ADAPTIVE CONTROL WITH CONVEX SATURATION CONSTRAINTS

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1 Proceedings of the ASME 3 Dynamic Systems and Control Conference DSCC3 October -3, 3, Palo Alto, California, USA DSCC ADAPTIVE CONTROL WITH CONVEX SATURATION CONSTRAINTS Jin Yan Dept of Aerospace Engineering University of Michigan Ann Arbor, MI yanjin@umichedu Davi Antônio dos Santos Instituto Tecnolõgico de Aeronáutica Divisão de Engenharia Mecânica-Aeronáutica 8-9 São José dos Campos, SP, Brazil davists@itabr Dennis S Bernstein Dept of Aerospace Engineering University of Michigan Ann Arbor, MI dsbaero@umichedu ABSTRACT This paper applies retrospective cost adaptive control (RCAC) to command following in the presence of multivariable convex input saturation constraints To account for the saturation constraint, we use convex optimization to minimize the quadratic retrospective cost function The use of convex optimization bounds the magnitude of the retrospectively optimized input and thereby influences the controller update to satisfy the control bounds This technique is applied to a tiltrotor with constraints on the total thrust magnitude and inclination of the rotor plane 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 nonlinearity [] The effects of saturation are traditionally addressed through antiwindup strategies [, 3] Within the context of modern multivariable control, techniques for dealing with saturation are presented in [4 7] Saturation within the context of adaptive control is addressed in [8 ] In the case of multiple control inputs, it is usually the case that individual control inputs are subject to independent saturation [] 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 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 heli- Address all correspondence to this author copter by constraining the total thrust magnitude and inclination in order to restrict the vehicle acceleration To address this problem, we revisit the problem of retrospective cost adaptive control (RCAC) under constraints [] RCAC can be used for adaptive command following and disturbance rejection for possibly nonminimum-phase systems under minimal modeling information [ 5] Unlike [], the present paper uses convex optimization to perform the retrospective input optimization [6] The use of convex optimization bounds the magnitude of the retrospectively optimized input and thereby influences the controller update to satisfy the control bounds We demonstrate this technique on illustrative numerical examples involving single and multiple inputs We then apply this approach to trajectory control for a multi-rotor helicopter We use the convex programming code [7] for the numerical optimization A related technique was used within the context of RCAC in [8] to address the problem of unknown nonminimum-phase zeros The contents of the paper are as follows In Section, we describe the command-following problem with input saturation nonlinearities In Section 3, we summarize the RCAC algorithm Numerical simulation results are presented in Section 4, and conclusions are given in Section 5 Problem Formulation Consider the MIMO discrete-time Hammerstein system x(k + ) = Ax(k) + BSat(u(k)) + D w(k), () y(k) = Cx(k) + D w(k), () z(k) = E x(k) + E w(k), (3) Copyright 3 by ASME

2 , for all k, x(k) R n, y(k) R l y, z(k) R l z, w(k) R l w, and u(k) R l u The signal u(k) is the commanded control input However, due to saturation, the actual control input is given by v(k) = Sat(u(k)), the saturation input nonlinearity is Sat : R l u U, and U R l u is the convex control constraint set We assume that the function Sat is onto, that is, Sat(R l u ) = U In particular, if U is rectangular, then Sat(u) = sat a,b (u ) sat alu,b lu (u lu ), (4) u = [u u lu ] T U = [a,b ] [a lu,b lu ] and sat : R [a,b] is defined as a, if u < a, sat a,b (u) = u, if a u b, b, if u > b The goal is to develop an adaptive output feedback controller that minimizes the command-following error z with minimal modeling information about the plant dynamics Note that w can represent either a command signal to be followed, an external disturbance to be rejected, or both For example, if D = and E, then the objective is to have the output E x follow the command signal E w On the other hand, if D and E =, then the objective is to reject the disturbance w from the performance variable E x The combined commandfollowing and disturbance-rejection problem is considered when D = [ D ], E = [ E ], and w(k) = [ w T (k) w T (k) ] T, the objective is to have E x follow E w while rejecting the disturbance w Finally, if D and E are zero matrices, then the objective is output stabilization, that is, convergence of z to zero 3 Retrospective Cost Adaptive Control In this section, we describe the constrained retrospective cost optimization algorithm 3 ARMAX Modeling Consider the ARMAX representation of () (3) given by z(k) = n i= α i z(k i) + n i=d β i Sat(u(k i)) + n i= (5) γ i w(k i), (6) α,,α n R, β,,β n R l z l u, γ,,γ n R l z l w, and d is the relative degree Next, let v(k) = Sat(u(k)), and define the transfer function G zv (q) = E (qi A) B = i=d q i H i = H d α(q) β(q), (7) q is the forward shift operator and, for each positive integer i, the Markov parameter H i of G zv is defined by H i = E A i B R l z l u (8) Note that, if d =, then H = β, as, if d, then β = = β d = H = = H d = (9) and H d = β d The polynomials α(q) and β(q) have the form α(q) = q n + α q n + + α n q + α n, () β(q) = q n d + β d+ q n d + + β n q + β n () Next, define the extended performance Z(k) R pl z and extended plant input V (k) R q cl u by z(k) Z(k) =, z(k p + ) v(k ) Sat(u(k )) V (k) = =, () v(k q c ) Sat(u(k q c )) the data window size p is a positive integer, and q c = n + p Therefore () can be expressed as W zw = Z(k) = W zw ϕ zw (k) + B f V (k), (3) α I lz αni lz lz lz lz lz γ γn lz lw lz lw lz lz lz lw lz lz lz lw lz lz lz lz α I lz αni lz lz lw lz lw γ γn R pl z [q c l z +(q c +)l w ], (4) Copyright 3 by ASME

3 and β β n lz l u lz l u B f = lz l u lz Rpl z q c l u, (5) l u lz l u lz l u β β n ϕ zw (k) = z(k ) z(k p n + ) w(k) w(k p n + ) R q cl z +(q c +)l w (6) Note that W zw includes modeling information about the poles of G zv and the exogenous signals, while B f includes modeling information about the zeros of G zv 3 Controller Construction The commanded control u(k) is given by the exactly proper time-series controller u(k) = n c i= M i (k)u(k i) + n c j= N j (k)z(k j), (7), for all i =,,n c, M i (k) R l u l u, and, for all j =,,n c, N j (k) R l u l z We express (7) as u(k) = θ(k)ϕ(k ), (8) θ(k) = [ M (k) M nc (k) N (k) N nc (k) ] R l u (n c l u +(n c +)l z ) (9) 33 Retrospective Performance Define the retrospective performance Ẑ(k) R pl z by B f = Ẑ(k) = W zw ϕ zw (k) + B f V (k) + B f [Û(k) U(k)], () lz (d )lu H d Hm lz lu lz lu lz lu lz lu lz (d )lu lz lu lz (d )lu lz lu lz lu H d Hm lz lu lz lu R pl z q c l u () is the retrospective input matrix with the model information of G zv Specifically, H,, H m in () are estimates of the Markov parameters of G zv, m Z + Next, define the extended commanded control U(k) R q cl u and the retrospectively optimized extended control vector Û(k) R q cl u by u(k ) U(k) =, Û(k) = u(k q c ) û k (k ) û k (k q c ), (3) û k (k i) R l u is a recomputed control Subtracting (3) from () yields Ẑ(k) = Z(k) + B f [Û(k) U(k)] (4) 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 commandfollowing, the command-following error is needed but the command w need not be separately measured The model information matrix B f is discussed in Section Retrospective Cost and RLS Controller Update Law 34 Retrospective Cost We define the retrospective cost function and u(k ) ϕ(k ) = u(k n c ) z(k) R n cl u +(n c +)l z () z(k n c ) J(Û(k),k) = Ẑ T (k)r(k)ẑ(k) + η(k)û(k) T Û(k), (5), for all k >, η(k) is a scalar and R(k) R pl z pl z is a positive-definite performance weighting The goal is to determine retrospectively optimized controls Û(k) that would have provided better performance than the controls U(k) that were applied to the plant The retrospectively optimized controls Û(k) are subsequently used to update the controller Using (4), (5) 3 Copyright 3 by ASME

4 can be rewritten as J(Û(k),k) = Û(k) T A(k)Û(k) +B(k)Û(k) +C(k), (6) A(k) = B T f R(k) B f + η(k)i qc l 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 to Next, we consider the problem of minimizing (5) subject 35 Model Information B f For SISO, minimum-phase, asymptotically stable linear plants, using the first nonzero Markov parameter in B f yields asymptotic convergence of z to zero [3, ] In this case, let m = d and H d = H d in () Furthermore, if the open-loop linear plant is nonminimum-phase and the absolute values of all nonminimum-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 [3] Alternatively, a phase-matching condition with η > is given in [, 3] to construct B f For MIMO Lyapunov-stable linear plants, an extension of the phase-matching-based method is given in [4] For unstable, nonminimum-phase plants, knowledge of the locations of the nonminimum-phase zeros is needed to construct B f For details, see [3, 5] In this paper, we consider only the case the zeros of G zv are either minimum-phase or on the unit circle Therefore, we set p = and let B f = [ z (d )l u H d z (n d)l u ] R l z nl u Û(k) U U (7) 34 Cumulative Cost and RLS Update Define the cumulative cost function J cum (θ,k) = k i=d+ λ k i ϕ T (i d )θ(i ) û k (i d) + λ k [θ(k) θ()] T P [θ(k) θ()], (8) is the Euclidean norm, P R l u[n c l u +(n c +)l z ] [n c l u +(n c +)l z ] is positive definite, and λ (,] is the forgetting factor The next result follows from standard recursive least-squares (RLS) theory [9, ] 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 optimization The numerical examples are constructed such that the objective is to minimize the performance z = y w, with ϕ(k) given by () In all simulations, we set λ = and initialize θ() to zero Example 4 Step command following for mass-springdamper structure with single-direction force actuation Consider the mass-spring-damper structure shown in Figure modeled by equation (3) Lemma 3 For each k d, the unique global minimizer of the cumulative retrospective cost function (8) is given by θ(k) = θ(k ) + P(k) = λ P(k )ϕ(k d)ε(k ) λ + ϕ T (k d)p(k )ϕ(k d), (9) [ P(k ) P(k )ϕ(k ] d)ϕt (k d)p(k ) λ + ϕ T, (k d)p(k )ϕ(k d) (3) P() = P, and ε(k ) = ϕ T (k d )θ(k ) û(k d) Figure Example 4 Mass-spring-damper structure with singledirection force actuation mẍ + cẋ + kx = v, (3) m, c, k are the mass, damping, and stiffness, respectively, 4 Copyright 3 by ASME

5 w is the command signal, and v is the force input given by v = sat(u) = { u if u, otherwise (3) Next, the state space representation of the mass-spring-damper structure can be written as [ ] [ ẋ = ẍ m k m c y = [ ][ x ẋ ][ ] [ ] x + ẋ v, (33) m ], (34) z = x w, (35) x and ẋ are the position and velocity, respectively, of the mass We discretize (33)-(35) using zero-order-hold and the plant transfer function from v to z, given by G zv (z) = 4(z + 85) z 38z + 6 (36) Our goal is to have the plant output y follow the step command w(k) We consider the saturation the input to the plant v = sat(u) is given by (5) Specifically, we let v min = and v max = when we compute the retrospectively optimized control û(k) in (7) The adaptive controller (7) with known saturation bounds is implemented in feedback with n c = 5, η =, P = I, and B f = 4 Note that, v min, v max, and B f are the only required model information for the adaptive controller Furthermore, we initialize the adaptive controller gain matrix θ() to zero Figure shows the time history of w, y, u, and v for the case v min = and v max = in (5) The adaptive controller is able to follow the step commands with single-direction force actuation In particular, by an appropriate modification of the retrospectively optimized control û(k), excessive adaptation of the unsaturated control signal u is prevented, that is, u(k) for all k This problem is closely related to the classical problem of controllability using positive controls considered in [6 8] However, Brammers theorem given in [6, 7] assumes that B = I, which is not the case in this example Example 4 Square wave command following for a minimum-phase, asymptotically stable plant with saturation Consider the asymptotically stable, minimum-phase plant trans- 3 Command signal w Output signal y Actual control u(k) Plant input v(k) z(k) Controller θ(k) Time Step k Figure Example 4 Step command following for mass-springdamper structure with single-direction force actuation We consider the saturation constraints the input to the plant v = sat(u) is given by (5) Specifically, we let v min = and v max = when we compute the retrospectively optimized control û(k) in (7) The adaptive controller (7) with known saturation bounds is implemented in feedback with n c = 5, η =, P = I, and B f = 4 Note that v min, v max, and B f are the only required model information for the adaptive controller Furthermore, we initialize the adaptive controller gain matrix θ() to zero The adaptive controller is able to follow the step commands with onedirection force actuator input In particular, by an appropriate modification of the retrospectively optimized control û(k), excessive adaptation of the unsaturated control signal u is prevented, that is, u(k) for all k fer function from v to z, given by G zv (z) = (z 5)(z 9) (z 7)(z 5 j5)(z 5 + j5) (37) Our goal is to have the plant output y follow the square-wave command w(k) = w s (k) We consider the unsaturated case, that is, v min = and v max =, together with four levels of saturation Let u ss,max = 3 and u ss,min = 3 be maximum and minimum steady-state command values that drive the performance z to zero in steady-state (ie z ss = ), respectively Next, we define four saturation levels, specifically, v max,% = 9u ss,max, v max,% = 8u ss,max, v max,4% = 6u ss,max, v max,8% = u ss,max, v min,% = 9u ss,min, v min,% = 8u ss,min, v min,4% = 6u ss,min, and v min,8% = u ss,min In all the four cases, the adaptive controller (7) with known saturation bounds in (7) is implemented in feedback with n c = 3, η =, P = I, and B f = Note that, v min, v max, and B f are the only required model information for the adaptive controller Furthermore, the adaptive controller gain matrix θ() is initialized to zero That is, no baseline controller is used to initialize RCAC Figure 3 shows the time history of w, y, u, and v for the case 5 Copyright 3 by ASME

6 without saturation as well as the case the control output u has %, %, 4%, and 8% saturation For the case without saturation, y follows the command w with zero steady-state error For the case with saturation, note that the adaptive controller is not able to follow the commands with zero steady-state error because the saturation makes this impossible However, the unsaturated control signal u does not exhibit integrator windup, and the output y is able to match w without phase lag No Sat % Sat % Sat 4% Sat 8% Sat Command signal w(k) (dashed) Output signal y(k) (solid) Commanded control u(k) (dashed) Plant input v(k) (solid) Figure 3 Example 4 Square wave command following for a minimum-phase, asymptotically stable plant with saturation The adaptive controller (7) is implemented in feedback with n c = 3, η =, P = I, and B f = Note that, v min, v max, and B f are the only required model information for the adaptive controller The adaptive controller gain matrix θ() is initialized to zero We consider the case without saturation as well as the case the control output u has %, %, 4%, and 8% saturation For the case without saturation, y follows the command w with zero steady-state error For the case with saturation, note that the adaptive controller is not able to follow the commands with zero steady-state error because the saturation makes this impossible However, the unsaturated control signal u does not exhibit integrator windup, and the output y is able to match w without phase lag Example 43 Triangle wave command following for a minimum-phase, Lyapunov stable plant with saturation Consider the Lyapunov stable, minimum-phase plant transfer function from v to z, given by G zv (z) = z (38) Our goal is to have the plant output y follow the triangle-wave command w(k) = w t (k) We consider the unsaturated case, that is, v min = and v max =, together with four levels of saturation Let u ss,max and u ss,min be maximum and minimum steady-state command values that drive the performance z to zero in steady-state (ie z ss = ), respectively Next, we define four saturation levels, specifically, v max,% = 9u ss,max, v max,% = 8u ss,max, v max,4% = 6u ss,max, v max,8% = u ss,max, v min,% = 9u ss,min, v min,% = 8u ss,min, v min,4% = 6u ss,min, and v min,8% = u ss,min In all the four cases, the adaptive controller (7) with known saturation bounds in (7) is implemented in feedback with n c = 3, η =, P = I, and B f = Note that, v min, v max, and B f are the only required model information for the adaptive controller Furthermore, the adaptive controller gain matrix θ() is initialized to zero That is, no baseline controller is used to initialize RCAC Figure 4 shows the time history of w, y, u, and v for the case without saturation as well as the case the control output u has %, %, 4%, and 8% saturation For the case without saturation, y follows the command w Each time the slope of w changes sign, the control u experiences a transient This is because RCAC adapts itself to follow the new command and the output y(k) reaches zero steady-state error For the case with saturation, note that the adaptive controller is not able to follow the commands with zero steady-state error because the saturation makes this impossible However, the unsaturated control signal u does not exhibit integrator windup and remains bounded In particular, the closed-loop system maintains stability Example 44 Command following for a multi-rotor helicopter The translational motion of a multi-rotor helicopter is described by q = m u +, (39) g q = [ q q q 3 ] T R 3 denotes the position of the vehicle center of mass resolved in the Earth frame, q and q denote horizontal displacements, while q 3 denotes the vertical displacement The initial conditions are q() = [ ] T and q() = [ ] T u = [ u u u 3 ] T R 3 is the control force, g = 98 m/s is the gravitational acceleration, and m = 5 kg is the mass of the vehicle 6 Copyright 3 by ASME

7 Command signal w(k) (dashed) Output signal y(k) (solid) Commanded control u(k) (dashed) Plant input v(k) (solid) denote the position command, and define the tracking error z R 3 by No Sat z = q w (4) % Sat % Sat Let the positive real numbers φ max = deg and u max = 6 N denote the maximum allowable values of φ and u, respectively The control problem is thus to construct a feedback control law for u that minimizes z subject to u + u u sinφ max, (43) 4% Sat u 3, (44) 8% Sat and u u max, (45) Figure 4 Example 43 Triangle wave command following for a minimum-phase, Lyapunov stable plant with saturation The adaptive controller (7) is implemented in feedback with n c = 3, η =, P = I, and B f = Note that, v min, v max, and B f are the only required model information for the adaptive controller The adaptive controller gain matrix θ() is initialized to zero We consider the case without saturation as well as the case the control output u has %, %, 4%, and 8% saturation For the case without saturation, y follows the command w Each time the slope of w changes sign, the control u experiences a transient and the output y(k) reaches zero steady-state error For the case with saturation, note that the adaptive controller is not able to follow the commands with zero steady-state error because the saturation makes this impossible However, the unsaturated control signal u does not exhibit integrator windup and remains bounded Define the inclination angle φ of u to be (43)-(45) form the convex control constraint set U shown in Figure 5 The problem of minimizing the retrospective cost function on U can thus be rewritten as the following secondorder cone programming (SOCP) problem: subject to min J(Û(k), k) (46) PÛ(k) QÛ(k) and Û(k) 6, (47) P = and Q = tan(φ max ) [ ] T The nonlinear programming method SOCP in the CVX toolbox [7] is used to solve the optimization problem (46) and (47) φ = cos u 3 u, (4) u denotes the Euclidean norm of u Let w(t) = cos(t) sin()t 3t + R 3 (4) Next, a state space representation of the multi-rotor heli- 7 Copyright 3 by ASME

8 Sat (u,u,u 3 ) = { u if u Sat 3 (u 3 )tanφ max sinϑ, Sat 3 (u 3 )tanφ max sinϑ otherwise, (53) Sat 3 (u 3 ) = sat,6 (u 3 ), (54) and ϑ = atan(u,u ) = arctan u (55) u + u + u Figure 5 Example 44 The convex control constraint set U formed by (43)-(45) copter is given by [ ] [ ][ ] [ ] q 3 3 I = 3 3 q q q m I v g, (48) y = [ ] [ ] q I , (49) q z = y w, (5) v = Sat(u) U is the saturated control input given by v = v v = Sat(u) = Sat (u,u,u 3 ) Sat (u,u,u 3 ), (5) v 3 Sat 3 (u 3 ) Sat (u,u,u 3 ) = { u if u Sat 3 (u 3 )tanφ max cosϑ, Sat 3 (u 3 )tanφ max cosϑ otherwise, (5) Next, we discretize (48)-(5) using zero-order-hold The adaptive controller (7) with knowledge of the saturation (47) is implemented in feedback with n c = 8, η =, P = I, d =, H = I 3 3, and we let B f = [ ] I Figure 6 shows the time history of y, y, and y 3 of the helicopter The transient especially along y 3 direction is due to the fact that (48)-(5) is unstable, the discretized equivalent of (48)-(5) has nonminimum-phase zeros at, and the gravitational acceleration g is unmodeled Furthermore, we initialize the adaptive controller at θ() = Note that the commanded control signal u does not exhibit integrator windup and remains bounded as shown in Figure 7, the black dots represent the control constraint set U, and the blue crosses represent the commanded control u Note that the blue crosses outside the control constraint set U (black dots region) are caused by the transient behavior of RLS update in (9) and (3) Figure 8 shows the distance between the unsaturated commanded control u(k) (blue crosses that are outside the control constraint set U in Figure 7) and the saturated control v(k) at each time step 5 Conclusion Adaptive control based on constrained retrospective cost optimization was applied to command following for Hammerstein systems with input saturation We numerically demonstrated that RCAC can improve the tracking performance when following square-wave and triangle-wave commands in the presence of saturation, provided that the saturation boundary is known RCAC was used with limited modeling information In particular, RCAC uses knowledge of the first nonzero Markov parameter of the linear system and the saturation bounds We also applied this technique to a multi-rotor helicopter command-following problem by formulating the multi-input constrained retrospective cost function as a second-order cone optimization (SOCP) problem With this approach, RCAC is shown to adapt to these constraints Future research will include a stability analysis of RCAC under saturation input nonlinearity 8 Copyright 3 by ASME

9 Command signal w 5 Output signal y Command signal w 5 Output signal y Command signal w 3 Output signal y Commanded control u Saturated control v Commanded control u Saturated control v θ(k) Commanded control u 3 Saturated control v Time (sec) Figure 6 Example 44 Command following for a multi-rotor helicopter The adaptive controller (7) with the saturation bounds in (47) is implemented in feedback with n c = 8, η =, P = I, B f = [I ], and θ() = Note that the outputs of y, y, y 3 follow the commands w, w, and w 3 after the transient ACKNOWLEDGMENT We wish to thank Jesse Hoagg and Alexey Morozov for helpful discussions REFERENCES [] Bernstein, D S, and Michel, A N, 995 A chronological bibliography on saturating actuators Int J Robust and Nonlinear Control, 5, pp [] Zaccarian, L, and Teel, A R, Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation Princeton University Press [3] Hippe, P, 6 Windup in Control: Its Effects and Their Prevention Springer [4] Hu, T, and Lin, Z, Control Systems with Actuator Saturation: Analysis and Design Birkhäuser [5] Glattfelder, A, and Schaufelberger, W, 3 Control Systems with Input and Output Constraints Springer [6] Kapila, T, and Grigoriadis, K, Actuator Saturation Control CRC Press [7] Tarbouriech, S, Garcia, G, da Silva Jr, J M G, and Queinnec, I, Stability and Stabilization of Linear Systems with Saturating Actuators Springer [8] Annaswamy, A M, and Wong, J-E, 998 Adaptive Figure 7 Example 44 The adaptive controller (7) with known saturation bounds in (47) is implemented in feedback with n c = 8, η =, P = I, B f = [I ], and θ() = The black dots represent the constraint set in (43) and (45), and the blue dots represent the unsaturated controller output u The blue crosses outside the boundary of the constraint (black dots region) are due to the transient behavior of RLS update in (9) and (3) u(k) Sat(u(k)) Time (sec) Figure 8 Example 44 At each time step, the distance between the unsaturated commanded control u(k) (blue crosses that are outside the control constraint set U in Figure 7) and the saturated control Sat(u(k)) control in the presence of saturation non-linearity Int J Adaptive Contr Sig Proc,, pp 3 9 [9] Teo, J, and How, J P, 9 Anti-windup compensation for nonlinear systems via gradient projection: Application to adaptive control In Proc IEEE Conf Dec Contr, pp [] Coffer, B J, Hoagg, J B, and Bernstein, D S, Cu- 9 Copyright 3 by ASME

10 mulative retrospective cost adaptive control of systems with amplitude and rate saturation In Proc Amer Contr Conf, pp [] Tyan, F, and Bernstein, D S, 999 Global stabilization of systems containing a double integrator using a saturated linear controller Int J Robust Nonlinear Control, 9(5), pp [] Hoagg, J B, Santillo, M A, and Bernstein, D S, 8 Discrete-time adaptive command following and disturbance rejection for minimum phase systems with unknown exogenous dynamics IEEE Trans Autom Contr, 53, pp 9 98 [3] Santillo, M A, and Bernstein, D S, Adaptive control based on retrospective cost optimization AIAA J Guid Contr Dyn, 33, pp [4] Hoagg, J B, and Bernstein, D S, Retrospective cost model reference adaptive control for nonminimumphase systems AIAA J Guid Contr Dyn, 35, pp [5] Hoagg, J B, and Bernstein, D S, Retrospective cost adaptive control for nonminimum-phase discrete-time systems part : The ideal controller and error system; part : The adaptive controller and stability analysis In Proc IEEE Conf Dec Contr, pp [6] D Amato, A M, and Bernstein, D S, Adaptive forward-propagating input reconstruction for nonminimum-phase systems In Proc Amer Contr Conf, pp [7] Grant, M, and Boyd, S, 3 CVX: Matlab software for disciplined convex programming, version March [8] Morozov, A, D Amato, A M, Hoagg, J B, and Bernstein, D S, Retrospective cost adaptive control for nonminimum-phase systems with uncertain nonminimumphase zeros using convex optimization In Proc Amer Contr Conf, pp [9] Åström, K J, and Wittenmark, B, 995 Adaptive Control Addison-Wesley [] Goodwin, G C, and Sin, K S, 984 Adaptive Filtering, Prediction, and Control Prentice Hall [] D Amato, A M, Sumer, E D, and Bernstein, D S, Frequency-domain stability analysis of retrospective-cost adaptive control for systems with unknown nonminimumphase zeros In Proc IEEE Conf Dec Contr, pp 98 3 [] Sumer, E D, D Amato, A M, Morozov, A M, Hoagg, J B, and Bernstein, D S, Robustness of retrospective cost adaptive control to Markov-parameter uncertainty In Proc IEEE Conf Dec Contr, pp [3] Sumer, E D, Holzel, M H, D Amato, A M, and Bernstein, D S, FIR-based phase matching for robust retrospective-cost adaptive control In Proc Amer Contr Conf, pp 77 7 [4] Sumer, E D, and Bernstein, D S, Retrospective cost adaptive control with error-dependent regularization for mimo systems with unknown nonminimum-phase transmission zeros In Proc AIAA Guid Nav Contr Conf AIAA--47 [5] Hoagg, J B, and Bernstein, D S, Cumulative retrospective cost adaptive control with rls-based optimization In Proc Amer Contr Conf, pp 46 4 [6] Brammer, R F, 97 Controllability in linear autonomous systems with positive controllers SIAM J Control,, pp [7] Jacobson, D H, Margin, D H, Pachter, M, and Geveci, T, 98 Extensions of Linear-Quadratic Control Theory Springer [8] Leyva, H, Sols-Daun, J, and Suárez, R, 3 Global CLF stabilization of systems with control inputs constrained to a hyperbox SIAM J Control Optim, 5, pp Copyright 3 by ASME