Predictive Iterative Learning Control using Laguerre Functions

Transcription

1 Milano (Italy) August 28 - September 2, 211 Predictive Iterative Learning Control using Laguerre Functions Liuping Wang Eric Rogers School of Electrical and Computer Engineering, RMIT University, Victoria 3, Australia, ( liuping.wang@rmit.edu.au) School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK, ( etar@ecs.soton.ac.uk) Abstract: This paper develops a predictive iterative learning control algorithm starting from some recent results in the area of predictive repetitive control. Although similar in spirit to the original norm-optimal iterative learning control, the algorithm developed employs receding horizon control and Laguerre functions to parameterize the future control trajectory. As a result, difficulties encountered by the norm-optimal iterative control algorithm are overcome. The stability of the predictive iterative learning system is analyzed and conditions on error convergence are established. In addition, a strategy for learning the unknown plant initial conditions and the reduction of the initial plant errors is developed. A simulation example to illustrate the algorithm is also given. Keywords: iterative learning control, predictive control, Laguerre functions 1. INTRODUCTION Repetitive control has been developed for cases where the processorplantoutputisrequiredtotrackagivenperiodic signal, where the novel feature is that information from previous periods or trials is used to modify the control signal Hara et al. [1988]. In a general sense, the reference signal considered in a repetitive control is captured by a periodic function in time. Closely related is iterative learning control Bristow et al. [26], Ahn et el. [27], which is applicable to systems operating in a repetitive manner. However, in iterative learning control, the task is to follow a pre-defined reference trajectory over a specific finite time interval. When one cycle, or trial, of the task is completed, the process is reset to the starting position for the start of the next one. The central idea in iterative learning control is to sequentially improve the tracking performance by using the information obtained from the previous trials, possibly in combination with the current trial information Bristow et al. [26], Ahn et el. [27]. One extensively studied class of iterative learning control algorithms is based on the minimization of an objective function constructed from the addition of two sums of squares terms. The first of these is formed from the current trial error, that is, the difference between the supplied reference signal and the current trial output, and the second from the difference between the control signals used on successive trials. This class of algorithms istermednormoptimalandexperimentalverificationofits performance has also been reported Amann et al. [1996], Rogersetal.[21].Fortheliterature,onestartingpoint is the survey articles Bristow et al. [26], Ahn et el. [27]. This paper considers the design of a predictive iterative learning controller that uses a similar cost function that in norm optimal iterative learning control but with the referencesignalmodelembeddedinthelearningcontroller, and the receding horizoncontrolprinciple. The idea of embedding the reference signal information in the controller wassuccessfullyusedin for,example,wangetal.[21a,b] predictive repetitive control. This paper shows that the reference signals provide a wealthy of information that can also be embedded in the iterative learning control. Using the internal model controlprinciple, it is shown that this embedded reference model has advantages in terms of along the trial dynamics. In iterative learning control, when one cycle, or trial, of the task is completed, the process is reset to the starting positionforthestartofthenextone,andhencetheprocess initial conditions play an important role in the error reduction from trial-to-trial. Instead of an arbitrary reset, this paper also develops a method for reliably selecting the plant state initial vector to avoid large amplitude control signals in the initial phase of the new trial. This paper is organized as follows. Section 2 discusses the embedded disturbance model. Section 3 develops the control law and Section 4 analyzes the stability of the controlled system and the convergence conditions for the errors between the reference trajectory and the trial output. Section 5 presents a simulation study to demonstrate efficacy of the new algorithm and the paper is completed in Section 6 with conclusions and suggestions for further research. 2. EMBEDDING THE REFERENCE SIGNAL IN ITERATIVE LEARNING CONTROL In the general setting of iterative learning control (ILC), the reference signal and the trial initial conditions are pre- Copyright by the International Federation of Automatic Control (IFAC) 5747

2 Milano (Italy) August 28 - September 2, 211 defined for the trials. In some applications, the reference signal is permitted to change from trial-to-trial Freeman et al. [211]. Suppose that the plant to be controlled has m inputs and outputs with state-space model x m (k +1)=A m x m (k)+b m u(k)+ω m µ(k), (1) y(k)=c m x m (k), (2) where x m (k) is the n 1 1 state vector, u(k) and y(k) are the m 1 input and output vectors respectively, and µ(k) is the m 1 is the vector used to embed the chosen reference signal frequency components in the model used for design. In the multiple-input multiple-output (MIMO) case, for each entry in µ a z-transform description Wang et al. [21b] exists with poles on the unit circle that model the frequency components to be included. Once this is completed for each entry a least common denominator D(z) of the m z transfer-functions can be formed that has the form D(z)=(1 z 1 )Π l (1 2cos(lω)z 1 +z 2 ) =1+d 1 z 1 +d 2 z 2 +d 3 z d γ z γ. (3) corresponding to frequency components of zero and hω, h = 1,2,...,l, for some chosen positive integer l. In the time domain and in the steady-state, µ(k) is described by the following difference equation in the backward shift operator q 1 D(q 1 )µ(k) = O D, (4) where O D denotes the m 1 zero vector. Also define the following auxiliary vectors using the disturbance model x s (k) = D(q 1 )x m (k), u s (k) = D(q 1 )u(k) (5) where x s (k) and u s (k) are the filtered state and control vectors. Then applying the operator D(q 1 ) to both sides of the state equation in (2) gives D(q 1 )x m (k +1)=A m D(q 1 )x m (k)+b m D(q 1 )u(k), or +Ω m D(q 1 )µ(k), (6) x s (k +1) = A m x s (k)+b m u s (k), (7) where the relation D(q 1 )µ(k) = O D has been used. Similarly, application of the operator D(q 1 ) to both sides of the output equation in (2) gives D(q 1 )y(k +1)=C m x s (k +1) =C m A m x s (k)+c m B m u s (k), (8) and expanding the right-hand side of (8) gives y(k +1)= d 1 y(k) d 2 y(k 1)... d γ 1 y(k γ +1) d γ y(k γ)+c m A m x s (k)+c m B m u s (k). (9) Introduce the new state vector x(k) = [ x T s (k) y T (k) y T (k 1)... y T (k γ) ] T to obtain the following the augmented state-space model to be used in controller design x(k +1)=Ax(k)+Bu s (k) y(k)=cx(k) (1) where A m O O... O O C m A m d 1 I d 2 I... d γ 1 I d γ I O I O... O O A = O O... I O O O O... O I O B = B m C m B m O. O O C = [O I O... O O] and from this point onwards, except where explicitly stated, and I denote the zero and identity matrices, respectively, of compatible dimensions. The system matrix A in the augmented deign model here is block lower diagonal and its characteristic equation is where det(zi A) = det(zi A m )det(zi A d ) = (11) d 1 I d 2 I... d γ 1 I d γ I O I O... O A d = O O... I O In the ILC setting the trial number is denoted by j =,1,2,...,, and the process state-space model (1) is rewritten as x j (k +1)=Ax j (k)+bu j s (k) y j (k)=cx j (k) (12) where x j (k), u j s (k) and yj (k) denote the state, filtered control and output vectors, respectively, at sampling instant k on trial j. The state vector x j (k) is formed from two sub-vectors, the first of which is x j s (k) and is equal to the process state vector x j m(k) filtered by D(z 1 ). In the steady-state this vector is zero. If it is assumed that the process has reached the steady-state before a trial commences, the state initial vector on each trial can be assumed to be zero. The second sub-vector y j (k) and its initial condition initial condition on trial j is known since this vector is a measured output. To complete the ILC problem formulation, let r(k) denote the reference vector to be tracked. Then the error on trial j is e j (k) = y j (k) r(k) In the ILC analysisthat followsin this paper, x j (k) in (12) is formed as x(k) = [ x T s (k) et (k) e T (k 1)... e T (k γ) ] T 5748

3 Milano (Italy) August 28 - September 2, PREDICTIVE ITERATIVE LEARNING CONTROL The analysis in this paper is for MIMO systems but, for ease of presentation only, a single input is considered until the introduction of the cost function below. On trial j + 1 and sampling instant k i, the future state vector along this trial is predicted as m 1 x j+1 (k i +m k i ) = A m x j+1 (k i )+ A m i 1 B s (i). Letus j+1 (i)for i m 1bedescribedbytheexpansion of Laguerre functions with coefficients represented using the vector form us j+1 (i) = L T (i)η j+1 (13) i= Also it is assumed that the Laguerre function vector L(i) remains unchanged from trial-to-trial and satisfies the difference equation L(i+1) = A l L(i) (14) where matrix A l has dimensions N N, and has the structure a β a... A l =. aβ β a N 2 β a N 3 β... β a with β = (1 a 2 ), and initial condition L T () = β [ 1 a a 2 a 3... ( 1) N 1 a N 1] The Laguerre functions are defined once the scaling factor a and the number of term N are selected. With the control trajectory represented by a Laguerre polynomial, the predicted state variable vector can be expressed as x j+1 (k i +m k i ) = A m x j+1 (k i )+φ T (m)η j+1 (15) where φ T (m) = m 1 i= Am i 1 BL T (m), and this term is invariant from trial-to-trial. The cost function for predictive ILC is selected to be of the form J = m=1 + x j+1 (k i +m k i ) T Qx j+1 (k i +m k i ) ( m= s (m) u j s(m)) T R( s (m) u j s(m)) (16) where Q and R are symmetric positive semi-definite and positive definite matrices, respectively, and also the difference between the control signals on the current trial previous trials is penalized. The motivation for this is to achieve trial-to-trialerrorreduction without unduely large changes in the amplitudes of the control signals required. The previous trial filtered state vector is also parameterized in the form of (13) with a long prediction horizon, N p 1 m= N p 1 m= N p 1 m= u j s(m) T Ru j s(m)=(η j ) T R L η j (17) s (m) T Ru j s (m)=(ηj+1 ) T R L η j (18) s (m) T R s (m)=(η j+1 ) T R L η j+1 (19) where the orthonormal property of the Laguerre functions has been used, that is, m= L(m)T L(m) = I, and R L is and N N diagonal matrix. Substituting (15) and (17) (19) into (16) gives where Ω = J =(η j+1 ) T Ωη j+1 +2(η j+1 ) T Ψx j+1 (k i ) m=1 2(η j+1 ) T R L η j +(η j ) T R L η j (2) φ(m)qφ T (m)+r L, Ψ = m=1 φ(m)qa m The minimum value of this cost function occurs when η j+1 = Ω 1 (Ψx j+1 (k i ) R L η j ) (21) With the receding horizon control, only the first sample of the optimal control trajectory is implemented, which is constructed as the filtered control signal on trial j +1 at sample k i, s (k i ) = L T ()η j+1 (22) If there is no resetting, the control law becomes predictive repetitive control of the form s (k i ) = L T ()Ω 1 Ψx j+1 (k i ) (23) 4. ANALYSIS OF CLOSED-LOOP STABILITY AND CONVERGENCE OF THE ERROR Consider the case of repetitive-predictive control given by (23). Then the resulting closed-loop system is guaranteed to be stable for a large prediction horizon and a large N in the Laguerre expansion, and an exponential weight α such that the control law is identical to the solution of a Linear Quadratic Regulator (LQR) problem Wang. [29]. In the ILC case it is necessary to obtain conditions under which the controlled system stable and trial-to-trial error converges. Substituting η j+1 in the control law (21) gives s (k)= L() T Ω 1 Ψx j+1 (k)+l() T Ω 1 R L η j (24) and, to simplify the notation, introduce K mpc = L() T Ω 1 Ψ For j =, with the assumption that η 1 is a zero vector, the filtered control signal is u 1 s(k) = K mpc x 1 (k) (25) and with the control applied, x 1 (k +1) = (A BK mpc )x 1 (k) = A cl x 1 (k) (26) 5749

4 Milano (Italy) August 28 - September 2, 211 For j = 1, the filtered control signal is u 2 s(k) = K mpc x 2 (k) K 1 x 1 (k) (27) where K 1 = L() T Ω 1 R L Ω 1 Ψ, and x 2 (k +1) = A cl x 2 (k) BK 1 x 1 (k) (28) For an induced matrix norm, and since all eigenvalues of A BK mpc ] have modulus strictly less than unity, there exist constants < M < and < λ < 1 such that (A BK mpc ) k Mλ k. Hence x 1 (k) Mλ k x 1 () (29) When j = 1, the filtered control signal is u 2 s (k) = K mpcx 2 (k) K 1 x 1 (k) (3) 5. A CASE STUDY Thissection isbasedonatwo-inputandtwo-outputmodel obtainedfromfrequencydomaintestsonanthropomorphic robotic arm see Fig. 1, undertaking a pick and place operation. Its end-effector travels between the pick and place locations in a straight line using joint reference trajectories that minimize the end-effector acceleration. Having reached the place location, the robot then returns back to the starting location. Positional and velocity control loops have been implemented around each joint to provide baseline performance. The overall system model is described by the transfer-function matrix G(s) of (32). which was sampled at.5 sec. The reference trajectories and x 2 (k +1) = (A BK mpc )x 2 (k) BK 1 x 1 (k) =(A BK mpc )x 2 (k) BK 1 (A BK mpc ) k x 1 () (31) where (26) has been used. With the initial condition x 2 () given, the solution of (31) is x 2 (k)=(a BK mpc ) k x 2 () (A BK mpc ) k i 1 BK 1 (A BK mpc ) i x 1 () k 1 i= and x 2 (k) is bounded by x 2 (k) Mλ k x 2 () Mλ k i 1 BK 1 Mλ i x 1 () k 1 + i= =Mλ k x 2 () +M 2 kλ k 1 BK 1 x 1 () and this formulaextends in anaturalmanner foranyvalue of j and x j (k) is bounded by x j (k) Mλ k x j () +M 2 kλ k 1 BK 1 x j 1 () Hence the initial conditions for each trial affect the error. Also stability of the closed-loop system, and the rate of trial-to-trial error convergence, is determined by the value of λ. The initial condition vector x j () is unknown since the entries in x j s() are unknown. Hence it is necessary to estimate the initial conditions for the controller and set the plant operating conditions for the next trial, where ideally x j () should be as small as possible. In order to reduce the error in the initial conditions, one approach is to design the ILC such that it operates in a learning mode when the process is resetting prior to the start ofthe next trial. Assuming that there are N T samples along the trial, the referencevectorfor this learningphase is r L (k) = r(n T k), where k = 1,2,..., N T. Once this operation is completed, the process will hold its operational conditions learned in the reverse, and the system is ready to begin the next trial. Fig. 1. Picture of the robot arm with the pick and place locations marked. for the pick and place tasks are of 4 seconds duration (4 samples) and are shown as the first 4 samples in Fig. 3. The second 4 samples shown are used during the resetting phase between successive trials to construct the initial state vector for the new trial as described in the previous section. Using Fourier analysis of the 4 sample period reference signals leads to the inclusion of three frequencies and D(z)=(1 z 1 )(1 2cos(2π/4)z 1 z 2 ) (1 2cos(4π/4)z 1 z 2 ) =1 6.7z z z z z z 6 z 7. (33) Consider now the case when the weighting matrix Q in the cost function has the form [ ] o1 o Q = 2 o 3 I 2 where o 1, o 2 and o 3 are zero matrices with appropriate dimensions, and I 2 is an identity matrix with dimension The weighting matrix R = I, the prediction horizon is 4, the Laguerre scaling factor is a =.6, and the number of terms used for both inputs is given by N = 18. Suppose also that the system is designed Anderson et al. [1971] to place all eigenvalues of A BK mpc inside the circle of radius λ =.7 in the complex plane (see also Wang. [29]) as shown in Fig. 2. To generate the simulation results shown below, all entries in the plant state initial vector where selected to be unity, and all other initial conditions were set to zero. Table 1 compares the sum of squared errors over 4 trials and from this table it is seen that the learning process during the 575

5 Milano (Italy) August 28 - September 2, 211 [ ] [ y1 (s) = y 2 (s) G 11 (s) G 21 (s)g 22 (s) G 12 (s) ][ ] u1 (s) u 2 (s) (32).16s s s e4s e5s e6s e7s e8s e8s + 7.6e8 G 11(s) = 5.25e 5s s s s s e4s e5s e6s e7s e7s e8s e8s + 7.6e8 G 22(s) =.22s s s s 4 1.6e4s e4s 2 G 12(s) = 5.25e 5s s s 8 + 2s s s e4s e5s e6s e6s e6.16s 7 8.7s 6 194s s e4s e4s 2 G 21(s) = 5.25e 5s s s s s s e4s e5s e6s e6s + 5.3e6.27s s s s e5s e6s e7s e7s e8s + 7.6e8 5.25e 5s s s s s e4s e5s e6s e7s e7s e8s e8s + 7.6e Imag Real Time (sec) 4 2 Fig. 2. Locations of the eigenvalues of A BK mpc relative to the unit circle for the design example with λ =.7. Table 1. Sum of squared errors for the errors in each channel against trial number. j (iteration) e e resetting finds the correct initial plant condition and rapid trial-to-trial error convergence is clearly occurring. In Fig. 3 the first 4 samples in each plot show the outputs and error for the 2 channels on trial 4 and confirm the conclusion on the performance of this design. Figure 4 shows the corresponding control inputs required. It is essential to note that the performance of this design critically depends on the choice of λ, where, for example, the performance on trial 4 is much degraded if λ where chosen such that a complex conjugate pair of eigenvalues of A BK mpc have magnitude CONCLUSIONS This paper has developed a predictive ILC design that uses a similar cost function to the that used in the normoptimal ILC. In its design,the dominant frequencies in the reference signals are embedded and receding horizon control principle is used to construct the control applied on each trial. It is shown that stability and trial-to-trial error convergence is critically dependent on the prescribed degree of stability < λ < 1 imposed on the eigenvalues of Time (sec) Fig. 3. Plots of the outputs and errorsfor the two channels on trial 4. Top to bottom: y 1, e 1, y 2, and r 2, respectively. the state matrix of the controlled system. The paper has also proposed a strategy to reduce the initial errors for the trials by tracking the reference signals in reverse when the system operates in resetting mode. A simulation study on a robot arm executing a pick and place operation that represents many systems to which ILC is applicable has demonstrated the efficacy of the design. Future research should include the development of tools to assist in the selection of λ and experimental verification. Moreover, the design should be extended to allow constraints on, for example, the amplitudes of the allowed control signals. REFERENCES B. D. O. Anderson and J. M. Moore, Linear Optimal Control, Prentice-Hall, Hemel Hempstead,

6 Milano (Italy) August 28 - September 2, Time (sec) Fig. 4. Control inputs on trial 4, top plot u 1, bottom u 2. N. Amann, D. H. Owens, and E. Rogers. Iterative learning control using optimal feedback and feedforward actions. International Journal of Control, volume 65, number 2, pages , S. Hara,Y. Yamamoto,T. Omata, andm. Nakano. Repetitive control system: a new type servo system for periodic exogenoussignals.ieee Transaction on Automatic Control, volume 33, number 7, pp , D. A. Bristow, M. Tharayil, and A. G. Alleyne. A survey of iterative learning control. IEEE Control Systems Magazine, volume 26, number 3, pp , 26. H. S. Ahn, Y. Chen, and K. L. Moore. Iterative learning control: brief survey and categorization. IEEE Transactions on Systems, Man and Cybernetics, Part C, volume 37, number 6, pp , 27. C. T. Freeman, Z. Cai, E. Rogers, and P. L. Lewin. Iterative learning control for multiple point to point tracking. IEEE Transactions on Control Systems Technology, 21, (In Press). E. Rogers, D. H. Owens, H. Werner, C. T. Freeman, P. L. Lewin, S. Kichhoff, C. Schmidt, and G. Lichtenberg. Normoptimaliterativelearningcontrolwithapplication to problems in accelerator based free electron lasers and rehabilitation robotics. European Journal of Control, volume 16, number 5, pages , 21. L. Wang, P. Gawthrop, D. H. Owens, and E. Rogers. Switched linear predictive controllers for periodic exogenous signals. International Journal of Control, volume 83, number 4, pp , 21. L.Wang,S.Chai,andE.Rogers, Predictiverepetitivecontrol based on frequency decomposition. 21 American Control Conference, July 21, pp Model Predictive Control System Design and Implementation Using MATLAB, 1st ed. Springer London,