Adaptive Control of Uncertain Hammerstein Systems with Monotonic Input Nonlinearities Using Auxiliary Nonlinearities

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1 5st IEEE Conference on Decision and Control December -3, Mai, Hawaii, USA Adaptive Control of Uncertain Hammerstein Systems with Monotonic Inpt Nonlinearities Using Axiliary Nonlinearities Jin Yan, Anthony M D Amato, E Dogan Smer, Jesse B Hoagg, and Dennis S Bernstein Abstract We extend retrospective cost adaptive control (RCAC) to command following for ncertain Hammerstein systems We assme that only one Markov parameter of the linear plant is known and that the inpt nonlinearity is monotonic bt otherwise nknown Axiliary nonlinearities are sed within RCAC to accont for the effect of the inpt nonlinearity I INTRODUCTION In many practical applications, an inpt nonlinearity precedes the linear plant dynamics; systems with this strctre are called Hammerstein systems [ 3] The inpt nonlinearity may represent properties of an actator, sch as satration to reflect magnitde restrictions on the control inpt, deadzone to represent actator stiction, and a signm nonlinearity to represent on-off behavior Adaptive control of Hammerstein systems with ncertain inpt nonlinearities and linear dynamics is considered in [ 6] Unlike [ 6], however, we make no attempt to identify and invert the inpt nonlinearity Instead, we apply retrospective-cost adaptive control (RCAC), which can be sed for plants that are possibly MIMO, nonminimm phase (NMP), and nstable [7 3] This approach relies on knowledge of Markov parameters and, for NMP openloop-nstable plants, estimates of the NMP zeros This information can be obtained from either analytical modeling or system identification [] In the present paper we consider a command-following problem for SISO Hammerstein plants limited modeling information is available concerning the inpt nonlinearity and the linear dynamics For the linear dynamics, we assme that one nonzero Markov parameter is known In addition, we consider plants that are open-loop asymptotically stable and ths, as shown in [, 3], knowledge of the NMP zeros is not needed We also assme that the inpt nonlinearity is monotonic bt not necessarily continos The novel contribtion of the present paper is the agmentation of RCAC with two axiliary nonlinearities that accont for the presence of the ncertain inpt nonlinearity N The axiliary nonlinearity N is a satration nonlinearity, which is chosen to tne the transient response of the closed-loop system and which may depend on estimates of the range of the inpt nonlinearity N and the gain of the linear dynamics In contrast, the axiliary nonlinearity N is chosen so that This work was spported in part by NSF Grant and NASA Grant NNX8A69A J B Hoagg is with the Department of Mechanical Engineering, University of Kentcky, Lexington, KY The remaining athors are with the Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 89- the composite nonlinear fnction N N is nondecreasing Therefore, if N is nondecreasing, then N is not needed If, however, N is nonincreasing, then N can be chosen sch that N N is nondecreasing Note that N need not be oneto-one or onto This approach extends the techniqe sed in [5] for Hammerstein systems with amplitde and rate satration In [ 6], the inpt nonlinearities are assmed to be piecewise linear The present paper does not impose this restriction Nmerical examples involving cbic, deadzone, satration, and on-off inpt nonlinearities are presented II HAMMERSTEIN COMMAND-FOLLOWING PROBLEM Consider the SISO discrete-time Hammerstein system x(k +) = Ax(k)+BN((k))+D w(k), () y(k) = Cx(k), () x(k) R n, (k), y(k) R, w(k) R d, N : R R, and k We consider the Hammerstein commandfollowing problem with the performance variable z(k) = y(k) r(k), (3) z(k), r(k) R The goal is to develop an adaptive otpt feedback controller that minimizes the commandfollowing error z with minimal modeling information abot the dynamics, distrbance w, and inpt nonlinearity N We assme that measrements of z(k) are available for feedback; however, measrements of v(k) = N((k)) are not available A block diagram for ()-(3) is shown in Figre Fig Adaptive command-following problem for a Hammerstein plant We assme that measrements of z(k) are available for feedback; however, measrements of v(k) = N((k)) and w(k) are not available The feedforward path is optional III ADAPTIVE CONTROL FOR THE HAMMERSTEIN COMMAND-FOLLOWING PROBLEM For the Hammerstein command-following problem, we assme that G is ncertain except for an estimate of a single //$3 IEEE 8

2 nonzero Markov parameter The inpt nonlinearity N is also ncertain To accont for the presence of the inpt nonlinearity N, the RCAC controller in Figre ses two axiliary nonlinearities The axiliary nonlinearity N modifies c to obtain the regressor inpt r, while the axiliary nonlinearity N modifies r to prodce the Hammerstein plant inpt The axiliary nonlinearities N and N are chosen based on limited knowledge of the inpt nonlinearity N, as described below Fig Hammerstein command-following problem with the RCAC adaptive controller and axiliary nonlinearities N and N A Axiliary Nonlinearity N Define the satration fnction sat p,q by p, if c < p, N ( c ) = sat p,q ( c ) = c, if p c q, () q, if c > q, p R and q R are the lower and pper satration levels, respectively For minimm-phase plants, the axiliary nonlinearity N is not needed, and ths the satration levels p and q are chosen to be large negative and positive nmbers, respectively For NMP plants, the satration levels are sed to tne the transient behavior In addition, the satration levels are chosen to provide the magnitde of the control inpt in order to follow the command r These vales depend on the range of the inpt nonlinearity N as well as the gain of the linear system G at freqencies in the spectra of r and w B Axiliary Nonlinearity N If N is nondecreasing, then N is not needed We ths consider the case in whichn is monotonically nonincreasing on the finite interval I = [p,q] Since the range of N is [p,q], we need to consider only r [p,q] If N is nonincreasing on I, then we define N ( r ) = p+q r I for all r I Ths, N is a piecewise-linear fnction that replaces N by its mirror image, which is nondecreasing in I Let R I (f) denote the range of f with argments in I Proposition 3: Assme that N is constrcted by the above rle Then the following statements hold: i) N N is nondecreasing ii) R I (N N ) = R I (N) Proof If N is nondecreasing on I, then N is the identity fnction and ths i) holds Now, assme that N is nonincreasing on I, and let r,, r, I, r, r, Then, = p+q r, = p+q r, Therefore, sincen is nonincreasing oni and, it follows that N(N ( r, )) = N( ) N( ) = N(N ( r, )) Ths, i) holds To prove ii), assme that N is nondecreasing on I Since N ( r ) = r for all r I, it follows that N (I) = I, that is, N : I I is onto Alternatively, assme that N is nonincreasing on I so that N ( r ) = p + q r Note that N (p i ) = q i, N (q) = p, and N is continos and decreasing on I Therefore, N (I i ) = I i, and ths N : I I is onto Hence, R I (N N ) = R I (N) As an example, consider the nonincreasing inpt nonlinearity = sat 5,5 ( 5) Let N ( c ) = sat p,q ( c ), p =,q =, and N ( r ) = r + for all r [,] according to Proposition 3 Figre 3(c) shows that the composite nonlinearity N N is nondecreasing on [,] Note that R I (N N ) = R I (N) = [ 5, 5] N N (c) r N(r) r Fig 3 Inpt nonlinearity = sat 5,5 ( 5) Axiliary nonlinearity N ( r ) = r + for r [,] (c) Composite nonlinearity N N Note that N N is nondecreasing and R(N N ) = R(N) = [ 5,5] Knowledge of only the monotonicity of N and the interval I are needed to modify the controller otpt r so thatn N is nondecreasing It ths follows that N N preserves the signs of the Markov parameters of the linearized Hammerstein system For details, see [3] IV RETROSPECTIVE-COST ADAPTIVE CONTROL For i, define the Markov parameter H i = E A i B For example, H = E B and H = E AB Let l be a positive integer Then, for all k l, x(k) = A l x(k l)+ and ths l A i BN(N (N ( c (k i)))), i= z(k) = E A l x(k l) E r(k)+ HŪ(k ), (5) 8

3 and Ū(k ) = H = [ H H l ] R l N(N (N ( c (k )))) N(N (N ( c (k l)))) Next, we rearrange the colmns of H and the components of Ū(k ) and partition the reslting matrix and vector so that HŪ(k ) = H U (k )+HU(k ), (6) H R (l l U), H R l U, U (k ) R l l U, and U(k ) R l U Then, we can rewrite (5) as z(k) = S(k)+HU(k ), (7) S(k) = E A l x(k l) E r(k)+h U (k ) (8) Next, for j =,,s, we rewrite (7) with a delay of k j time steps, k k k s, in the form z(k k j ) = S j (k k j )+H j U j (k k j ), (9) (8) becomes S j (k k j ) = E A l x(k k j l)+h ju j(k k j ) and (6) becomes HŪ(k k j ) = H ju j(k k j )+H j U j (k k j ), H j R (l l U ) j, H j R l U j, U j (k k j ) R l l U j, and Uj (k k j ) R l U j Now, by stacking z(k k ),,z(k k s ), we define the extended performance Therefore, Z(k) = Z(k) = S(k) = Ũ(k ) has the form Ũ(k ) = z(k k ) z(k k s ) R s () S(k)+ HŨ(k ), () S (k k ) S s (k k s ) R s, N(N (N ( c (k q )))) N(N (N ( c (k q lũ)))) R lũ,, fori =,,lũ,k q i k s +l, and H R s lũ is constrcted according to the strctre ofũ(k ) The vector Ũ(k ) is formed by stacking U (k k ),,U s (k k s ) and removing copies of repeated components Next, for j =,,s, we define the retrospective performance ẑ j (k k j ) = S j (k k j )+H j Û j (k k j ), () the past controls U j (k k j ) in (9) are replaced by the retrospective controls Ûj(k k j ) In analogy with (), the extended retrospective performance for () is defined as Ẑ(k) = and ths is given by ẑ (k k ) ẑ s (k k s ) R s Ẑ(k) = S(k)+ HˆŨ(k ), (3) the components of ˆŨ(k ) R lũ are the components of Û(k k ),,Ûs(k k s ) ordered in the same way as the components of Ũ(k ) Sbtracting () from (3) yields Ẑ(k) = Z(k) HŨ(k )+ HˆŨ(k ) () Finally, we define the retrospective cost fnction J(ˆŨ(k ),k) = ẐT (k)r(k)ẑ(k), (5) R(k) R s s is a positive-definite performance weighting The goal is to determine refined controls ˆŨ(k ) that wold have provided better performance than the controls U(k) that were applied to the system The refined control vales ˆŨ(k ) are sbseqently sed to pdate the controller Next, to ensre that (5) has a global minimizer, we consider the reglarized cost J(ˆŨ(k ),k) = ẐT (k)r(k)ẑ(k) +η(k)ˆũ T (k )ˆŨ(k ), (6) η(k) Sbstitting () into (6) yields J(ˆŨ(k ),k) = ˆŨ(k ) T A(k)ˆŨ(k ) +B(k)ˆŨ(k )+C(k), A(k) = H T R(k) H+η(k)I lũ, B(k) = H T R(k)[Z(k) HŨ(k )], C(k) = Z T (k)r(k)z(k) Z T (k)r(k) HŨ(k ) +ŨT (k ) H T R(k) HŨ(k ) If either H has fll colmn rank or η(k) >, then A(k) is positive definite In this case, J(ˆŨ(k ),k) has the niqe global minimizer ˆŨ(k ) = A (k)b(k) (7) 83

4 A Controller Constrction The control (k) is given by the strictly proper time-series controller of order n c given by n c n c (k) = M i (k)(k i)+ N i (k)z(k i) i= i= n c + Q i (k)r(k i), (8) i=, for all i =,,n c, M i (k) R, N i (k) R, and Q i (k) R The control (8) can be expressed as and (k) = θ(k)φ(k ), θ(k) = [M (k) M n c (k) N (k) N n c (k) Q (k) Q n c (k)] R l 3n c φ(k ) = [(k ) (k n c) z(k ) z(k n c) r(k ) r(k n c )] T R 3n c Next, letdbe a positive integer sch thatũ(k ) contains (k d) and define the cmlative cost fnction J R (θ,k) = k i=d+ λ k i φ T (i d )θ T (k) û T (i d) +λ k (θ(k) θ )P (θ(k) θ ) T, (9) is the Eclidean norm, and λ (,] is the forgetting factor Minimizing (9) yields θ T (k) = θ T (k )+β(k)p(k )φ(k d ) [φ T (k d)p(k )φ(k d )+λ(k)] [φ T (k d )θ T (k ) û T (k d)], β(k) is either zero or one The error covariance is pdated by P(k) = β(k)λ P(k )+[ β(k)]p(k ) β(k)λ P(k )φ(k d ) [φ T (k d )P(k )φ(k d)+λ] φ T (k d )P(k ) We initialize the error covariance matrix as P() = αi 3nc, α > Note that when β(k) =, θ(k) = θ(k ) and P(k) = P(k ) Therefore, setting β(k) = switches off the controller adaptation, and ths freezes the control gains When β(k) =, the controller is allowed to adapt V NUMERICAL EXAMPLES In all examples, we assme that at least one nonzero Markov parameter of G is known For convenience, each example is constrcted sch that the first nonzero Markov parameter H d =, d is the relative degree of G RCAC generates a control signal c (k) that attempts to minimize the performance z(k) in the presence of the inpt nonlinearity N In all cases, we initialize the adaptive controller to be zero, that is, θ() = We let λ = for all examples Example 5: We consider the asymptotically stable, minimm-phase plant G(z) = (z 5)(z 9) (z 7)(z 5 j5)(z 5+j5), () with the cbic inpt nonlinearity = 3, () which is nonincreasing, one-to-one, and onto and has the offsetn() = Note thatd = andh d = We consider the sinsoidal command r(k) = sin(θ k), θ = π/5 rad/sample To illstrate the effect of the nonlinearities on the closed-loop command-following performance, we first remove the inpt nonlinearity and simlate the openloop system for the first time steps Then, at k =, we trn the adaptation on and let RCAC adapt to the linear system for 3 time steps Next, at k =, we stop the adaptation and introdce the inpt nonlinearity Conseqently, from k = to k = 7, we se the frozen gain matrix θ() as the feedback gain withot adaptation in order to demonstrate the performance degradation de to the inpt nonlinearity Finally, at k = 7, we restart the adaptation and let RCAC adapt to the Hammerstein system As shown in Figre, we choose N ( c ) = sat p,q ( c ), p = 6 and q = 6 in () Since N is decreasing for all [ 6, 6 ], we let N ( r ) = r Note that knowledge of only the monotonicity of N is sed to choose N We let n c =, P = I 3nc, η =, and H = H Figre shows the reslting time history of the commandfollowing performance z, while Figre (c) shows the time history of the control and linear plant inpt v Finally, Figre (d) shows the time history of the controller gain vector θ Example 5: We consider the asymptotically stable, NMP plant G(z) = z 5 (z 8)(z 6), () with the deadzone inpt nonlinearity +5, if < 5, =, if 5 5, (3) 5, if > 5, which is not one-to-one bt onto and satisfies N() = Note that d = and H d = We consider the two-tone sinsoidal command r(k) = sin(θ k)+5sin(θ k), θ = π/ rad/sample, andθ = π/ rad/sample As shown in Figre 5, since is nondecreasing for all R, we choose N ( c ) = sat p,q ( c ), p = a, q = a, and N ( r ) = r We let n c =, P = I 3nc, η =, and H = H, and we vary the satration level a for the NMP plant () Figre 5(bi) shows the time history of the performance z with a =, the transient behavior is 8

5 θ(k) performance z(k) Control (k) Plant inpt v(k) Ne() N N β = β = β = 5 5 time step k β = β = β = 5 5 time step β = β = β = 5 5 time step (c) β = β = β = time step (d) Fig Example 5 shows the inpt nonlinearity N given by () shows the closed-loop response to the sinsoidal command r(k) = sin(πk) of the asymptotically stable minimm-phase plant G given by () The vale of β indicates whether the controller is frozen or adapting (c) shows the time history of the control and the plant inpt v with and withot the inpt nonlinearity N present (d) shows the time history of the controller gain vector θ with and withot N present poor Figre 5(bii) shows the time history of the performance z with a =, the transient performance is improved and z reaches steady state in abot 3 time steps Finally, we frther redce the satration level Figre 5(biii) shows the time history of the performance z with a = ; in this case, RCAC cannot follow the command de to fact that a = is not large enogh to provide the control otpt c needed to drive z to a small vale (i) Time Step (k) (ii) Time Step (k) β = β = β = β =, a = β =, a = β =, a = 6 8 (iii) Time Step (k) Fig 5 Example 5 shows the deadzone inpt nonlinearity given by (3) shows the closed-loop response of the asymptotically stable NMP plant G given by () with the two-tone sinsoidal command r(k) = sin(θ k)+5sin(θ k), θ = π/ rad/sample, and θ = π/ rad/sample Figre 5(bi) shows the time history of the performance z with a =, the transient behavior is poor Figre 5(bii) shows the time history of the performance z with a = Note that the transient performance is improved and z reaches steady state in abot 3 time steps Finally, we frther redce the satration level Figre 5(biii) shows the time history of the performance z with a = ; in this case, RCAC cannot follow the command de to the fact that a = is not large enogh to provide the control otpt c needed to drive z to a small vale Example 53: We consider the asymptotically stable, NMP plant () with the satration inpt nonlinearity 8, if <, =, if, () 8, if >, which is nondecreasing and one-to-one bt not onto, and satisfies N() = We consider the two-tone sinsoidal command r(k) = 5sin(θ k) + 5sin(θ k), θ = π/5 rad/sample and θ = π/ rad/sample for the Hammerstein system with the inpt nonlinearity N As shown in Figre 6, since is nondecreasing for all R, we choose N ( c ) = sat p,q ( c ), p = and q = in (), and N ( r ) = r We let n c =, P = I 3nc, η =, and H = H The Hammerstein system rns openloop for time steps, and RCAC is trned on at k = Figre 6 shows the time history of the performance z 85

6 with the inpt nonlinearity present Note that z does not converge to zero de to the distortion introdced by the inpt nonlinearity N β = β = Time Step (k) Fig 6 Example 53 shows the satrating inpt nonlinearity given by () shows the closed-loop response of the stable NMP plantg given by () with the two-tone sinsoidal commandr(k) = 5sin(θ k)+ 5sin(θ k), θ = π/5 rad/sample, and θ = π/ rad/sample Example 5: We consider the nstable doble integrator plant z G(z) = (z ) (5) with the piecewise-constant inpt nonlinearity = [sign( )+sign(+)] (6) Note that can assme only the vales,, and Note that d = and H d = We let the command r(k) be zero, and consider stabilization sing RCAC with the inpt relay nonlinearity given by (6) As shown in Figre 7, the relay nonlinearity is monotonically nondecreasing for all R, and we ths choose N ( c ) = sat p,q ( c ), p = 3, q = 3, and N ( r ) = r We let n c =, P = I 3nc, η =, and H = H The closed-loop performance approaches ± in abot 5 time steps Figre 7 shows the time history of z with the initial condition x = [ 5 ] T VI CONCLUSIONS Retrospective cost adaptive control (RCAC) was applied to a command-following problem for Hammerstein systems with nknown distrbances RCAC was sed with limited modeling information In particlar, the inpt nonlinearity is assmed to be monotonic bt is otherwise nknown, and RCAC ses knowledge of only the first nonzero Markov parameter of the linear dynamics To handle the effect of the inpt nonlinearity, RCAC was agmented by axiliary nonlinearities chosen based on the monotonicity of the inpt nonlinearity 86 performance z(k) time step k Fig 7 Example 5 Closed-loop response of the plant G given by (5) with the initial condition x = [ 5, ] T The system rns open loop for time steps, and the adaptive controller is trned on at k = with the inpt relay nonlinearity given by (6) The closed-loop performance z approaches ± in abot 5 time steps REFERENCES [] W Haddad and V Chellaboina, Nonlinear control of hammerstein systems with passive nonlinear dynamics, IEEE Trans Atom Contr, vol 6, no, pp 63 63, [] L Zaccarian and A R Teel, Modern Anti-windp Synthesis: Control Agmentation for Actator Satration Princeton, [3] F Giri and E W Bai, Block-Oriented Nonlinear System Identification Springer, [] M C Kng and B F Womack, Discrete-time adaptive control of linear dynamic systems with a two-segment piecewise-linear asymmetric nonlinearity, IEEE Trans Atom Contr, vol 9, no, pp 7 7, 98 [5], Discrete-time adaptive control of linear systems with preload nonlinearity, Atomatica, vol, no, pp 77 79, 98 [6] G Tao and P V Kokotović, Adaptive Control of Systems with Actator and Sensor Nonlinearities Wileyr, 996 [7] R Vengopal and D S Bernstein, Adaptive distrbance rejection sing ARMARKOV system representations, IEEE Trans Contr Sys Tech, vol 8, pp 57 69, [8] J B Hoagg, M A Santillo, and D S Bernstein, Discrete-time adaptive command following and distrbance rejection for minimm phase systems with nknown exogenos dynamics, IEEE Trans Atom Contr, vol 53, pp 9 98, 8 [9] M A Santillo and D S Bernstein, Adaptive control based on retrospective cost optimization, AIAA J Gid Contr Dyn, vol 33, pp 89 3, [] J B Hoagg and D S Bernstein, Retrospective cost adaptive control for nonminimm-phase discrete-time systems part : The ideal controller and error system; part : The adaptive controller and stability analysis, Proc IEEE Conf Dec Contr, pp 893 9, December [], Retrospective cost model reference adaptive control for nonminimm-phase discrete-time systems, part : The ideal controller and error system; part : The adaptive controller and stability analysis, Proc Amer Contr Conf, pp , Jne [] A M D Amato, E D Smer, and D S Bernstein, Retrospective cost adaptive control for systems with nknown nonminimm-phase zeros, Proc AIAA Gid Nav Contr Conf, pp AIAA 63, Agst [3], Freqency-domain stability analysis of retrospective-cost adaptive control for systems with nknown nonminimm-phase zeros, Proc IEEE Conf Dec Contr, pp 98 3, December [] M S Fledderjohn, M S Holzel, H Palanthandalam-Madapsi, R J Fentes, and D S Bernstein, A comparison of least sqares algorithms for estimating markov parameters, Proc Amer Contr Conf, pp , Jne [5] B C Coffer, J B Hoagg, and D S Bernstein, Cmlative retrospective cost adaptive control with amplitde and rate satration, Proc Amer Contr Conf, pp 3 39, Jne