Adaptive control with convex saturation constraints JinYan, Davi Antônio dos Santos, Dennis S. Bernstein

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1 Published in IET Control Theory Applications Received on 4th June 13 Revised on 19th January 14 Accepted on 4th February 14 doi: 1149/iet-cta1358 ISSN Adaptive control with convex saturation constraints JinYan, Davi Antônio dos Santos, Dennis S Bernstein Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI, USA yanjin@umichedu Abstract: This study applies retrospective cost adaptive control to comm following in the presence of multi-variable convex input saturation constraints To account for the saturation constraint, the authors use convex optimisation to minimise the quadratic retrospective cost function The use of convex optimisation bounds the magnitude of the retrospectively optimised input thereby influences the controller update to satisfy the control bounds This technique is applied to a multi-rotor helicopter with constraints on the total thrust magnitude inclination of the rotor plane 1 Introduction All real-world control systems must operate subject to constraints on the allowable control inputs These constraints typically have the form of a saturation input non-linearity [1] The effects of saturation are addressed through antiwindup strategies [ 7] Within the context of modern multi-variable control, techniques for dealing with saturation are presented in [8 1] Saturation within the context of adaptive control is addressed in [13 16] In the case of multiple control inputs, it is usually the case that individual control inputs are subject to independent saturation [17] However, in many applications, a saturation constraint may constrain multiple control inputs This is the case, for example, if the control inputs are produced by common hardware, such as a single power supply, amplifier or actuator In the present paper, we consider an adaptive control for problems, in which multiple control inputs may be subject to dependent saturation constraints In particular, we are motivated by the problem of safely controlling the trajectory of a multi-rotor helicopter by constraining the total thrust magnitude inclination in order to restrict the vehicle acceleration To address this problem, we revisit the problem of retrospective cost adaptive control (RCAC) under constraints [15] RCAC can be used for adaptive comm following disturbance rejection for possibly non-minimumphase systems under minimal modelling information [18 1] Unlike [17], the present paper uses convex optimisation to perform the retrospective input optimisation [] The use of convex optimisation bounds the magnitude of the retrospectively optimised input thereby influences the controller update to satisfy the control bounds We demonstrate this technique on illustrative numerical examples involving single multiple inputs We then apply this approach to trajectory control for a multi-rotor helicopter We use the convex programming code [3] for the numerical optimisation A related technique was used within the context of RCAC in [4] to address the problem of unknown non-minimum-phase zeros The contents of the paper are as follows In Section, we describe the comm-following problem with input saturation non-linearities In Section 3, we summarise the RCAC algorithm Numerical simulation results are presented in Section 4, conclusions are given in Section 5 Problem formulation Consider the multiple-input multiple-output (MIMO) discrete-time Hammerstein system x(k + 1) = Ax(k) + BSat(u(k)) + D 1 w(k) (1) where, for all k, x(k) R n, y(k) R ly, z(k) R lz, w(k) R lw u(k) R lu The signal u(k) is the commed control input, w(k) is exogenous signal However, because of saturation, the actual control input is given by v(k) = sat(u(k)), where the saturation input non-linearity is sat : R lu U, U R lu is the convex control constraint set We assume that the function Sat is onto, that is, sat(r lu ) = U In particular, if U is rectangular, then sat a1,b1 (u 1 ) Sat(u) = () sat alu,b lu (u lu ) where u =[u 1 u lu ] T U =[a 1, b 1 ] [a lu, b lu ] sat : R [a, b] is defined as a, if u < a sat a,b (u) = u, if a u b (3) b, if u > b We consider the Hammerstein comm-following disturbance rejection problem with the performance variable z(k) = E 1 x(k) + E w(k) (4) 196 IET Control Theory Appl, 14, Vol 8, Iss 1, pp The Institution of Engineering Technology 14 doi: 1149/iet-cta1358

2 we assume that measurements of z(k) are available for feedback; however, measurements of sat(u(k)) are not available The goal is to develop an adaptive output feedback controller that minimises the performance error z(k) with minimal modelling information about the plant dynamics, exogenous signal w input saturation non-linearity sat Note that w can represent either a comm signal to be followed, an external disturbance to be rejected or both For example, if D 1 = E =, then the objective is to have the output E 1 x follow the comm signal E won the other h, if D 1 = E =, then the objective is to reject the disturbance w from the performance variable E 1 x The combined comm-following disturbancerejection problem is considered when D 1 =[D 11 ], E =[ E ] w(k) =[w T 1 (k) wt (k)]t, where the objective is to have E 1 x follow E w while rejecting the disturbance w 1 Finally, if D 1 E are zero matrices, then the objective is output stabilisation, that is, convergence of z to zero 3 Retrospective cost adaptive control In this section, we describe the constrained retrospective cost optimisation algorithm 31 ARMAX modelling Consider the autoregressive-moving-average model with exogenous inputs (ARMAX) representation of (1) (4) given by z(k) = n α i z(k i) + i=1 + n β i sat(u(k i)) i=d n γ i w(k i) (5) i= where α 1,, α n R, β 1,, β n R lz lu, γ,, γ n R lz lw d is the relative degree Next, let v(k) Sat(u(k)), define the transfer function G zv (q) E 1 (qi A) 1 B = i=d q i H i = H d α(q) β(q) where q is forward shift operator, for each positive integer i, the Markov parameter H i of G zv is defined by (6) H i E 1 A i 1 B R lz lu (7) Note that, if d = 1, then H 1 = β 1, whereas, if d, then β 1 = =β d 1 = H 1 = =H d 1 = (8) H d = β d The polynomials α(q) β(q) have the form α(q) = q n 1 + α 1 q n 1 + +α n 1 q + α n (9) β(q) = q n d + β d+1 q n d 1 + +β n 1 q + β n (1) Next, define the extended performance Z(k) R plz extended plant input V (k) R qclu by z(k) v(k 1) Z(k), V (k) z(k p + 1) v(k q c ) sat(u(k 1)) = (11) sat(u(k q c )) where the data window size p is a positive integer, q c n + p 1 Therefore (11) can be expressed as where (see (13)) Z(k) = W zw φ zw (k) + B f V (k) (1) β 1 β n lz lu lz lu B f lz lu Rplz qclu lz l u lz lu lz lu β 1 β n (14) φ zw (k) z(k 1) z(k p n + 1) w(k) w(k p n + 1) R qclz+(qc+1)lw (15) Note that W zw includes modelling information about the poles of G zv the exogenous signals, whereas B f includes modelling information about the zeros of G zv For the open-loop system (5), we make the following assumptions: (A1) The relative degree d is known (A) The first non-zero Markov parameter H d is known (A3) There exists an integer n such that n < n n is known (A4) If ζ C, ζ > 1, β(ζ) =, then the spectral radius of A is less than 1 (A5) The performance variable z(k) is measured available for feedback (A6) Each component of the function Sat is monotonically non-decreasing in each component in u with the remaining components of u fixed α 1 I lz α n I lz lz lz lz lz γ γ n lz lw lz lw W zw lz lz lz l w lz l z lz l w lz lz lz lz α 1 I lz α n I lz lz lw lz lw γ γ n Rplz [qclz+(qc+1)lw] (13) IET Control Theory Appl, 14, Vol 8, Iss 1, pp doi: 1149/iet-cta1358 The Institution of Engineering Technology 14

3 (A7) The exogenous signal w(k) is generated by x w (k + 1) = A w x w (k) (16) w(k) = C w x w (k) (17) where x w R lw all of the eigenvalues of A w are on the unit circle do not coincide with the transmission zeros of G zv (A8) There exists an integer n w such that n w < n w n w is known (A9) The exogenous signal w(k) is not measured (A1) α(q), β(q), n x() are unknown The Assumption 31 is motivated by Yan Bernstein [5], where it is shown that monotonicity of the input non-linearity preserves the signs of the Markov parameters of the linearised system 3 Controller construction The commed control u(k) is given by the exactly proper time-series controller n c n c u(k) = M i (k)u(k i) + N j (k)z(k j) (18) i=1 where, for all i = 1,, n c, M i (k) R lu lu,, for all j =,, n c, N j (k) R lu lz We express (18) as where j= u(k) = θ(k)φ(k 1) (19) θ(k) [ M 1 (k) M nc (k) N (k) N nc (k) ] R lu (nclu+(nc+1)lz) () u(k 1) u(k n φ(k 1) c ) z(k) R nclu+(nc+1)lz (1) z(k n c ) 33 Retrospective performance Define the retrospective performance Ẑ(k) R plz Ẑ(k) W zw φ zw (k) + B f V (k) + B f [Û(k) U(k)] () where (see (3)) is the retrospective input matrix with the model information of G zv Specifically, H 1,, H m in (3) are estimates of the Markov parameters of G zv, where m Z + Next, define by the extended commed control U(k) R qclu the retrospectively optimised extended control vector Û(k) R qclu by u(k 1) U(k) u(k q c ) û k (k 1) Û(k) (4) û k (k q c ) where û k (k i) R lu is a recomputed control Subtracting (1) from () yields Ẑ(k) = Z(k) + B f [Û(k) U(k)] (5) Note that the retrospective performance Ẑ(k) does not depend on W zw or the exogenous signal w For disturbance rejection, we do not assume that the disturbance is known; for comm-following, the comm-following error is needed but the comm w need not be separately measured The model information matrix B f is discussed in Section Retrospective cost RLS controller update law 341 Retrospective cost: We define the retrospective cost function J (Û(k), k) Ẑ T (k)r(k)ẑ(k) + η(k)û(k) T Û(k) (6) where, for all k >, η(k) is a scalar R(k) R plz plz is a positive-definite performance weighting The goal is to determine retrospectively optimised controls Û(k) that would have provided better performance than the controls U(k) that were applied to the plant The retrospectively optimised controls Û(k) are subsequently used to update the controller Using (5), (6) can be rewritten as J (Û(k), k) = Û(k) T A(k)Û(k) + B(k)Û(k) + C(k) (7) where A(k) B T f R(k) B f + η(k)i qcl u B(k) B T f R(k)[Z(k) B f U(k)] C(k) Z T (k)r(k)z(k) Z T (k)r(k) B f U(k) + U(k) T B T f R(k) B f U(k) Note that if either B f has full rank or η(k) >, then A(k) is positive definite Next, we consider the problem of minimising (6) subject to Û(k) U U (8) The following result follows from the Weierstrass theorem Lemma 31: If U is compact, then (6) has at least one minimiser If, in addition, U is convex, then (6) has a unique lz (d 1)lu H d H m lz lu lz lu lz lu lz lu B f lz (d 1)lu lz lu Rplz qclu (3) lz (d 1)lu lz lu lz lu H d H m lz lu lz lu 198 IET Control Theory Appl, 14, Vol 8, Iss 1, pp The Institution of Engineering Technology 14 doi: 1149/iet-cta1358

4 minimiser In particular, if U = R lu, then the unique global minimiser of J (Û(k), k) is Û(k) = 1 A 1 (k)b(k) (9) 34 Cumulative cost RLS update: Define the cumulative cost function J cum (θ, k) k λ k i φ T (i d 1)θ(i 1) û k (i d) i=d+1 + λ k [θ(k) θ()] T P 1 [θ(k) θ()] (3) where is the Euclidean norm, P R lu[nclu+(nc+1)lz] [nclu+(nc+1)lz] is positive definite, λ (, 1] is the forgetting factor The next result follows from stard recursive least-squares (RLS) theory [6, 7] Lemma 3: For each k d, the unique global minimiser of the cumulative retrospective cost function (3) is given by θ(k) = θ(k 1) + where P(k) = 1 λ P(k 1)φ(k d)ε(k 1) λ + φ T (k d)p(k 1)φ(k d) (31) [ P(k 1) P(k 1)φ(k ] d)φt (k d)p(k 1) λ + φ T (k d)p(k 1)φ(k d) (3) P() = P, ε(k 1) φ T (k d 1)θ(k 1) û(k d) 35 Model information B f For soft-input soft-output, minimum-phase, asymptotically stable linear plants, using the first non-zero Markov parameter in B f yields asymptotic convergence of z to zero [19, 8] In this case, let m = d H d = H d in (3) Furthermore, if the open-loop linear plant is non-minimum-phase the absolute values of all non-minimum-phase zeros are greater than the plant s spectral radius, then a sufficient number of Markov parameters can be used to approximate the nonminimum-phase zeros [19] Alternatively, a phase-matching condition with η>is given in [9, 3] to construct B f For MIMO Lyapunov-stable linear plants, an extension of the phase-matching-based method is given in [31] For unstable, non-minimum-phase plants, knowledge of the locations of the non-minimum-phase zeros is needed to construct B f For details, see [19, 3] In this paper, we consider only the case where the zeros of G zv are either minimum-phase or on the [ unit circle Therefore] we set p = 1 let B f = 1z (d 1)l u H d 1z (n d)lu R l z nl u 4 Numerical examples In this section, we present numerical examples to illustrate the response of RCAC for plants with input saturation based on constrained retrospective optimisation The numerical examples are constructed such that the objective is to minimise the performance z = y w, with φ(k) given by Fig 1 Example 41 Mass-spring structure with single-direction force actuation (1) In all simulations, we set λ = 1 initialise θ() to zero Example 41: Comm following for an undamped massspring structure with single-direction force actuation Consider the mass-spring-damper structure shown in Fig 1 modelled by m q + kq = v (33) where m = 1 kg k = 3 N/m are the mass stiffness, q q are the position velocity, respectively, of the mass The saturated control v is given by v = sat(u) = sat,5 (u) (34) The discrete-time transfer function with sampling time T s = 1 s is given by 4(z + 1) G zv (z) = (35) z 177z + 1 The goal is to bring the mass to rest at q = We consider (3) with v = sat,5 (u) Note that this problem is related to the classical problem of controllability using positive controls considered in [33 35] However, 33 s theorem given in [33, 34] assumes that B = I, which is not the case in this example The adaptive controller (18) with known saturation bounds is implemented in feedback with n c = 5, η = 1, P = 1I B f =[4 ] The goal is to bring the mass to q = with singledirection force actuation Fig shows the response with q() = 3 m q() = 5 m/s Note that, by constraining the retrospectively optimised control û(k), the commed control u(k) is non-negative for all k > 5 Example 4: Position comm following for a multi-rotor helicopter Consider the multi-rotor helicopter illustrated in Fig 6 The body frame S B ={X B, Y B, Z B } is attached to the vehicle at its centre of mass (CM) with the Z B axis normal to the rotor plane The reference frame S R ={X R, Y R, Z R } is fixed on the ground at point O with the Z R axis aligned with the local vertical The vehicle has six degrees of freedom, three of which are of rotation the other three are of translational motion The present example is concerned only with the translational motion, which can be described in S R by [ ] q = 1 m u + (36) g where q =[q 1 q q 3 ] T R 3 describes the position of CM, with q 1 q denoting horizontal position q 3 representing the altitude, u =[u 1 u u 3 ] T R 3 is the IET Control Theory Appl, 14, Vol 8, Iss 1, pp doi: 1149/iet-cta1358 The Institution of Engineering Technology 14

5 4 z(k) Controller θ(k) Time Step k Time step k Commed control u(k) Saturated control v(k) Time step k Fig Example 41 Comm-following for an undamped mass-spring-damper structure with single-direction force actuation The adaptive controller (18) is implemented with n c = 5, η = 1, P = 1I B f = [4 ] The goal is to bring the mass to q = with single-direction force actuation with q() = 3 m q() = 5 m/s Fig 3 Example 4 The multi-rotor helicopter the body reference frame total thrust vector, m = 5z kg is the mass of the vehicle, g = 98 m/s is the gravitational acceleration (Fig 3) Consider the initial conditions q() =[ ] T q() =[ ] T Define the inclination angle ϕ of the rotor plane to be ϕ cos 1 u 3 (37) u where u denotes the Euclidean norm of u Fig 6 shows that ϕ is the angle between the thrust vector u the Z R axis (local vertical) In order to implement the so-obtained controller in practice, the vehicle system is required to feature two low-level control loops: a total thrust controller an attitude controller In this case, the thrust magnitude u serves as a comm to the total thrust controller, whereas the thrust Fig 4 Example 4 The convex control constraint set U formed by (4) (4) direction u/ u is used to generate an attitude comm to the attitude controller In this example, both low-level controllers are assumed to be accurate significantly faster than the translational dynamics, in such a way that their comm u can be considered equal to its corresponding actual value Define the tracking error z R 3 by z q w (38) 11 IET Control Theory Appl, 14, Vol 8, Iss 1, pp The Institution of Engineering Technology 14 doi: 1149/iet-cta1358

6 Comm signal w 1 5 Output signal y Comm signal w 3 5 Output signal y Comm signal w 3 Output signal y Commed control u 1 Saturated control v Commed control u Saturated control v Commed control u Saturated control v θ(k) Time (sec) Fig 5 Example 4 Comm following for the multi-rotor helicopter The adaptive controller (18) with the saturation bounds in (44) is implemented in feedback with n c = 8, η =, P = 1I, B f = [1I ] θ() = Note that y 1,y,y 3 follow the comms w 1,w w 3 after the transient where w =[w 1 w w 3 ] R 3 denotes a position comm Consider w(t) = [ ] cos(1t) sin(1t) 3t + 1 (39) Let ϕ max 9 u max > denote the maximum allowable values of ϕ u, respectively Consider ϕ max = u max = 6N The control problem is to construct a feedback law for u that minimises z subject to u 1 + u sin ϕ max (4) u u 3 (41) u u max (4) The inequalities (4) (4) form the conic convex control constraint set U illustrated in Fig 4 The problem of minimising the retrospective cost function on U can thus be rewritten as the following second-order cone programming (SOCP) problem subject to min J (Û(k), k) (43) PÛ(k) QÛ(k) Û(k) 6 (44) where [ ] 1 1 P (45) Q tan(ϕ max ) [ 1 ] T (46) The non-linear programming method SOCP available in the CVX toolbox [3] is used to solve the above optimisation problem IET Control Theory Appl, 14, Vol 8, Iss 1, pp doi: 1149/iet-cta1358 The Institution of Engineering Technology 14

7 Fig 6 Example 4 The adaptive controller (18) with known saturation bounds in (44) is implemented in feedback with n c = 8, η =, P = 1I, B f =[1I ] θ() = The black dots represent the constraint set in (4) (4), the blue dots represent the unsaturated controller output u The blue crosses outside the boundary of the constraint (black dots region) are because of the transient behaviour of RLS update in (31) (3) Next, a state space representation of the multi-rotor helicopter is given by [ q q] = [ ][ ] [ ] 3 3 I q q 1 I m 3 3 y = [ I 3 3 ] [ ] q 3 3 q v + d 1 d g (47) (48) z = y w (49) where the horizonal wind velocity d 1 = d = 1 m/s v = Sat(u) U is the saturated control input given by v = [ v1 ] v = Sat(u) = v 3 [ ] Sat1 (u 1, u, u 3 ) Sat (u 1, u, u 3 ) Sat 3 (u 3 ) (5) where (see (51) (5)) Sat 3 (u 3 ) sat,6 (u 3 ) (53) ϑ atan(u, u 1 ) = arctan u u 1 + u + u 1 (54) Note that the function Sat in (5) satisfies 31 Next, we discretise (47) (49) using zero-order-hold with sampling time T s = 1 s The adaptive controller (18) with knowledge of the saturation (44) is implemented in feedback with n c = 8, η =, P = 1I, d = 1, H 1 = I 3 3 we let B f =[1I ] Fig 5 shows the time history of y 1, y y 3 of the helicopter The transient especially along y 3 direction is owing to the fact that (47) (49) is unstable, the discretised equivalent of (47) (49) has non-minimum-phase zeros at 1, the horizontal wind velocity d 1, d the gravitational acceleration g are unmodelled Furthermore, we initialise the adaptive controller at θ() = Note that the commed control signal u does not exhibit integrator windup { u1 if u Sat 1 (u 1, u, u 3 ) 1 Sat 3 (u 3 ) tan ϕ max cos ϑ Sat 3 (u 3 ) tan ϕ max cos ϑ otherwise { u if u Sat (u 1, u, u 3 ) sat 3 (u 3 ) tan ϕ max sin ϑ Sat 3 (u 3 ) tan ϕ max sin ϑ otherwise (51) (5) 11 IET Control Theory Appl, 14, Vol 8, Iss 1, pp The Institution of Engineering Technology 14 doi: 1149/iet-cta1358

8 Time (sec) Fig 7 Example 4 At each time step, the distance between the commed control u(k) (blue crosses that are outside the control constraint set U in Fig 6 the saturated control sat(u(k)) remains bounded as shown in Fig 6, where the black dots represent the control constraint set U, the blue crosses represent the commed control u Note that the blue crosses outside the control constraint set U (black dots region) are caused by the transient behaviour of RLS update in (31) (3) Fig 7 shows that at each time step, the distance between the unsaturated commed control u(k) (blue crosses that are outside the control constraint set U in Fig 6 the saturated control v(k) 5 Conclusions Adaptive control based on constrained retrospective cost optimisation was applied to comm following for Hammerstein systems with multi-variable convex input saturation We numerically demonstrated that convex optimisation applied to the retrospective cost can improve the tracking performance when following comms in the presence of saturation We also applied this technique to a multi-rotor helicopter comm-following problem by formulating the multi-input constrained retrospective cost function as a second-order cone optimisation problem With this approach, RCAC was shown to adapt to these constraints The numerical results motivate future research on the stability analysis of RCAC under input saturation 6 Acknowledgment We wish to thank Jesse Hoagg Alexey Morozov for helpful discussions 7 References 1 Bernstein, DS, Michel, AN: A chronological bibliography on saturating actuators, Int J Robust Nonlinear Control, 1995, 5, pp Zaccarian, L, Teel, AR: Modern anti-windup synthesis: control augmentation for actuator saturation (Princeton University Press, 11) 3 Hippe, P: Windup in control: its effects their prevention (Springer, 6) 4 Hanus, R, Kinnaert, M, Henrotte, JL: Conditioning technique, a general anti-windup bumpless transfer method, Automatica, 1987, 3, (6), pp Åström, KJ, Rundqwist, L: Integrator windup how to avoid it, Proc American Control Conf, Pittsburgh, PA, June 1989, pp Zheng, A, Kothare, MV, Morari, M: Anti-windup design for internal model control, Int J Control, 1993, 6, pp Tarbouriech, S, Turner, M: Anti-windup design: an overview of some recent advances open problems, IET Control Theory Appl, 9, 3, (1), pp Hu, T, Lin, Z: Control systems with actuator saturation: analysis design (Birkhäuser, 1) 9 Glattfelder, A, Schaufelberger, W: Control systems with input output constraints (Springer, 3) 1 Kapila, T, Grigoriadis, K: Actuator saturation control (CRC Press, ) 11 Tarbouriech, S, Garcia, G, dasilvajr, JMG, Queinnec, I: Stability stabilization of linear systems with saturating actuators (Springer, 11) 1 Zaccarian, L, Teel, AR: Modern anti-windup synthesis: control augmentation for actuator saturation (Princeton University Press, 11) 13 Annaswamy, AM, Wong, J-E: Adaptive control in the presence of saturation non-linearity, Int J Adaptive Control Sig Proc, 1998, 11, pp Teo, J, How, JP: Anti-windup compensation for nonlinear systems via gradient projection: Application to adaptive control, Proc IEEE Conf on Dec Control, Shanghai, China, December 9, pp Coffer, BJ, Hoagg, JB, Bernstein, DS: Cumulative retrospective cost adaptive control of systems with amplitude rate saturation, Proc American Control Conf, San Francisco, CA, June 11, pp Do, HM, Başar, T, Choi, JY: An anti-windup design for single input adaptive control systems in strict feedback form, Proc American Control Conf, Boston, MA, June 4, pp Tyan, F, Bernstein, DS: Global stabilization of systems containing a double integrator using a saturated linear controller, Int J Robust Nonlinear Control, 1999, 9, (15), pp Hoagg, JB, Santillo, MA, Bernstein, DS: Discrete-time adaptive comm following disturbance rejection for minimum phase systems with unknown exogenous dynamics, IEEE Trans Autom Control, 8, 53, pp Santillo, MA, Bernstein, DS: Adaptive control based on retrospective cost optimization, AIAA J Guid Control Dyn, 1, 33, pp Hoagg, JB, Bernstein, DS: Retrospective cost model reference adaptive control for nonminimum-phase systems, AIAA J Guid Control Dyn, 1, 35, pp Hoagg, JB, Bernstein, DS:, Retrospective cost adaptive control for nonminimum-phase discrete-time systems part 1: The ideal controller error system; 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