vol. 31, pp Leibfritz, F. (21). An LMI-based Algorithm for designing Sboptimal Static Otpt Feedback Controllers, SIAM Jornal on Control and Optimization, Vol. 39, No. 6, pp Mäkilä, P. M.(1985). Parametric LQ control, International Jornal of Control, Vol. 41, No. 6, pp Rgh, W. and Shamma, Jeff S. (2). A Srvey of Research on Gain-Schedling, Atomatica, pp , September. Silvestre, C. (2). Mlti-Objective Optimization Theory with Applications to the Integrated Design of Controllers / Plants for Atonomos Vehicles, PhD thesis, Department of Electrical Engineering and Compter Science, Institto Sperior Técnico, Lisbon, Portgal. Zho, Kemin, Doyle, John C., and Glover, Keith (1995). Robst and Optimal Control, Prentice Hall, New Jersey, USA.