Multivariable Robust LTR-i Controller for a Ship

256 yayv and roll changes. Cargo, passenger and naval vessels usually employ steering and stabilisation systems in order to provide improved manoeuvering characteristics and motion control. Roll is certainly the most severe angular motion experienced by a ship. Large roll angles can make working on the ship difficult and can lead to motion sickness. The reasons for introducing active roll stabilisation systems in ships are basically: 1) security conditions, 2) transport costs reduction, 3) passenger comfort, 4) personnel efficiency; and additionally in naval vessels: 5) stable weapon platform maintenance and 6) stable platform for helicopter landing on the ship. If only the motions of roll. sway, yaw and surge are considered, the system is reduced to a problem of four degrees of freedom. The ship model described by Kallstrom and Ottosson (1982) has been used in the simulations carried out in this work. This model has demonstrated to be of great utility for evaluating the control algorithms by simulation, as a previous phase to sea trials (Kallstrom and Ottosson, 1982; Messer and Grimble, 1992). The ship model is a non-linear multivariable model, and the motion equations are (Kallstrom and Ottosson, 1982): [ all 0 0 0 0 a22 a23 a2-1 0 an a33 a3'! 0 a-u a'!3 aH 'Where "tot" indicates the total forces and torques acting on the ship, due to the following effects: hydrodynamics , wind, waves and current. The state variables are respectively: .El = ~~ (transversal speed), .E2 =~, .E3 = -0, .E4 = </J and .E;, = 1l :. Actuators dynamic are modelled as: b = (be -b)/TH , et = (Qc -Q)/Tl- , I et I~ etma% , I Q I~ Qmu The control magnitudes are aCt) and bet), the angles of the fins and rudder respectively, and the magnitudes to be controlled are </J(t) and 1l !( t), the angles of roll and heading. To design the controller, a linearized model has been chosen for nominal conditions of cruising speed F = lO . 8m/ s. Forces and torques expressions, hydrodynamic derivatives and coefficients are taken from Kallstrom and Ottosson (1982). Fig , L Feedback control configuration 3 CONTROL ALGORITHM Consider the control system of fig. 1. It consists of the plant (C), controller (K), pre-compensator P, reference signal (T), measurement noise (It), and disturbancies (d;, do). All signals are multivariable, and nominal mathematical models for C, 1(, P are LT!. The control objetives can be expressed at different levels of demanding: 1) Nominal stability (NS): bounded outputs for all bounded disturbancies, and bounded reference inputs. 2) Nominal performance (NP): small errors in the presence of disturbancies d;, do and reference inputs T. 3) Robust stability (RS): consider the feedback system in fig. 1. Suppose that the plant is not precisely known, and is modelled as belonging to a class of possible transfer matrices 9. A controller /\,- satisfies the robust stability condition if K stabilizes all C' E 9. 4) Robust performance (RP): this requirement is said to be met if the performance specifications are satisfied for all possible plants C' E 9. The LTR (Loop Transfer Recovery) design methodology seeks to define the MnvIO compensator K( s) so that the stability robustness and performance specifications are met to the possible greatest extent. This involves two basic steps: 1) vVe generate a MIMO target loop transfer function (TLTF) . 2) A special compensator K( s) is used, so that performance of the feedback system in fig. 1 approximates the performance of the TLTF established in step one. The degree of approximation (or recovery) depends on characteristics of the plant. If the plant is minimum phase, then the degree of recovery of the TLTF can be arbitrarily good (Stein and Athans, 1987). If the plant is nonminimum phase and the frequencies of the unstable zeros are beyond the bandwidth of the TLTF, the recovery will take place in low frequencies, and for all practical purposes the presence of far-away non minimum phase zeros does not degrade the low frequency characteristics of the design. Different approaches have been suggested in the control literature. obtain the TLTF . One of these is based on K alman filter techniques (which generates the LTR-o procedure (Athans , 1986). An- Fig. 2. TLTF synthesis other one is based on th e l·inear qu(tdrat-I.c regulator (LQa, or also known as LQSF: linear quadratic state feedback) theory, and it generates the LTR- i procedure (Zhang and Freudenberg, 1990; Maciejowski, 1989). In this work we have employed the latter one: LTR-i. Target Loop Transfer Function Synthesis Consider the plant model (which includes the scaling of the variables and augmentation dynamics that the designer has appended to meet specifications): .i;(t) Ax(t) + Bu(t) y = Cx(t) The transfer function matrix of the plant is: G(s) = Cif>(s)B, where if>(s) = (sI -A)-I, and we assume that [A, B] is stabilizable and that [A, C] is detectable. The structure of the TLTF is shown in fig. 2. It is simply defined by the parameters Band if> (s) of the plant model , and by a constant matrix Kc (optimal state feedback matrix). If we break the loop at the input of the plant we obtain the TLTF: For stability robustness to hold, in the face of multiplicative uncertainties at the input of the plant (G ' = (I + E)G, 0'( E) < e( u:) , the interconnection system (Morari and Zafiriou, 1989) is in this case M (s) = Tc (s)), the following inequality must be true for all J.-. (small gain theorem): or /1 . [T c (j:..:)] < 1j e (J.-·) in the case of structured uncertainties (diagonal structure); where 0' is the maximum singular value and /1 . represents the structured singular value [MoZa89]. Control demand , command-following and disturbance-rejection can be evaluated from fig. 2 for the matrix Kc obtained . Frequency-domain analysis is made and the temporal responses of the system are obtained by simulating the TLTF in fig. 2, in order to prove if design specifications are sa tisfied. To obtain matrix Kc we solve the LQR problem, which consists of meeting the control signal 'which will minimize the cost: with: Q = QT ~ O,R c = R~ > O, Qc = MTQM . The solution is u = -Kcx, and Kc is given by: where Pc = PI' ~ 0 satisfies the algebraic Riccati e quation: Some remarkable characteristics of the TLTF obtained in thi s way ar e: 1) optimal control law, 2) O'(T c) :S 2, 3) O'(Sc) :S 1, 4) at least 60° of phas e margin in each input channel, and infinite gain margin ; if the loop is conditionally stable it has a margin of at least 6dB against gain reductions (Stein and Athans, 1987; Maciejowski, 1989). LTR procedures Once the TLTF has been obtained, we can ask ourselves if ""ould be possible to construct a compensator K(s) in fig. 1 with the property that the feedback system of fig. 1 approximates the behaviour of the TLTF in fig. 2. This would happen if the following equality were true (where K(s)G(s) is the loop transfer function LTF): K(s)G(s) = Hc{s). However, for the purposes of design it is not necessary for us to have exact equality. Indeed, if we are interested in finding K(s) so that the approximate relation over the band of interest frequencies is satisfied. This is the point of view of the LTR-i method presented in this work. 'Ve now examine two procedur es to obtain the LTR controller K(s) , one observer based , and the other non observer based. The respective structures are shown in fig. 3 and fig. 4. As we c an see fig . 3 shows the conventional LQG observer based controllers structure (OBC), and fig. 4 illustrates the compensator structure developed by 257 258 Fig . .3 . LTR-i (OBe) structure Fig. 4. LTR-i (NOBC) structure Chen et al. (1991) (NOBC). The difference between them is that the NOBC removes the link from the control signal 11 to the observer via the control dis tribution matrix B, which is outside the realm of observer theory and hence the separation principle is no longer valid. In this case to guarantee the closed-loop stabilit y J(o must be such that A - J(0C' has all its eigenvalues in the left complex half plan. The respective controllers are: The procedure to obtain the matrix I{o is the same in both cases. One way is that proposed by Doyle and Stein (1981), and is based on the Kalman filter problem (KBF). For this the following algebraic Riccati equation is solved: where: and the Kalman filter gain matrix is obtained from: If we obtain J(o( q) by choosing the covariance matrix Qo as: it can be proved [DoSt81] for the minimum phase plant that lim J((s)G(s) = Hc(s) qoc · Therefore : LTF q~ , TLTF The NO BC characteristics for q ~ qo (the value of qo must be calculated in each case) are that (Chen et aI., 1991, 1992): 1) The compensator is openloop stable , 2) closed-loop stability is guaranteed and above all c) much smaller values of gain recovery gain q are required than the conventional OBC for the same degree of recovery. This fact implies that the compensator band-width is much smaller than that of the conventional controller and thus we have the advantage of avoiding, in some circumstancies, the saturation of the actuators as well as an improvement in the insensitivity to noise or other high-frequency disturbances. The approach followed in this work is based on the following points: 1) 'We are only interested in a partial recovery in the interest frequency range (low and medium frequencies). 2) At high frequencies the singular values of Hc(j:..:) roll-off at- 20 dB/dec , ,vhile those of J((j:..:)G(j:.,;) roll-off at- 40 dB/dec. Thus, LTR loops offer some additional robustness to high frequency unmodelled dynamics as compared to the TLTF. 3) The commandfollowing and disturbance rejection performance in the low frequency region between the TLTF and the LTF with LTR will be essentially the same. 4 Sn"IULATION STUDIES First we design a LTR-i controller to achieve adequate responses to changes in the reference signal. For this we use the linearized nominal model of the ship for Y = 7. 72 m/sand we employ the following design parameters: significant wave height of 4m with 40° relative to ship reference course is chosen in the simulations. \-Ve can see that there is a remarkable improvement in roll damping with the LTR-i MIMO controller. Figure 11 shows heading and roll for nonnominal speed conditions (9. 0m/s and 8m/s) ; we can see that the behaviour is adequate, which is another proof of the controller robustness. In order to improve performance chara cteristics a gain scheduling controller can be used ,vith the speed of the ship as the auxiliary variable. Due to plant and regulators dynamics, we can implement the controller directly in a digital computer with a sample time of 0.1 seconds, withou t explicitly taking into account the sample - data character of the system. All the algorithm implementations used in the simulations with the non-linear model of the ship are realized in this way. ,) CONCLUDING RElvlARKS Multivariable controllers based on LTR-i (Loop Transfer Recovery at the input of the plant) have been developed: a) for course changing, with considerable decrease in the coupling roll angle and b) for ship steering and roll regulation , with a considerable decrease in roll angle due to waves. The controller uses a non observer based control structure , and a partial recovery procedure over the band of interest frequencies. 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