28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June -3, 28 FrA4.4 L Adaptive Controller for Multi Input Multi Output Systems in the Presence of Unmatched Disturbances Chengyu Cao and Naira Hovakimyan Abstract L adaptive control architecture introduced in [ [4 ensures that the input and the output of an uncertain linear system track the input and output of a desired linear system during the transient phase, in addition to asymptotic tracking. In this paper, we extend the L adaptive controller to multi input multi output systems in the presence of unmatched time-varying disturbances. Simulation results illustrate the theoretical findings. I. INTRODUCTION The paper extends the results of [ [4 to a class of multi input multi output MIMO uncertain systems in the presence of time-varying parameters, unknown high-frequency gain, unmatched time-varying unknown disturbances that cannot be attenuated by backstepping type controllers. The control signal is defined as the output of a low-pass filter that appropriately attenuates the high-frequencies in the control signal typical for large adaptation rates. The L norm bounds for the error signals between the closed-loop adaptive system and the closed-loop reference system can be systematically reduced by fast adaptation. The paper is organized as follows. Section II states some preliminary definitions, and Section III gives the problem formulation. Section IV introduces the L adaptive control architecture. Stability and uniform transient tracking bounds of the L adaptive controller are presented in Section V. Section VI discusses design details. In Section VII simulation results are presented, while Section VIII concludes the paper. All proofs are in the Appendix. II. PRELIMINARIES Throughout this paper, I indicates the identity matrix of appropriate dimension, Hs L denotes the L gain of Hs, x L denotes the L norm of xt, x t L denotes the truncated L norm of xt at the time instant t, and x 2 and x indicate the 2- and - norms of the vector x respectively. The next lemma extends the results of Example.2 [6, page 99 to general MIMO systems. Lemma : For a stable proper MIMO system Hs with input rt R m and output xt R n, we have x t L Hs L r t L, t. III. PROBLEM FORMULATION Consider the following multi-input multi-output system: ẋt = A m xt + Bθtxt + ωut + σt, yt = C xt, x = x, Research is supported by AFOSR under Contract No. FA9---7. The authors are with Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, VA 246-23, e-mail: chengyu, nhovakim@vt.edu where x R n is the system state vector measurable, u R m is the control signal, y R m is the regulated output, B,C R n m are known constant matrices, A m is a known n n Hurwitz matrix, A m,b is controllable, θt R m n is a matrix of time-varying unknown parameters, ω R m m is a constant non-singular unknown matrix, and σt R n is a vector of time-varying unmatched disturbances. We assume conservative knowledge of unknown parameters and uncertainties, i.e. there exist compact sets Ω, Θ, and such that ω Ω, θt Θ, σt, for all t. We further assume θt and σt are differentiable with bounded derivatives, i.e. there exist finite d θ and d σ such that tr θ t θt d θ, σt 2 d σ, t. 2 Let H o s = C Hs, Hs = si A m B. Assumption : The poles of Ho s are located in the left half plane. The control objective is to design an adaptive controller to ensure that yt tracks a given bounded continuous reference signal rt R m both in transient and steady state, following a given reference model ys H o sk g rs, 3 where K g R m m is a constant matrix. IV. L ADAPTIVE CONTROLLER In this section, we develop a novel adaptive control architecture that permits complete transient characterization for both system s both input and output signals. We consider the following state predictor or passive identifier for generation of the adaptive laws: ˆxt = A mˆxt+bˆθtxt+ ˆωtut+ ˆσt, ˆx = x. 4 The adaptive estimates ˆωt, ˆθt, ˆσt are defined as: ˆωt = ΓProjˆωt, x tpb u t, ˆθt = ΓProjˆθt, x tpb x t, ˆσt = ΓProjˆσt, x tp, in which xt = ˆxt xt is the tracking error between the system dynamics in and the state predictor in 4, Γ > is the adaptation gain, and P = P > is the solution of the algebraic Lyapunov equation A mp + PA m = Q, Q >, while Proj, denotes the projection operator [8, defined over the sets Θ, Ω and. The control signal is generated through gain feedback of the following system: χs = Ds rs, us = Kχs, 6 978--4244-279-7/8/$2. 28 AACC. 4 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.
where K R m m, rt = ˆθtxt + ˆωtut + σt K g rt, σs = Ho sc si A m ˆσs, 7 while Ds is a matrix of m by m transfer functions. Letting Fs = ωki + DsωK Ds, 8 the choice of Ds and K needs to ensure that i Fs is strictly proper and stable with F = I; 9 ii FsHo s is proper and stable; iii Ḡs L L <, where L = max θ L = max θ Θ i j θ ij, 2 Ḡs = si A m BI Fs. 3 The complete L adaptive controller consists of 4,, 6 subject to 9-. A. Design of Control Law In this section, we address how to choose Ds and K to satisfy 9-. For simplicity, we consider the following most common choice of Ds as Ds = /si, which leads to Fs = ωksi + ωk. 4 Hence, the requirement in 9 is equivalent to determining a K that renders ωk Hurwitz. Using 4, it can be derived straightforwardly that the impulse response for I Fs is Iδt ωke ωkt, where δt denotes the impulse function. Let λ max ωk be the maximum eigenvalue of the Hurwitz matrix ωk. If K is chosen to have arbitrarily small λ max ωk, then ωke ωkt can approximate the delta function Iδt arbitrarily closely. Hence, we have lim I Fs L =. λ max ωk It follows from Lemma that Ḡs L si A m B L I Fs L. 6 Since si A m B L and L are constants, it follows from and 6 that can always be satisfied by making λ max ωk small via appropriate choice of K. If ω is a diagonal matrix, the design of K is straightforward. Assuming that the signs of the diagonal elements ω i, i =,..,m are known, K is selected to be a diagonal matrix with its i th element K i having the same sign as ω i. Hence, it follows from 4 that Fs is a diagonal matrix with its i th diagonal element being F i s = K i ω i /s + K i ω i. 7 We notice that 9 is obviously satisfied, while can be verified if we increase K i. Instead of the first order lowpass filter in 7, we can increase the relative degree of Fs by choosing Ds with larger relative degree as Ds = s I or Ds = 3a2 s+a 3 2 s 3 +3as I, a >. Hence, FsH 2 o s can always be made proper via the choice of Ds. Once FsHo s is proper, its stability follows from Assumption straightforwardly. If m =, then Fs = Kω/s + ωk, where K is scalar. We notice that in this case the condition in degenerates into the one for SISO systems stated in [ [4. Also, Assumption degenerates into conventional minimum phase requirement for H o s. V. ANALYSIS OF L ADAPTIVE CONTROLLER The analysis of the above stated control architecture is an extension of the results from [ [4. The main difference as compared to [ [4 is that the system in this paper has multiple inputs and multiple outputs and unmatched timevarying disturbances. Let F s = HsFsHo sc, 8 F 2 s = ω FsHo sc. 9 It follows from that both F s and F 2 s are proper and stable and hence their L gains are finite. A. Closed-loop Reference System We consider the following closed-loop LTI reference system with its control signal and system response being defined as follows: ẋ ref t = A m x ref t + Bθtx ref t + ωu ref t +σt, x ref = x 2 y ref t = C x ref t 2 u ref s = ω Fsη ref s, 22 where η ref s is the Laplace transformation of the signal η ref t = θtx ref t σ ref t + K g rt, σ ref s = Ho sc si A m σs. 23 We note that the condition in implies that the transfer function between σt and u ref t is proper and stable, and there is no singularity or differentiation involved in the generation of u ref t. Since u ref t uses unknown signals and parameters ω, θt and σt, this closed-loop reference system is not implementable and is only used for analysis purposes. The next Lemma establishes stability of the closedloop system in 2-22. Lemma 2: If K and Ds verify 9-, the reference system in 2-22 is stable. B. Guaranteed Transient Performance of L Adaptive Controller Let where θ m 4 λ maxp λ min Q +4 γ = θ m /λ min PΓ, 24 d θ max θ Θ tr θ θ + d σ max σ σ 2 max θ Θ trθ θ + max ω Ω trω ω + max σ σ σ. 2 46 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.
The performance of L adaptive controller is stated in the following Theorem. Theorem : Given the system in, the reference system in 2-22, and the L adaptive controller defined via 4, and 6, subject to 9-, we have: x L γ, 26 x x ref L γ, 27 y y ref L C L γ, 28 u u ref L γ 2, 29 where γ is defined in 24 and γ = F s L γ / Ḡs L L, 3 γ 2 = ω Fs L Lγ + F 2 s L γ. 3 Theorem states that xt, yt and ut follow x ref t, y ref t and u ref t not only asymptotically but also during the transient, provided that the adaptive gain is selected sufficiently large. Thus, the control objective is reduced to designing K and Ds to ensure that the closed-loop reference system in 2-22 has the desired response specified in 3. VI. DESIGN OF L ADAPTIVE CONTROLLER In this section, we address how to determine K and Ds to ensure that the reference system in 2-22 can achieve the control objective stated in 3. Consider the following desired system response ẋ des t = A m x des t + BK g rt,y des t = C x des t,x des = x, which is independent of uncertainties. Obviously y des s = H o sk g rs + C si A m x. 32 First we prove that for a step input the steady state error between the reference system and this desired system is zero, if the unknown parameters are constant. Lemma 3: For the reference system in 2-22 and the desired system in 32, subject to conditions 9-, if rt, σt, θt are constant, then lim y reft y des t =. 33 t Next, we characterize the performance bounds between the reference system and the desired system during the transient. Let F 3 s = H o si Fs and F 4 s = H o si FsHo sc si A m. We note that both F 3 s and F 4 s are proper and stable transfer functions. It follows from Lemma 2 that x ref L is finite for any bounded rt. Lemma 4: For the reference system in 2-22 and the desired system in 32, subject to conditions 9-, we have: y ref y des L F 3 s L L x ref L + K g r L + F 4 s L σ L. 34 Theorem and Lemma 4 imply that the output yt of the system in will follow y des t both in transient and steady state with quantifiable bounds, given in 28 and 34. Since lim I F 3s L = and lim I F 4s L =, Fs I Fs I we have lim y ref y des L =. Using, we notice Fs I that if we increase the bandwidth of Fs to achieve arbitrarily close approximation of the identity matrix I, we can ensure that y ref t tracks y des t arbitrarily closely both in transient and steady state. However, increasing the bandwidth of Fs will lead to reduced time-delay margin and hurt the robustness of the closed-loop system as proved in [4. Remark : We notice that Assumption and the condition in are new as compared to earlier results in [ [4. These are required to compensate for the effects of the unmatched disturbance σt on the system output yt. If the disturbance σt can be factored as Bσt, where σt R m instead of σt R n, both Assumption and condition can be dropped. Remark 2: We notice that due to the unmatched nature of the disturbance, its effect on the performance of the system states cannot be compensated by the choice of the control signal. The control signal will only compensate for the effect of the disturbance on the system output. However, since the control signal is defined as an output of a low-pass system with appropriately banded frequencies, the large adaptive gain required in the analysis will not lead to adverse effects on the system states. s+3 VII. SIMULATION RESULTS [ Consider the system: ẋt = A m xt+b + [ θtxt ω ut + σt, yt = C ω xt, x =, 2 where A m = 3, B =, C = 2. We note here that θt = [θ t θ 2 t θ 3 t is a row [ vector. It can be derived straightforwardly that H o s = s+. We notice that the desired system response is characterized by[ y des s = H o sk g rs with K g = C A m B =, H 3 o sk g = [ s+. The L adaptive controller is de- 3 s+3 signed based on 4, and 6 as follows: ˆxt = [ [ ˆω t A mˆxt + B + ut + ˆθtxt ˆω 2 t ˆσt, ˆωi t = ΓProjˆω i t,[ x tpb u t {i,i}, ˆθt = ΓProjˆθt,[ x tpb x t {2,:}, ˆσt = ΓProjˆσt, x tp, where [X {i,i} indicates the the i th row i th column element of the matrix X, while [X {2,:} indicates the 2 nd row vector of the matrix X. The control signal is given by: χs = Ds rs, us = Kχs, where Ds = /s, K = ki 2 2, k >, rt = [ ˆθtxt + [ ˆω t ˆω 2 t ut + σt K g rt, σs = H o sc si A m ˆσs = 47 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.
2 2 2 2 4 6 8 a y t solid, r t dashed and y des t dotted 2 2 3 a y t solid, r t dashed and y des t dotted 2 2 2 4 6 8 b y 2 t solid, r 2 t dashed and y des2 t dotted 2 2 3 b y 2 t solid, r 2 t dashed and y des2 t dotted 4 3 2 3 2 2 2 4 6 8 c Time-history of ut Fig.. Performance of L adaptive controller for r = [2 2 3 2 2 3 c Time-history of ut Fig. 3. Performance of L adaptive controller for rt = [ + sin.4t cost. 4 2 3 2 2 4 6 8 a y t solid, r t dashed and y des t dotted 2 2 3 a y t solid, r t dashed and y des t dotted 4 3 2 2 4 6 8 b y 2 t solid, r 2 t dashed and y des2 t dotted 8 6 4 2 2 4 6 8 c Time-history of ut Fig. 2. Performance of L adaptive controller for r = [4 4 [ ˆσs. Let ω =, ω 2 =., and θt = [.sin.t sin.3t., σt = [ + sin2t sin3t sint. We assume the following con- 2 2 3 b y 2 t solid, r 2 t dashed and y des2 t dotted 3 2 2 3 2 2 3 c Time-history of ut Fig. 4. Performance in the presence of disturbance defined in 3. servative bounds for the unknown time-varying signals for the implementation of the projection operator: ω i [. for i =,2, and θ i t, σ i t, for i =,2,3. It 48 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.
can be derived straightforwardly that L = 3, while Ḡs L can be calculated numerically. Letting k =, it can be verified numerically that the L -gain upper bound is satisfied for all possible ω i [,.. The simulation results of the L adaptive controller are shown in Figures and 2 for constant reference inputs r = [2 2 and r = [4 4, respectively. We note that it leads to scaled control inputs and scaled system outputs for scaled reference inputs. Figure 3 shows the system response and the control signal for the reference input rt = [+sin.4t cost, without any retuning of the controller. In Fig. 4, we consider a different signal σt = [ + 2sint 2sint sint. 3 We note that the L adaptive controller guarantees smooth and uniform transient performance in the presence of different unknown time-varying disturbances without any retuning of the controller. VIII. CONCLUSION In this paper, we extend the L adaptive controller to multi input multi output systems in the presence of unmatched time-varying disturbances. By appropriate modification of the control architecture, the L adaptive controller compensates for the effects of the unmatched disturbances on the system outputs. REFERENCES [ C. Cao and N. Hovakimyan. Design and analysis of a novel L adaptive control architecture, Part I: Control signal and asymptotic stability. In Proc. of American Control Conference, pages 3397 342, 26. [2 C. Cao and N. Hovakimyan. Design and analysis of a novel L adaptive control architecture, Part II: Guaranteed transient performance. In Proc. of American Control Conference, pages 343 348, 26. [3 C. Cao and N. Hovakimyan. Guaranteed transient performance with L adaptive controller for systems with unknown time-varying parameters: Part I. American Control Conference, pages 392 393, 27. [4 C. Cao and N. Hovakimyan. Stability margins of L adaptive controller: Part II. American Control Conference, pages 393 3936, 27. [ P. Ioannou and J. Sun. Robust Adaptive Control. Prentice Hall, 996. [6 H. K. Khalil. Nonlinear Systems. Prentice Hall, Englewood Cliffs, NJ, 22. [7 K. Zhou and J. C. Doyle. Essentials of Robust Control. Prentice Hall, Englewood Cliffs, NJ, 998. [8 J.B. Pomet and L. Praly. Adaptive nonlinear regulation: Estimation from the Lyapunov equation. IEEE Trans. Autom. Contr., 376:729 74, June 992. APPENDIX Proof of Lemma 2. The control law in 22 leads to the following closed-loop dynamics: x ref s = Gsrs + Ḡsηs HsFs σ refs +si A m σs + si A m x, 36 where Gs = HsFsK g, η s is the Laplace transformation of signal η t θtx ref t. Since η t L L x reft L, it follows from 23 and Lemma that x reft L Ḡs L L x reft L 37 + η 2t L + F ssi A m L σ t L, where η 2t has the following Laplace transformation: η 2s = Gsrs + si A m σs + si A m x. We note that the conditions 9- ensure that F ssi A m L is finite. The condition in 9 implies that Gs and Hs are stable, which further implies that η 2 L is finite, since rt, x and σt are bounded. Hence, it follows from 37 that x ref L η 2 L + F ssi A m L σ L Ḡs L L, which implies that x ref t is bounded and completes the proof. Proof of Theorem. Letting θt = ˆθt θt, σt = ˆσt σt, ωt = ˆωt ω, the following error dynamics can be derived from and 4 xt = A m xt + B θtxt + ωtut + σt 38 with x =. Consider the following candidate Lyapunov function: V xt, θt, ωt, σt = x tp xt+γ tr θ θ+ Γ tr ω ω + Γ σ σ. Using the projection based adaptation laws from, one has the following upper bound: V t x tq xt + 2Γ tr θ t θt + σ t σt. 39 The projection algorithm ensures that ˆθt Θ, ˆωt Ω and ˆσt for all t, and therefore max t tr θ t θt + tr ω t ωt + σ t σt 4 max θ Θ trθ θ + max ω Ω trω ω + max σ σ σ. 4 If V t θ m/γ at some t, then it follows from 4 that x tp xt 4 λmaxp Γλ min d Q θ max θ Θ tr θ θ + d σ max σ σ 2, and hence x tq xt λ minq x tp xt/λ maxp 4Γ d θ max tr θ θ + d σ max σ 2. 4 θ Θ σ The upper bounds in 2 along with the projection based adaptive laws lead to the following upper bound: tr θ t θt + σ t σt 2d θ max θ Θ tr θ θ + d σ max σ σ 2. 42 Hence, if V t θ m/γ, then it follows from 39, 4, 42 that V t. 43 Since we have set ˆx = x, we can verify that V 4 max θ Θ trθ θ + max ω Ω trω ω + max σ σ σ /Γ < θ m/γ. It follows from 43 that V t θ m/γ for all t. Since λ minp xt 2 x tp xt V t, then xt 2 θ m 2, which along with 24 implies that λ minpγ and proves 26. Let xt 2 γ, t 44 rt = ωtut + θtxt, ξ t = θtxt. 4 It follows from 6 that χs = Dsωus + σs + ξ s K grs + rs, where rs and ξ s are the Laplace transformations of signals rt and ξ t respectively. Consequently χs = I + DsωK Dsξ s + σs K grs + rs, us = KI+DsωK Dsξ s+ σs K grs+ rs. Using the definition of Fs from 8, we can write ωus = Fsξ s + σs K grs + rs, 46 and the system in consequently takes the form: xs = Gsrs + Ḡsξs HsFs rs HsFs σs + si A m σs + si A m x. 47 49 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.
Let et = xt x ref t. Then, using 36 and 47, we have es = Ḡsξ2s ξ3s ξ4s, 48 where ξ 3s = HsFs rs, ξ 4s = HsFs σs σ ref s, and ξ 2 s is the Laplace transformation of the signal ξ 2t = θtet. From 7 and 23, we have ξ 4s = F ssi A m σs, where F s is defined in 8. The definition of ξ 3s implies that ξ 3s = HsFsC Hs C Hs rs = HsFsHo sc Hs rs = F shs rs. From the relationship in 38 we have xs = Hs rs + si A m σs, 49 which leads to ξ 3s + ξ 4s = F s xs. Using 44, we can upper bound ξ 3 + ξ 4 L F s L x L F s L γ. For the error dynamics in 48, Lemma implies that e t L Ḡs L ξ 2t L + ξ 3 + ξ 4 t L. Using the definition of L in 2, one can verify easily that ξ 2t L L e t L, which along with and implies that e t L Ḡs L L e t L + F s L γ. 2 It follows from and 2 that e t L F s L γ for all Ḡs L L t, which proves 27. The upper bound in 28 follows from Lemma and the definition of yt and y ref t directly. To prove the bound in 29, we notice that from 22 and 46 that us u ref s = ω Fsξ 2s + η 4s + rs, where rt is defined in 4, η 4t = σt σ ref t. Hence, we have u u ref L ω Fs L L e L + η L, 3 where η s = ω Fsη 4s + rs. From 7 and 23, we have η s = ω Fs rs ω FsHo sc si A m σs = ω FsC Hs C Hs rs ω FsHo sc si A m σs = ω FsHo sc Hs rs +si A m σs which along with 49 implies that η s = ω FsH o sc xs. 4 and hence x ref s = I Ḡsθ Gsr/s HsFsHo sc si A m σ/s +si A m σ/s + si A m x. 6 It follows from the end value theorem that lim dest t = lim sy des s s = lim sh s osk gr/s = H ok gr, 7 lim reft t = lim sy ref s. s 8 Eq. 6 and the definition of y ref = C x ref imply that lim y reft = lim sy ref s = C I Ḡθ Gr t s + A m σ HFHo C A m σ. 9 Since 9 implies that F = I, we have G = HK g and Ḡ =. Using the definition of H o = C H and the relationship in 9, we can write: lim sy refs = C HK s gr + C A m σ H o Ho C A m σ = H o K g r. 6 The limiting expression in 33 follows from 7 and 6 directly. Proof of Lemma 4. It follows from 23 and 36 that x ref s = Gsrs + Ḡsηs and hence HsFsHo sc si A m σs +si A m σs + si A m x, y ref s = H osfsk grs + H osi Fsη s H osfsho sc si A m σs +C si A m σs + C si A m x, = y des s H osi FsK grs + H osi Fsη s + H osi FsHo sc si A m σs. It follows from Lemma and the definitions of F 3s, F 4s that y ref y des L F 3s L L x ref L + K gr L + F 4s L σ L, 6 which concludes the proof. It follows from 44 and 4 that η L ω FsH o sc L γ. Hence, the relationships in 27, 3, and the definition of F 2s in 9 imply that u u ref L ω Fs L Lγ + F 2s L γ, which proves 29. Proof of Lemma 3. If θ, r and σ are constant, it follows from 23 and 36 that x ref s = Gsr/s + Ḡsθx refs HsFsHo sc si A m σ/s +si A m σ/s + si A m x, 4 Authorized licensed use limited to: UNIVERSITY OF CONNECTICUT. Downloaded on February 2, 29 at 9:44 from IEEE Xplore. Restrictions apply.