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Design of Nonlinear Predictive Control for Pneumatic Muscle Actuator Based on Echo State Gaussian Process*
Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July 2017 The International Federation of Automatic Control Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 1952–1957 Design of Nonlinear Predictive Control for Design of Nonlinear Predictive Control for Design of Nonlinear Predictive Control for Design of Nonlinear Predictive Control for Pneumatic Muscle Actuator Based on Echo Pneumatic Muscle Actuator Based on Echo Pneumatic Muscle Actuator Based on Echo Pneumatic Muscle Actuator Based on Echo State Gaussian Process State Gaussian Process State Gaussian Process State Gaussian Process ∗ ∗,† ∗ ∗
Cao Jian Huang Huang ∗,† Gangzheng Gangzheng Ding Ding ∗∗ Yongji Yongji Wang Wang ∗∗ Cao ∗∗ Jian Cao ∗ Jian Huang ∗,† ∗,† Gangzheng Ding ∗ Yongji Wang ∗ Cao Jian Huang Gangzheng Ding Yongji Wang ∗ School of Automation, Huazhong University of Science and ∗ School of Automation, Huazhong University of Science and Automation, Huazhong University of Science and Technology, 430074 China (Tel:+86-27-87558472; ∗ School of Wuhan Technology, 430074 China P.R. P.R. (Tel:+86-27-87558472; School of Wuhan Automation, Huazhong University of Science and Technology, Wuhan 430074 China P.R. (Tel:+86-27-87558472; e-mail: huang [email protected]). e-mail: 430074 huang [email protected]). Technology, Wuhan China P.R. (Tel:+86-27-87558472; e-mail: huang [email protected]). e-mail: huang [email protected]). Abstract: Recently, the application Abstract: Recently, the application of of Pneumatic Pneumatic Muscle Muscle Actuators Actuators (PMAs) (PMAs) for for driving driving Abstract: Recently, the become application of Pneumatic Muscle AActuators (PMAs) for driving rehabilitation robots has a matter of great concern. traditional control algorithm, rehabilitation robots has become a matter of great concern. AActuators traditional control for algorithm, Abstract: Recently, the application of Pneumatic Muscle (PMAs) driving rehabilitation robots has become a matter high-precision of great concern. A traditional control algorithm, such as cannot achieve satisfactory performance in tracking such as PID, PID, robots cannot has achieve satisfactory performance in trajectory trajectory tracking rehabilitation become a matter high-precision of great concern. A traditional control algorithm, such as for PID, cannot achieve satisfactory high-precision performance in trajectory tracking problem PMAs, due to PMAs’ features of nonlinear effects, slow response time, time-varying problem for PMAs, dueachieve to PMAs’ features ofhigh-precision nonlinear effects, slow response time, time-varying such as PID, cannot satisfactory performance in trajectory tracking problem for PMAs, due towe PMAs’ features of nonlinear effects, slow response time, time-varying parameters. In this study proposed a nonlinear predictive control strategy SNN-ESGP, which parameters. In this study we proposed a nonlinear predictive control strategy SNN-ESGP, which problem for PMAs, due to PMAs’ features of nonlinear effects, slow response time, time-varying parameters. Inofthis studymodel we proposed a nonlinear predictiveprocess control(ESGP) strategy that SNN-ESGP, which is comprised a novel called echo state Gaussian is suitable for is comprisedInofthis a novel model called echo state Gaussian process (ESGP) that is suitable for parameters. study we proposed a nonlinear predictive control strategy SNN-ESGP, which is comprised of a novel model called as echo state Gaussiantheir process (ESGP) that issingle suitable for modeling nonlinear unknown systems well as measuring uncertainties, and a neural modeling nonlinear unknown systems as well as measuring their uncertainties, and a single neural is comprised of a novel model called echo state Gaussian process (ESGP) that is suitable for modeling(SNN) nonlinear unknown systems as well as measuring their uncertainties, and a single neural network serves as the controller of the system. To analyze the system convergence, we network (SNN) serves as thesystems controller of the system. Totheir analyze the system convergence, we modeling nonlinear unknown as well as measuring uncertainties, and a single neural network (SNN) serves as the controller of deduce the system. To analyze the system convergence, we utilize the gradient descent algorithm and the iterative rules of the weights of SNN. The utilize the(SNN) gradient descent algorithm and deduce the iterative rulesthe of the weights of SNN. The network serves as the controller of the system. To analyze system convergence, we utilize theofgradient descent algorithm and deduce the iterative rulestheorem. of the weights of SNN. The stability the closed-loop system is analyzed by using Lyapunov Finally, studies of stability ofgradient the closed-loop system is analyzed by the using Lyapunov theorem. Finally, studies of utilize the descent algorithm and deduce iterative rules of the weights of SNN. The stability experiments of the closed-loop system is analyzed bycontrol using strategy Lyapunovwith theorem. Finally, trajectory studies of physical illustrate the validity of the high-precision physical experiments illustrate the validity of the control strategy with high-precision trajectory stability of the closed-loop system is analyzed by using Lyapunov theorem. Finally, studies of physical experiments illustrate the validity of the control strategy with high-precision trajectory performance. performance. physical experiments illustrate the validity of the control strategy with high-precision trajectory performance. performance. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: predictive control, Keywords: Pneumatic Pneumatic muscle muscle actuators, actuators, nonlinear nonlinear predictive control, echo echo state state gaussian gaussian Keywords: Pneumatic muscle actuators, nonlinear predictive control, echo state gaussian process, single neural network process, single neural network Keywords: Pneumatic muscle actuators, nonlinear predictive control, echo state gaussian process, single neural network process, single neural network 1. most 1. INTRODUCTION INTRODUCTION most utilized utilized algorithm, algorithm, owing owing to to its its easy easy implementation implementation 1. INTRODUCTION most utilized algorithm, owing to its easy implementation and preferable performance (Sun, M. et (2014)). and preferable performance (Sun, M.easy et al. al. (2014)). But But 1. INTRODUCTION most utilized algorithm, owing to its implementation and preferable performance (Sun, M. et al. (2014)). But it is difficult for PID controller to achieve a satisfactory Robot-assisted therapies have been applied to improve it is preferable difficult forperformance PID controller to M. achieve a(2014)). satisfactory (Sun, et al. But Robot-assisted therapies have been applied to improve and it is difficult for PID controller to achieve a satisfactory result in a high-precision trajectory tracking task, Robot-assisted therapies process have been applied to improve result in a high-precision trajectory tracking task, and the overall in years(Hussain and it is difficult for PID controller to achieve a satisfactory the overall rehabilitation rehabilitation process in recent recent years(Hussain Robot-assisted therapies have been applied to improve result in a high-precision trajectory tracking task, supand a PID controller suffers aa lack of the overall rehabilitation process in recent years(Hussain et al. (2013)). Compared with rehabilitative PID in controller suffers from from lack tracking of theoretical theoretical supa high-precision trajectory task, and et al. (2013)). Comparedprocess with traditional traditional rehabilitative aresult the overall rehabilitation in recent years(Hussain a PIDforcontroller suffers fromstability. a lack ofThe theoretical support proving the system sliding mode et al. (2013)). Compared with traditional rehabilitative therapies, the robot-assisted therapies can increase the proving the system sliding mode PIDfor controller suffers fromstability. a lack ofThe theoretical suptherapies, the robot-assisted therapies can rehabilitative increase the aport et al. (2013)). Compared with traditional port for proving the system stability. The sliding mode (SMC) technique is common method for therapies, the robot-assisted therapiesstaff can and increase the control intensity, save the enhance control is another another common method for for(SMC) provingtechnique the system stability. The sliding mode intensity, save the efforts efforts of of medical medical staff and enhance therapies, save the robot-assisted can and increase the port control (SMC) technique is another common method for nonlinear control (e.g., Cai et al. (2003)). Our previous intensity, the efforts of therapies medical staff enhance rehabilitated efficiency (Chen, S. H. et al. (2016)). Meannonlinear control (e.g., Cai et al. (2003)). Our previous (SMC) technique is another common method for rehabilitated efficiency (Chen, S. H. et staff al. (2016)). Mean- control intensity, save the efforts of medical and enhance nonlinear controla (e.g., Cai etdisturbance al. (2003)).observer Our previous work proposed non-linear based rehabilitated efficiency (Chen, S. H. et (PMAs) al. (2016)). Meanwhile, Pneumatic Muscle Actuators server as proposed a (e.g., non-linear disturbance observer based nonlinear control Cai et al. (2003)). Our previous while, Pneumatic Muscle Actuators (PMAs) server as work rehabilitated efficiency (Chen, S. H. et al. (2016)). Meanproposed a non-linear disturbance observer based sliding mode while depends the while, Pneumatic Muscle Actuators (PMAs) server as work compliant actuators that an role in slidingproposed mode control, control, while it itdisturbance depends on onobserver the identified identified a non-linear based compliant actuatorsMuscle that play play an important important roleserver in reharehawhile, Pneumatic Actuators (PMAs) as work sliding mode(Xing control, while it depends ondevelopment the identified PMA model et al. (2010)). With the of compliant actuators that play an important role in rehabilitation robot application (Noritsugu et al. (1997)), due PMA model (Xing et al. (2010)). With the development of mode control, while it depends on the identified bilitation robot application (Noritsugu et al. role (1997)), due sliding compliant actuators that play an important in rehamodel (Xing et al. (2010)). With the development of artificial intelligence, the neural networks have attracted bilitation robot application (Noritsugu et al. (1997)), due PMA to their high force-to-weight ratio, no mechanical parts, artificial intelligence, the neural networks have attracted PMA model (Xing et al. (2010)). With the development of to their high force-to-weight ratio, no et mechanical parts, bilitation robot application (Noritsugu al. (1997)), due intelligence, the of neural networks have attracted great attention, because their excellent capabilities of to their high force-to-weight ratio,and no low mechanical parts, artificial lower compressed air consumption, cost (Caldwell great attention, because of their networks excellent have capabilities of artificial intelligence, the neural attracted lower compressed air consumption, and low cost (Caldwell to their high force-to-weight ratio, no mechanical parts, great attention, because of their excellent capabilities of modelling uncertain, nonlinear, and complex systems lower compressed air consumption, and low cost (Caldwell et al. And PMAs are to muscle modelling uncertain, nonlinear, andexcellent complexcapabilities systems (Han (Han attention, because of their of et al. (1995)). (1995)). Andair PMAs are analogous analogous to skeletal skeletal muscle great lower compressed consumption, and low cost (Caldwell modelling uncertain, nonlinear, and complex systems (Han et al. (2016)). In addition, due to model-free approximaet al.only (1995)). And PMAs are analogous to skeletal muscle not in size and force-output but the behavior that et al. (2016)). In addition, due and to model-free approximauncertain, nonlinear, complex systems (Han not only in size and force-output but the behavior that modelling et al. (1995)). And PMAs are analogous to skeletal muscle et al.and (2016)). In addition, due to model-free approximation stability analysis neutral networks, they not only in size decompressed and force-output but the behavior that the pressurized, processes of are just stability analysis of of neutral networks,approximathey have have et al.and (2016)). In addition, due to model-free the pressurized, decompressed processes of PMAs PMAs are that just tion not pressurized, only in size decompressed and force-output but the behavior and stability analysis of neutral networks, they have brought great convenience for modelling systems. the processes of PMAs are just tion similar to muscle contraction. Unlike conventional robotic brought great convenience for modelling systems. tion and stability analysis of neutral networks, they have similar to muscle contraction. Unlike conventional robotic the pressurized, decompressed processes of PMAs are just brought great convenience for modelling systems. similar to muscle contraction. Unlike conventional robotic actuators, PMAs can necessary compliance brought great convenience for modelling systems. actuators, PMAs contraction. can also also provide provide necessary compliance A remarkably efficient neural network structure for similar to muscle Unlike conventional robotic remarkably efficient neural network structure for rereactuators, PMAs can also provide necessary compliance A which makes the processes safer. A remarkably efficient neural network structure for rewhich makes the rehabilitation rehabilitation processes safer.compliance current neural networks (RNNs) was proposed by actuators, PMAs can also provide necessary current neural efficient networksneural (RNNs) was first first proposed by which makes the rehabilitation processes safer. A remarkably network structure for recurrent neural networks (RNNs) was first proposed by Jaeger (Jaeger,H. (2001)) called echo state network (ESN). which makes the rehabilitation processes safer. However, high-precision trajectory tracking tasks of the Jaeger (Jaeger,H. (2001)) called echo state network (ESN). neural networks (RNNs) was first proposed by However, high-precision trajectory tracking tasks of the current (Jaeger,H. (2001)) calledneural echo state network (ESN). Compared with conventional networks, ESN has However, high-precision trajectory trackingproblems, tasks of due the Jaeger PMA systems turn out to be challenging Compared with conventional neural networks, ESN has Jaeger (Jaeger,H. (2001)) called echo state network (ESN). PMA systems turn out to be challenging problems, due However, high-precision trajectory tracking tasks of the Compared with inconventional neural networks, ESN of hasa superior ability capturing dynamic behaviours PMA systems turnnature, out tohysteresis be challenging problems, due to their and paability capturing the the dynamic behaviours Compared with in neural networks, ESN of hasa to their nonlinear nonlinear nature, hysteresis and time-varying time-varying pa- superior PMA turnnature, out tohysteresis be challenging problems, due superior ability inconventional capturing the dynamic behaviours of a complex nonlinear system. But in practical applications, to theirsystems nonlinear andFor time-varying parameters (Andrikopoulos et al. (2014)). such kinds of complex nonlinear system. But in practical applications, superior ability in capturing the dynamic behaviours of a rameters (Andrikopoulos et al. (2014)). For such kinds of to their nonlinear nature, et hysteresis andFor time-varying panonlinear system. But in practical applications, it may be ill-posed and accompanied with large output rameters (Andrikopoulos al. (2014)). such kinds of complex unknown and nonlinear plants, the conventional proporit may benonlinear ill-posedsystem. and accompanied with large output complex But in practical applications, unknown and nonlinear plants, the conventional proporrameters (Andrikopoulos et al. (2014)). For such kinds of it may be ill-posed and accompanied training with large output weights since the frequently-used method of unknown and nonlinear plants, the conventional proportional integral derivative is weights since the most most frequently-used method of may be ill-posed and accompanied training with large output tional integral derivative (PID) (PID) controller controller is one one of of the the it unknown and nonlinear conventional since the most frequently-used training method of ESN is pseudoinverse (Li et al. (2012)). tional integral derivativeplants, (PID) the controller is oneproporof the weights ESN is pseudoinverse (Li et al. (2012)). weights since the most frequently-used training method of tional integral derivative (PID) controller is one of the This work was supported by the National Natural Science Foun This work was supported by the National Natural Science FounESN is pseudoinverse (Li et al. (2012)). ESN is pseudoinverse (Li et al. (2012)). Recently, a novel Bayesian approach towards reservoir dation of China under Grant 61473130, the Science Fund for DisThis work was supported by the National Natural Science FounRecently, a novel Bayesian approach towards reservoir dation China Grant the Natural Science Fund forFounDisThis of work was under supported by61473130, the National Science Recently, a isnovel Bayesian approach towards reservoir computing proposed by Demiris called state tinguished Young Scholars of Hubei Province (2015CFA047), the dation of China under Grant 61473130, the Science Fund for Discomputing proposed by Y. Y.approach Demiris towards called echo echo state tinguished Youngunder Scholars of 61473130, Hubei Province (2015CFA047), the Recently, a is novel Bayesian reservoir dation of China Grant the Science Fund for DisFundamental Research Funds the Province Central Universities (HUST: computing is proposed by Y. Demiris called echo state tinguished Young Scholars of for Hubei (2015CFA047), the Gaussian process (ESGP) that is essentially a fusion of Fundamental Research Funds for the Province Central Universities (HUST: Gaussian process (ESGP) that is essentially a fusion of computing is proposed by Y. Demiris called echo state tinguished Young Scholars of Hubei (2015CFA047), the 2015TS028) and the Program Century Universities Excellent Talents in Fundamental Research Funds for for New the Central (HUST: Gaussian process (ESGP) that isforessentially a Processes fusion of ESNs with Bayesian inference Gaussian 2015TS028) and the Program for New Century Excellent Talents in ESNs with Bayesian inference for Gaussian Processes Fundamental Research Funds for the Central Universities (HUST: Gaussian process (ESGP) that is essentially a fusion of University (NCET-12-0214). 2015TS028) and the Program for New Century Excellent Talents in ESNs with Bayesian inference for Gaussian Processes University (NCET-12-0214). 2015TS028) and the Program for New Century Excellent Talents in ESNs with Bayesian inference for Gaussian Processes University (NCET-12-0214). Yu Yu Yu Yu∗
University (NCET-12-0214). Copyright © 2017, 2017 IFAC 1988Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 1988 Copyright ©under 2017 responsibility IFAC 1988Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 1988 10.1016/j.ifacol.2017.08.390
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Yu Cao et al. / IFAC PapersOnLine 50-1 (2017) 1952–1957
(GPs). The ESGP combines both the merits of ESNs and GPs, and its training method can be easily carried out by using type-II maximum likelihood (Chatzis et al. (2011)). Thus, the ESGP is able to approximate the dynamic behaviours of an unknown model, as well as measure the model uncertainties. Though the ESGP has been presented for years, there are few researches concerning the application in control systems. On account of the above analysis, a nonlinear predictive control, which is based on a single neural network (SNN) controller together with predictive model ESGP, is developed for the PMA system. The activation function of SNN must be continuous bounded, and its coefficients and weights should be carefully selected. Fortunately, a couple of intelligent optimization methods have been developed for this kind of problems, such as Particle Swarm Optimization (PSO) (Clerc (1996)). With the accurate prediction of the plant output, SNN is sufficient for generating proper control signal.
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2.3 Echo State Gaussian Process The ESGP is implmented by utilizing a GP to regress against the augmented reservoir states, which can be presented as: φ(t) = [x(t); u(t)] ∈ (N +K)
(4)
And the outputs of ESN y ˆ(t) = [ˆ yj (t)]L j=1 can be rewritten as (5) yˆj (t) = wjT φ(t) Imposing a spherical Gaussian prior over the weights of ESN’s output layer, we have (6) wj ∼ N (0, I) Then, the mean and variance of ESN’s output yˆj (t) are obtained by applying probability theory. (7) E[ˆ yj (t)] = E[wjT φ(t)] = 0 E[ˆ yj (t1 )ˆ yj (t2 )] = φ(t1 )T E[wj wjT ]φ(t2 ) = φ(t1 )T φ(t2 ) (8)
2. PROBLEM STATEMENT AND PRELIMINARIES 2.1 Model Formulation In this paper, we adopt the generalized three-element model of PAM (Reynolds et al. (2003)). The dynamic behaviours of a PMA hanging vertically can be described as: M y¨ + B(P )y˙ + K(P )y = F (P ) − M g
Thus, yˆj (t1 ), yˆj (t2 ) are joint Gaussian distribution with zero mean and covariance given by φ(t1 )T φ(t2 ) for any j ∈ [1, ..., L] and ∀t1 , t2 . Based on the above analysis, the distribution of ESN’s outputs turns out to be a GP (9) [ˆ yj (t)]L j=1 ∼ N (0, Kr (Ψ, Ψ)) where Ψ = [φ(t1 ), . . . , φ(tM )] ∈ (N +K)×M φ(t1 )T φ(t1 ) . . . φ(t1 )T φ(tM ) .. .. .. Kr (Ψ, Ψ) = . . . φ(tM )T φ(t1 ) . . . φ(tM )T φ(tM )
(1)
where M is the mass of payloads, g denotes the acceleration of gravity, y represents the contractile length. The position of y = 0 corresponds to the fully deflated position. K(P ) = K0 + K1 P is the spring coefficient, B(P ) = B0 + B1 P is the damping coefficient, and F (P ) = F0 + F1 P is the effective force provided by the contractile element. Meanwhile, P is the air pressure, which acts as the input of the PMA system.
Applying the Gaussian Process Regression, the predictive mean and variance of the ESGP can be described as: uj∗ = φ(t∗ )T A−1 Ψ˜ yj (10) σ∗2 = σ 2 φ(t∗ )T A−1 φ(t∗ )
(11)
where A = ΨΨT + σ 2 INk ∈ (N +K)×(N +K)
2.2 Echo State Network The ESN contains K inputs, N neurons in the dynamic reservoir, and L neurons in the output layer. The state transformation and the outputs of ESN can be desribed as: x(k + 1) = f [Win u(k + 1) + Wx(k) + Wback y ˆ(k)] (2) y ˆ(k + 1) = Wout [x(k + 1); u(k + 1)]
(3)
where f (·) is a nonlinear activation function, often f (x) = tanh(x); Win ∈ N ×K , W ∈ N ×N , Wback ∈ N ×L and Wout ∈ L×(N +K) are the input, internal, output and feedback weight matrices; x(k) = [x1 (k), ..., xN (k)] ∈ N , u(k) = [u1 (k), ..., uK (k)] ∈ K and y ˆ(k + 1) = [ˆ y1 (k + 1), . . . , yˆL (k + 1)] ∈ L represent the reservoir states, network inputs and network outputs respectively. By updating the dynamic reservoir states and the outputs in accordance with equations (2), (3), the evolution of an ESN can be done.
and y˜j = yˆj + ε , ε ∼ N (0, σ 2 ) denotes that the outputs of ESN superimpose on an independent white Gaussian M noise signal. y ˜j = [˜ yj (tτ )]M means the outputs τ =1 ∈ of ESN at historical time points t1 , ..., tM . According to the inference theory of Bayesian rules, y ˜j is a set of certain values rather than random variables. Therefore, the historical states of ESN together with its inputs and the corresponding outputs serve as the training set, which ˜j (ti ))|i = 1, ..., M }. can be represented as D = {(φ(ti ), y Then, φ(t∗ ) acts as the testing input at time point t∗ . By combining the training set and the testing input, the predicted output uj∗ at time point t∗ can be attained. 3. ECHO STATE GAUSSIAN PROCESSES BASED PREDICTIVE CONTROL Fig. 1 shows the architecture for the control system. Due to the unknown PMA model, its dynamic behaviours are approximated by the ESGP, which utilizes the histories
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Proceedings of the 20th IFAC World Congress 1954 Yu Cao et al. / IFAC PapersOnLine 50-1 (2017) 1952–1957 Toulouse, France, July 9-14, 2017
yi (k)(i = 1, 2, 3, 4) are the inputs of SNN, which can be described by: y1 (k) = yr (k + p) − yc (k + p) y2 (k) = yr (k + p)
y3 (k) = yr (k + p) − yr (k + p − 1) y4 (k) = y1 (k) − y1 (k − 1)
where yr (k + p) means the reference output of the PMA at time k + p. Fig. 1. The architecture of the nonlinear control system of the PMA plant to obtain the distribution of outputs. In this way, t∗ and φ(t∗ ) can be regarded as a future time point and the correspoding testing input respectively, refering to t∗ = k + i and φ(t∗ ) = φ(k + i). Also the expression of the ESGP predicted mean uj∗ can be indicated as the predicted output of the system ym (k + i) ˆ out = at time point k + i where i = 1, 2, ..., p. Let W −1 T (A Ψ˜ yj ) , the predictive output of the ESGP can be obtained as: ˆ out φ(k + i) ym (k + i) = W (12) The p step prediction of outputs ym (k + 1), . . . , ym (k + p) of the PMA plant can be calculated by employing the historical outputs of the plant [y(tτ )]kτ=k−M +1 and the control signal uf (The system control signal is the first element of the ESGP’s input). At the same time, we can easily find that the mean function of predicted distribution can be divided into two parts from equation (10). The first part is the testing input referring to φ(t∗ ) = φ(k + p) = [x(k + p); u(k)], in which the control variables change once for p step in accordance with the single-value predictive control. Depends on the ESGP predictive model, we can compute x(k + p) by applying (2), (12) iteratively. The second part is the training set, referring to the equation A−1 Ψ˜ yj . Once the training data set is obtained from the histories of the system, the equation will be fixed during p step prediction. In other words, the training part is necessary to be calculated only once. In order to overcome errors between the predicted mean function and the plant’s actual outputs, we first compute the one-step prediction ym (k), and obtain the compensation term y(k) − ym (k). By combining the compensation with p step prediction, the actual prediction of p step can be expressed as: yc (k + p) = ym (k + p) + (y(k) − ym (k)) (13) As soon as the p step actual prediction is attained, the signal feeds back to the SNN controller. Fig 1 shows that a single-layer neural network controller consists of four input weights and one output neuron. The output can be obtained as: 4 uf (k + 1) = λh ku vi (k)yi (k) − ξ (14)
For the tracking problem, the actual predicted output yc (k + i)(i = 1, ..., p) is able to approximate the reference yr (k + i)(i = 1, ..., p) on purpose. In order to attain the proper parameters of SNN and satisfy the real-time requirement, without loss any of generality, we define the cost function as: 1 J = (yr (k + p) − yc (k + p))2 (16) 2 Then, the gradient descent algorithm is utilized to update the weights of SNN and generate the iterative rule. The function can be expressed as follow: vi (k + 1) = vi (k) − ηi
1 1 + e−x
(15)
(17)
The reference yr (k + p) can be seen as a known constant value, and we let e(k + p) = yr (k + p) − yc (k + p)
(18)
thus ∂yc (k + p) ∂J = − e(k + p) ∂vi (k) ∂vi (k) ∂y(k) ∂ym (k) ∂ym (k + p) + − ) = −e(k + p)( ∂vi (k) ∂vi (k) ∂vi (k) ∂ym (k + p) ∂uf (k + 1) = −e(k + p)( (19) ∂uf (k + 1) ∂vi (k) ∂ym (k) ∂uf (k + 1) ∂y(k) ∂uf (k + 1) − ) + ∂uf (k + 1) ∂vi (k) ∂uf (k + 1) ∂vi (k) ∂ym (k + p) ∂uf (k + 1) = −e(k + p) ∂uf (k + 1) ∂vi (k) due to ∂ym (k) ∂y(k) = 0, =0 ∂uf (k + 1) ∂uf (k + 1) According to equation (14)(15), we can get ∂uf (k + 1) = λku h(ku fu )[1 − h(ku fu )]yi (k) ∂vi (k)
(20)
where fu =
i=1
h(x) =
∂J ∂vi (k)
4 i=1
vi (k)yi (k) − ξ
(21)
On the other hand, we can derive the following equation:
where vi (k) represents the weights of SNN, h(·) means the activation function, λ, ku , ξ are the coefficients, and 1990
∂ym (k + p) ∂φ(k + p)T ∂ym (k + p) = ∂uf (k + 1) ∂uf (k + 1) ∂φ(k + p)
(22)
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Yu Cao et al. / IFAC PapersOnLine 50-1 (2017) 1952–1957
and
According to equation (10), we have T
−1
∂(φ(k + p) A Ψ˜ yj ) ∂ym (k + p) = = A−1 Ψ˜ yj ∂φ(k + p) ∂φ(k + p) Meanwhile ∂φ(k + p)T ∂x(k + p)T ∂φ(k + p)T = ∂uf (k + 1) ∂uf (k + 1) ∂x(k + p)
∂x(k + i)T ∂x(k + i − 1) ∂xN (k + i) ∂x1 (k + i) ∂x1 (k + i − 1) . . . ∂x1 (k + i − 1) .. .. .. = . . . ∂x1 (k + i) ∂xN (k + i) ··· ∂xN (k + i − 1) ∂xN (k + i − 1)
(23)
(24)
Since the input vector of the ESGP can be expressed as the following form: φ(k +p) = [x1 (k +p), · · · , xN (k +p),u1 (k + T p), · · · ,uK (k + p)] The derive term is given by: 1 ... 0 ... 0 T ∂φ(k + p) = ... . . . ... . . . ... ∂x(k + p) 0 ··· 1 ··· 0 IN ×(N +K) ∈ N ×(N +K) = ( IN ×N 0N ×K ) = ¯
(25)
n=1
+
L
j=1
∂xm (k + i) ∂xn (k + i − 1) L back out = (wmn + wmj w ˆjn )f (gn (k + i − 1))
According to (31) and (33), we can attain the equation: ∂x(k + i)T ¯ +W ¯ )Λ(k + i − 1) = (W ∂x(k + i − 1) where
L
(28)
1j
j=1
back yˆi (k) wji
0
in Λ(k) = WU
(26), (27)
(29)
back out wN ˆj1 j w
.. .. . . L back out ··· wN j w ˆjN ...
j=1
j=1
. . . wN 1 . .. . .. · · · wN N
Next, we can get the derivative term from (30), (34)
f (gN (k))
p
∂x(k + p)T ¯ +W ¯ )Λ(k + i − 1) = (W ∂x(k + 1) i=2
Λ(k) = diag{f (g1 (k)), · · · , f (gN (k))} in in T in T [WU ] = w11 , · · · , wN 1
Since we have ∂x(k + 2)T ∂x(k + p)T ∂x(k + p)T = · .. · ∂x(k + 1) ∂x(k + 1) ∂x(k + p − 1) p ∂x(k + i)T = ∂x(k + i − 1) i=2
L
(34)
and Λ(k + i − 1) = diag{f (g1 (k + i − 1)), · · · , f (gN (k + i − 1))}
where
jN
w11 .. ¯ W= . w1N
i=1
in , · · · , wN 1 f (gN (k)) ··· 0 .. .. . .
back out w ˆj1 w1j
j=1 .. ¯ W = . L ˆ out wback w
wji xi (k)
i=1
Then, we put equation (28) to in ∂x(k + 1)T = w11 f (g1 (k)) ∂uf (k + 1) f (g1 (k)) in .. in = w11 · · · wN 1 .
(33)
where m, n = 1, ..., N
where
i=1
(32)
j=1
(26)
where the control signal uf (k+1) is the first element of the ESN’s input. Rewriting the equation (2), we can express a new function as: g1 (k) (27) x(k + 1) = f ... gN (k) N
(31)
Then
∂x(k + 1)T ∂x(k + p)T ∂x(k + p)T = ∂uf (k + 1) ∂uf (k + 1) ∂x(k + 1)
in wji ui (k + 1) +
xm (k + i) N L back out = f( [(wmn + wmj w ˆjn )xn (k + i − 1)])
In addition, we have
K
Let equation (12) bring back to equation (2) , we can get the relationship between x(k + i − 1) and x(k + i)
where IN ×N is the identity matrix with N dimension, and 0N ×K is the zero matrix with N × K dimension.
gj (k) =
1955
(35)
Combining (22), (23), (24), (25), (26),(29), (35), we have ∂φ(k + p)T ∂ym (k + p) ∂ym (k + p) = ∂uf (k + 1) ∂uf (k + 1) ∂φ(k + p) p in ¯ +W ¯ )Λ(k + i − 1)} Λ(k){ (W = WU i=2
yj ·¯ IN ×(N +K) A−1 Ψ˜
(30)
Finally, it follows that 1991
(36)
Proceedings of the 20th IFAC World Congress 1956 Yu Cao et al. / IFAC PapersOnLine 50-1 (2017) 1952–1957 Toulouse, France, July 9-14, 2017
∂ym (k + p) ∂J = −e(k + p) ∂vi (k) ∂vi (k) = −λku h(ku fu )[1 − h(ku fu )]yi (k)e(k + p) p in ¯ +W ¯ )Λ(k + i − 1)} Λ(k){ (W ·WU
(37)
i=2
yj ·¯ IN ×(N +K) A−1 Ψ˜
4. CONVERGENCE ANALYSIS So as to demonstrate the convergence of the system conveniently, the discrete Lyapunov function can be defined as: 1 V (k) = e2 (k + p) (38) 2
Let the sum of the higher order terms O(k + p) equals to O(t) for short. Theorem 1. The closed-loop system is asymptotically stable when the sum of the higher order terms of the Taylor expansion is bounded and should be depicted as follow: O(t) + (1−QT (t)ηQ(t))e(t) (44) 2 < (1 − (QT (t)ηQ(t)) )e2 (t) and QT (t)ηQ(t) < 1 where
Q(t) = Let
where e(k +p) has been described above which denotes the modelling error of the ESGP referring to (18). The increasement of the Lyapunov function ∆V (k) can be obtained by: ∆V (k) = V (k + 1) − V (k) 1 = (e2 (k + p + 1) − e2 (k + p)) 2
(39)
By applying Taylor series expansion, the error can be decomposed into a known affine term plus a higher order term(Han et al. (2016))
p(t) = QT (t)ηQ(t)
(45)
Proof. Combining (41), (42), (43), (45), we can get 1 ∆V = −p(t)(1 − p(t))e2 (t) 2 1 +(1 − p(t))e(t)O(t) + O2 (t) (46) 2 < −p(t)(1 − p(t))e2 (t) 1 +(1 − p(t))e(t)O(t) + O2 (t) 2 Using (44),(45) and (46), it follows that
∂e(k + p) T e(k + p + 1) = e(k+p) + [ ] ∆v(k) + O(k+p)(40) ∂v(k) Then, we have 1 ∆V (k) = (e2 (k + p + 1) − e2 (k + p)) 2 1 = (e(k + p + 1) − e(k + p))(e(k + p + 1) + e(k + p)) 2 (41) 1 ∂e(k + p) T = ([ ] ∆v(k) + O(k+p)) 2 ∂v(k) ∂e(k + p) T ·(2e(k + p) + [ ] ∆v(k) + O(k+p)) ∂v(k)
∂ym (k + p) , e(t)=e(k + p) ∂v(k)
∆V < −p(t)(1 −p(t))e2 (t) +(1 − p(t))e(t)( (1 − p2 (t))e2 (t) − (1−p(t))e(t)) (47) 1 + ( (1 − p2 (t))e2 (t) − (1−p(t))e(t))2 2 =0 5. EXPERIMENT STRUDY The PMA system experiment platform that is comprised of the PMA plant and related devices is shown in Fig 2.
According to (13) and (18), it follows that ∂(yr (k + p) − yc (k + p)) ∂e(k + p) = ∂v(k) ∂v(k) ∂yc (k + p) ∂ym (k + p) =− =− ∂v(k) ∂v(k)
(42)
Also, the errors of SNN’s weights can be expressed in the following form: ∂J ∆v(k) = v(k + 1) − v(k) = −η ∂v(k) (43) ∂ym (k + p) ∂yc (k + p) =e(k + p)η = e(k + p)η ∂v(k) ∂v(k) T
where v(k) = [ v1 (k) v2 (k) v3 (k) v4 (k) ] , T ∂J ∂J ∂J ∂J ∂J = ∂v1 (k) ∂v2 (k) ∂v3 (k) ∂v4 (k) ∂v(k) and η = diag { η1 η2 η3 η4 } means the step size of the gradient descent algorithm.
Fig. 2. PMA system experiment platform An air compressor provides air pressure via the electromagnetic proportion valve. The measured displacement and force of the PMA are sampled and delivered to the computer through the data acquisition card. After that, the computer calculates control signals using SNN-ESGP strategy and feeds them into the electromagnetic proportion valve, which is able to regulate the input pressure.
1992
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Yu Cao et al. / IFAC PapersOnLine 50-1 (2017) 1952–1957
network. By analyzing the coefficients of SNN, we derived the iteration rules. Also, the Lyapunov function was put forward to illustrate the stability of the closed-loop system. The experiments were implemented to compare the performance of different methods. The results demonstrate that the SNN-ESGP can estimate the behaviours of the PMA plant and it is more accurate than PID as well as SMC.
0.04 Reference SNN-ESGP PID SMC
displacement (m)
0.035 0.03
1957
0.025 0.02 0.015 0.01
REFERENCES 4
6
8
10 time (s)
12
14
16
Fig. 3. Trajectory tracking control results of the model 3
× 10 -3 SNN-ESGP PID SMC
2
error (m)
1 0 -1 -2 -3
4
6
8
10 time (s)
12
14
16
Fig. 4. Comparison of the tracking error using SNN-ESGP, PID and SMC In our experiments, the MATLAB toolbox xPC Target, which provides an environment to construct the real-time system, is applied for the control strategies of the PMA plant. With the MATLAB/Simulink, C code is generated and downloaded into target computer for running. The PMA system is modeled by nonlinear autoregressive exogenous model y(k + 1) = f (u(k), y(k), y(k − 1)). Combining [u(k), y(k), y(k − 1)] and the dynamic reservoir states, the input of the predictive model using ESGP can be built and y(k+1) represents the predicted output of the ESGP. Together with the gradient descent algorithm, the SNN-ESGP approach has been implemented to evaluate the control performance. Since it is difficult to obtain optimal value of coefficients and weights, we apply PSO for the proper value. Then, the coefficients of the SNN are chosen as λ = 369598, ξ = 1.235, ku = 1.2. The position tracking result of the proposed control strategy is shown in Fig 3, and it clearly shows that it can track the reference with high-precision performance. Fig 4 gives the comparison of tracking errors of control strategies. It shows that the SMC is unable to get a satisfactory result, since it suffers from the inaccuracy of the model, and errors of the model and parameters are inevitable in identifying physical system. As a result, it is difficult for the SMC to get perferable performance. On the other hand, even though PID can achieve tracking the reference, it cannot meet the requirement of highprecision trajectory tracking, and the average error of SNN-ESGP is much smaller than the PID controller. With the application of the SNN-ESGP, it can satisfy the requirements of controlling rehabilitation robots. 6. DISCUSSION AND CONCLUSION In this paper, we proposed a predictive control based on an echo state Gaussian process with a single layer neural
Hussain, S, S. Q. Xie, and P. K. Jamwal Effect of cadence regulation on muscle activation patterns during robot assisted gait: a dynamic simulation study. IEEE Journal of Biomedical and Health Informatics, volume 17(2), pages 442-451, 2013. Chen S H, Lien W M, Wang W W, et al. Assistive Control System for Upper Limb Rehabilitation Robot. IEEE Transactions on Neural Systems and Rehabilitation Engineering, volume 24(11), pages 1199-1209, 2016. Noritsugu, Toshiro, and T. Tanaka Application of rubber artificial muscle manipulator as a rehabilitation robot. IEEE/ASME Transactions on Mechatronics, volume 2(4), pages 259-267, 1997. Caldwell, D. G., G. A. Medrano-Cerda, and M. Goodwin Control of pneumatic muscle actuators. IEEE Control Systems, volume 15(1), pages 40-48, 1995. Andrikopoulos, G., G. Nikolakopoulos, and S. Manesis Advanced Non-linear PID Based Antagonistic Control for Pneumatic Muscle Actuators. IEEE Transactions on Industrial Electronics, volume 61(12), pages 6926-6937, 2014. Sun, M., Zhang L, Wang Z, et al. PID pitch attitude control for unstable flight vehicle in the presence of actuator delay: Tuning and analysis. Journal of the Franklin Institute, volume 351(12), pages 5523-5547, 2014. Cai, D. and H. Yamaura A robust controller for manipulator driven by artificial muscle actuator. Proc. of the IEEE Conf. on Control Applications, pp: 57-64, 2003. Xing K, Huang J, Wang Y, et al. Tracking control of pneumatic artificial muscle actuators based on sliding mode and non-linear disturbance observer Control Theory & Applications, volume 4(10), pages 2058-2070, 2010. Han,H G, Zhang L, Hou Y, et al. Nonlinear Model Predictive Control Based on a Self-Organizing Recurrent Neural Network. IEEE Transactions on Neural Networks & Learning Systems, volume 27(2), pages 402-415, 2016. Jaeger,H The”echo state”approach to analysing and training recurrent neural networks German Nat. Research Center Information Technology, 2001. Li, D., M. Han, and J. Wang Chaotic time series prediction based on a novel robust echo state network IEEE Transactions on Neural Networks & Learning Systems , volume 23(5), pages 787-799, 2012. Chatzis, S. P. and Y. Demiris Echo State Gaussian Process IEEE Transactions on Neural Networks, volume 22(9), pages 1435-1445, 2011. Clerc, M., and Kennedy, J. The particle swarm - explosion, stability, and convergence in a multidimensional complex space IEEE Transactions on Evolutionary Computation, volume 6(1), pages 58-73, 2002. Reynolds, D. B., et al Modeling the dynamic characteristics of pneumatic muscle Annals of biomedical engineering , 310-317, 2003.
1993
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 1952–1957 Design of Nonlinear Predictive Control for Design of Nonlinear Predictive Control for Design of Nonlinear Predictive Control for Design of Nonlinear Predictive Control for Pneumatic Muscle Actuator Based on Echo Pneumatic Muscle Actuator Based on Echo Pneumatic Muscle Actuator Based on Echo Pneumatic Muscle Actuator Based on Echo State Gaussian Process State Gaussian Process State Gaussian Process State Gaussian Process ∗ ∗,† ∗ ∗
Cao Jian Huang Huang ∗,† Gangzheng Gangzheng Ding Ding ∗∗ Yongji Yongji Wang Wang ∗∗ Cao ∗∗ Jian Cao ∗ Jian Huang ∗,† ∗,† Gangzheng Ding ∗ Yongji Wang ∗ Cao Jian Huang Gangzheng Ding Yongji Wang ∗ School of Automation, Huazhong University of Science and ∗ School of Automation, Huazhong University of Science and Automation, Huazhong University of Science and Technology, 430074 China (Tel:+86-27-87558472; ∗ School of Wuhan Technology, 430074 China P.R. P.R. (Tel:+86-27-87558472; School of Wuhan Automation, Huazhong University of Science and Technology, Wuhan 430074 China P.R. (Tel:+86-27-87558472; e-mail: huang [email protected]). e-mail: 430074 huang [email protected]). Technology, Wuhan China P.R. (Tel:+86-27-87558472; e-mail: huang [email protected]). e-mail: huang [email protected]). Abstract: Recently, the application Abstract: Recently, the application of of Pneumatic Pneumatic Muscle Muscle Actuators Actuators (PMAs) (PMAs) for for driving driving Abstract: Recently, the become application of Pneumatic Muscle AActuators (PMAs) for driving rehabilitation robots has a matter of great concern. traditional control algorithm, rehabilitation robots has become a matter of great concern. AActuators traditional control for algorithm, Abstract: Recently, the application of Pneumatic Muscle (PMAs) driving rehabilitation robots has become a matter high-precision of great concern. A traditional control algorithm, such as cannot achieve satisfactory performance in tracking such as PID, PID, robots cannot has achieve satisfactory performance in trajectory trajectory tracking rehabilitation become a matter high-precision of great concern. A traditional control algorithm, such as for PID, cannot achieve satisfactory high-precision performance in trajectory tracking problem PMAs, due to PMAs’ features of nonlinear effects, slow response time, time-varying problem for PMAs, dueachieve to PMAs’ features ofhigh-precision nonlinear effects, slow response time, time-varying such as PID, cannot satisfactory performance in trajectory tracking problem for PMAs, due towe PMAs’ features of nonlinear effects, slow response time, time-varying parameters. In this study proposed a nonlinear predictive control strategy SNN-ESGP, which parameters. In this study we proposed a nonlinear predictive control strategy SNN-ESGP, which problem for PMAs, due to PMAs’ features of nonlinear effects, slow response time, time-varying parameters. Inofthis studymodel we proposed a nonlinear predictiveprocess control(ESGP) strategy that SNN-ESGP, which is comprised a novel called echo state Gaussian is suitable for is comprisedInofthis a novel model called echo state Gaussian process (ESGP) that is suitable for parameters. study we proposed a nonlinear predictive control strategy SNN-ESGP, which is comprised of a novel model called as echo state Gaussiantheir process (ESGP) that issingle suitable for modeling nonlinear unknown systems well as measuring uncertainties, and a neural modeling nonlinear unknown systems as well as measuring their uncertainties, and a single neural is comprised of a novel model called echo state Gaussian process (ESGP) that is suitable for modeling(SNN) nonlinear unknown systems as well as measuring their uncertainties, and a single neural network serves as the controller of the system. To analyze the system convergence, we network (SNN) serves as thesystems controller of the system. Totheir analyze the system convergence, we modeling nonlinear unknown as well as measuring uncertainties, and a single neural network (SNN) serves as the controller of deduce the system. To analyze the system convergence, we utilize the gradient descent algorithm and the iterative rules of the weights of SNN. The utilize the(SNN) gradient descent algorithm and deduce the iterative rulesthe of the weights of SNN. The network serves as the controller of the system. To analyze system convergence, we utilize theofgradient descent algorithm and deduce the iterative rulestheorem. of the weights of SNN. The stability the closed-loop system is analyzed by using Lyapunov Finally, studies of stability ofgradient the closed-loop system is analyzed by the using Lyapunov theorem. Finally, studies of utilize the descent algorithm and deduce iterative rules of the weights of SNN. The stability experiments of the closed-loop system is analyzed bycontrol using strategy Lyapunovwith theorem. Finally, trajectory studies of physical illustrate the validity of the high-precision physical experiments illustrate the validity of the control strategy with high-precision trajectory stability of the closed-loop system is analyzed by using Lyapunov theorem. Finally, studies of physical experiments illustrate the validity of the control strategy with high-precision trajectory performance. performance. physical experiments illustrate the validity of the control strategy with high-precision trajectory performance. performance. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: predictive control, Keywords: Pneumatic Pneumatic muscle muscle actuators, actuators, nonlinear nonlinear predictive control, echo echo state state gaussian gaussian Keywords: Pneumatic muscle actuators, nonlinear predictive control, echo state gaussian process, single neural network process, single neural network Keywords: Pneumatic muscle actuators, nonlinear predictive control, echo state gaussian process, single neural network process, single neural network 1. most 1. INTRODUCTION INTRODUCTION most utilized utilized algorithm, algorithm, owing owing to to its its easy easy implementation implementation 1. INTRODUCTION most utilized algorithm, owing to its easy implementation and preferable performance (Sun, M. et (2014)). and preferable performance (Sun, M.easy et al. al. (2014)). But But 1. INTRODUCTION most utilized algorithm, owing to its implementation and preferable performance (Sun, M. et al. (2014)). But it is difficult for PID controller to achieve a satisfactory Robot-assisted therapies have been applied to improve it is preferable difficult forperformance PID controller to M. achieve a(2014)). satisfactory (Sun, et al. But Robot-assisted therapies have been applied to improve and it is difficult for PID controller to achieve a satisfactory result in a high-precision trajectory tracking task, Robot-assisted therapies process have been applied to improve result in a high-precision trajectory tracking task, and the overall in years(Hussain and it is difficult for PID controller to achieve a satisfactory the overall rehabilitation rehabilitation process in recent recent years(Hussain Robot-assisted therapies have been applied to improve result in a high-precision trajectory tracking task, supand a PID controller suffers aa lack of the overall rehabilitation process in recent years(Hussain et al. (2013)). Compared with rehabilitative PID in controller suffers from from lack tracking of theoretical theoretical supa high-precision trajectory task, and et al. (2013)). Comparedprocess with traditional traditional rehabilitative aresult the overall rehabilitation in recent years(Hussain a PIDforcontroller suffers fromstability. a lack ofThe theoretical support proving the system sliding mode et al. (2013)). Compared with traditional rehabilitative therapies, the robot-assisted therapies can increase the proving the system sliding mode PIDfor controller suffers fromstability. a lack ofThe theoretical suptherapies, the robot-assisted therapies can rehabilitative increase the aport et al. (2013)). Compared with traditional port for proving the system stability. The sliding mode (SMC) technique is common method for therapies, the robot-assisted therapiesstaff can and increase the control intensity, save the enhance control is another another common method for for(SMC) provingtechnique the system stability. The sliding mode intensity, save the efforts efforts of of medical medical staff and enhance therapies, save the robot-assisted can and increase the port control (SMC) technique is another common method for nonlinear control (e.g., Cai et al. (2003)). Our previous intensity, the efforts of therapies medical staff enhance rehabilitated efficiency (Chen, S. H. et al. (2016)). Meannonlinear control (e.g., Cai et al. (2003)). Our previous (SMC) technique is another common method for rehabilitated efficiency (Chen, S. H. et staff al. (2016)). Mean- control intensity, save the efforts of medical and enhance nonlinear controla (e.g., Cai etdisturbance al. (2003)).observer Our previous work proposed non-linear based rehabilitated efficiency (Chen, S. H. et (PMAs) al. (2016)). Meanwhile, Pneumatic Muscle Actuators server as proposed a (e.g., non-linear disturbance observer based nonlinear control Cai et al. (2003)). Our previous while, Pneumatic Muscle Actuators (PMAs) server as work rehabilitated efficiency (Chen, S. H. et al. (2016)). Meanproposed a non-linear disturbance observer based sliding mode while depends the while, Pneumatic Muscle Actuators (PMAs) server as work compliant actuators that an role in slidingproposed mode control, control, while it itdisturbance depends on onobserver the identified identified a non-linear based compliant actuatorsMuscle that play play an important important roleserver in reharehawhile, Pneumatic Actuators (PMAs) as work sliding mode(Xing control, while it depends ondevelopment the identified PMA model et al. (2010)). With the of compliant actuators that play an important role in rehabilitation robot application (Noritsugu et al. (1997)), due PMA model (Xing et al. (2010)). With the development of mode control, while it depends on the identified bilitation robot application (Noritsugu et al. role (1997)), due sliding compliant actuators that play an important in rehamodel (Xing et al. (2010)). With the development of artificial intelligence, the neural networks have attracted bilitation robot application (Noritsugu et al. (1997)), due PMA to their high force-to-weight ratio, no mechanical parts, artificial intelligence, the neural networks have attracted PMA model (Xing et al. (2010)). With the development of to their high force-to-weight ratio, no et mechanical parts, bilitation robot application (Noritsugu al. (1997)), due intelligence, the of neural networks have attracted great attention, because their excellent capabilities of to their high force-to-weight ratio,and no low mechanical parts, artificial lower compressed air consumption, cost (Caldwell great attention, because of their networks excellent have capabilities of artificial intelligence, the neural attracted lower compressed air consumption, and low cost (Caldwell to their high force-to-weight ratio, no mechanical parts, great attention, because of their excellent capabilities of modelling uncertain, nonlinear, and complex systems lower compressed air consumption, and low cost (Caldwell et al. And PMAs are to muscle modelling uncertain, nonlinear, andexcellent complexcapabilities systems (Han (Han attention, because of their of et al. (1995)). (1995)). Andair PMAs are analogous analogous to skeletal skeletal muscle great lower compressed consumption, and low cost (Caldwell modelling uncertain, nonlinear, and complex systems (Han et al. (2016)). In addition, due to model-free approximaet al.only (1995)). And PMAs are analogous to skeletal muscle not in size and force-output but the behavior that et al. (2016)). In addition, due and to model-free approximauncertain, nonlinear, complex systems (Han not only in size and force-output but the behavior that modelling et al. (1995)). And PMAs are analogous to skeletal muscle et al.and (2016)). In addition, due to model-free approximation stability analysis neutral networks, they not only in size decompressed and force-output but the behavior that the pressurized, processes of are just stability analysis of of neutral networks,approximathey have have et al.and (2016)). In addition, due to model-free the pressurized, decompressed processes of PMAs PMAs are that just tion not pressurized, only in size decompressed and force-output but the behavior and stability analysis of neutral networks, they have brought great convenience for modelling systems. the processes of PMAs are just tion similar to muscle contraction. Unlike conventional robotic brought great convenience for modelling systems. tion and stability analysis of neutral networks, they have similar to muscle contraction. Unlike conventional robotic the pressurized, decompressed processes of PMAs are just brought great convenience for modelling systems. similar to muscle contraction. Unlike conventional robotic actuators, PMAs can necessary compliance brought great convenience for modelling systems. actuators, PMAs contraction. can also also provide provide necessary compliance A remarkably efficient neural network structure for similar to muscle Unlike conventional robotic remarkably efficient neural network structure for rereactuators, PMAs can also provide necessary compliance A which makes the processes safer. A remarkably efficient neural network structure for rewhich makes the rehabilitation rehabilitation processes safer.compliance current neural networks (RNNs) was proposed by actuators, PMAs can also provide necessary current neural efficient networksneural (RNNs) was first first proposed by which makes the rehabilitation processes safer. A remarkably network structure for recurrent neural networks (RNNs) was first proposed by Jaeger (Jaeger,H. (2001)) called echo state network (ESN). which makes the rehabilitation processes safer. However, high-precision trajectory tracking tasks of the Jaeger (Jaeger,H. (2001)) called echo state network (ESN). neural networks (RNNs) was first proposed by However, high-precision trajectory tracking tasks of the current (Jaeger,H. (2001)) calledneural echo state network (ESN). Compared with conventional networks, ESN has However, high-precision trajectory trackingproblems, tasks of due the Jaeger PMA systems turn out to be challenging Compared with conventional neural networks, ESN has Jaeger (Jaeger,H. (2001)) called echo state network (ESN). PMA systems turn out to be challenging problems, due However, high-precision trajectory tracking tasks of the Compared with inconventional neural networks, ESN of hasa superior ability capturing dynamic behaviours PMA systems turnnature, out tohysteresis be challenging problems, due to their and paability capturing the the dynamic behaviours Compared with in neural networks, ESN of hasa to their nonlinear nonlinear nature, hysteresis and time-varying time-varying pa- superior PMA turnnature, out tohysteresis be challenging problems, due superior ability inconventional capturing the dynamic behaviours of a complex nonlinear system. But in practical applications, to theirsystems nonlinear andFor time-varying parameters (Andrikopoulos et al. (2014)). such kinds of complex nonlinear system. But in practical applications, superior ability in capturing the dynamic behaviours of a rameters (Andrikopoulos et al. (2014)). For such kinds of to their nonlinear nature, et hysteresis andFor time-varying panonlinear system. But in practical applications, it may be ill-posed and accompanied with large output rameters (Andrikopoulos al. (2014)). such kinds of complex unknown and nonlinear plants, the conventional proporit may benonlinear ill-posedsystem. and accompanied with large output complex But in practical applications, unknown and nonlinear plants, the conventional proporrameters (Andrikopoulos et al. (2014)). For such kinds of it may be ill-posed and accompanied training with large output weights since the frequently-used method of unknown and nonlinear plants, the conventional proportional integral derivative is weights since the most most frequently-used method of may be ill-posed and accompanied training with large output tional integral derivative (PID) (PID) controller controller is one one of of the the it unknown and nonlinear conventional since the most frequently-used training method of ESN is pseudoinverse (Li et al. (2012)). tional integral derivativeplants, (PID) the controller is oneproporof the weights ESN is pseudoinverse (Li et al. (2012)). weights since the most frequently-used training method of tional integral derivative (PID) controller is one of the This work was supported by the National Natural Science Foun This work was supported by the National Natural Science FounESN is pseudoinverse (Li et al. (2012)). ESN is pseudoinverse (Li et al. (2012)). Recently, a novel Bayesian approach towards reservoir dation of China under Grant 61473130, the Science Fund for DisThis work was supported by the National Natural Science FounRecently, a novel Bayesian approach towards reservoir dation China Grant the Natural Science Fund forFounDisThis of work was under supported by61473130, the National Science Recently, a isnovel Bayesian approach towards reservoir computing proposed by Demiris called state tinguished Young Scholars of Hubei Province (2015CFA047), the dation of China under Grant 61473130, the Science Fund for Discomputing proposed by Y. Y.approach Demiris towards called echo echo state tinguished Youngunder Scholars of 61473130, Hubei Province (2015CFA047), the Recently, a is novel Bayesian reservoir dation of China Grant the Science Fund for DisFundamental Research Funds the Province Central Universities (HUST: computing is proposed by Y. Demiris called echo state tinguished Young Scholars of for Hubei (2015CFA047), the Gaussian process (ESGP) that is essentially a fusion of Fundamental Research Funds for the Province Central Universities (HUST: Gaussian process (ESGP) that is essentially a fusion of computing is proposed by Y. Demiris called echo state tinguished Young Scholars of Hubei (2015CFA047), the 2015TS028) and the Program Century Universities Excellent Talents in Fundamental Research Funds for for New the Central (HUST: Gaussian process (ESGP) that isforessentially a Processes fusion of ESNs with Bayesian inference Gaussian 2015TS028) and the Program for New Century Excellent Talents in ESNs with Bayesian inference for Gaussian Processes Fundamental Research Funds for the Central Universities (HUST: Gaussian process (ESGP) that is essentially a fusion of University (NCET-12-0214). 2015TS028) and the Program for New Century Excellent Talents in ESNs with Bayesian inference for Gaussian Processes University (NCET-12-0214). 2015TS028) and the Program for New Century Excellent Talents in ESNs with Bayesian inference for Gaussian Processes University (NCET-12-0214). Yu Yu Yu Yu∗
University (NCET-12-0214). Copyright © 2017, 2017 IFAC 1988Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 1988 Copyright ©under 2017 responsibility IFAC 1988Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 1988 10.1016/j.ifacol.2017.08.390
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Yu Cao et al. / IFAC PapersOnLine 50-1 (2017) 1952–1957
(GPs). The ESGP combines both the merits of ESNs and GPs, and its training method can be easily carried out by using type-II maximum likelihood (Chatzis et al. (2011)). Thus, the ESGP is able to approximate the dynamic behaviours of an unknown model, as well as measure the model uncertainties. Though the ESGP has been presented for years, there are few researches concerning the application in control systems. On account of the above analysis, a nonlinear predictive control, which is based on a single neural network (SNN) controller together with predictive model ESGP, is developed for the PMA system. The activation function of SNN must be continuous bounded, and its coefficients and weights should be carefully selected. Fortunately, a couple of intelligent optimization methods have been developed for this kind of problems, such as Particle Swarm Optimization (PSO) (Clerc (1996)). With the accurate prediction of the plant output, SNN is sufficient for generating proper control signal.
1953
2.3 Echo State Gaussian Process The ESGP is implmented by utilizing a GP to regress against the augmented reservoir states, which can be presented as: φ(t) = [x(t); u(t)] ∈ (N +K)
(4)
And the outputs of ESN y ˆ(t) = [ˆ yj (t)]L j=1 can be rewritten as (5) yˆj (t) = wjT φ(t) Imposing a spherical Gaussian prior over the weights of ESN’s output layer, we have (6) wj ∼ N (0, I) Then, the mean and variance of ESN’s output yˆj (t) are obtained by applying probability theory. (7) E[ˆ yj (t)] = E[wjT φ(t)] = 0 E[ˆ yj (t1 )ˆ yj (t2 )] = φ(t1 )T E[wj wjT ]φ(t2 ) = φ(t1 )T φ(t2 ) (8)
2. PROBLEM STATEMENT AND PRELIMINARIES 2.1 Model Formulation In this paper, we adopt the generalized three-element model of PAM (Reynolds et al. (2003)). The dynamic behaviours of a PMA hanging vertically can be described as: M y¨ + B(P )y˙ + K(P )y = F (P ) − M g
Thus, yˆj (t1 ), yˆj (t2 ) are joint Gaussian distribution with zero mean and covariance given by φ(t1 )T φ(t2 ) for any j ∈ [1, ..., L] and ∀t1 , t2 . Based on the above analysis, the distribution of ESN’s outputs turns out to be a GP (9) [ˆ yj (t)]L j=1 ∼ N (0, Kr (Ψ, Ψ)) where Ψ = [φ(t1 ), . . . , φ(tM )] ∈ (N +K)×M φ(t1 )T φ(t1 ) . . . φ(t1 )T φ(tM ) .. .. .. Kr (Ψ, Ψ) = . . . φ(tM )T φ(t1 ) . . . φ(tM )T φ(tM )
(1)
where M is the mass of payloads, g denotes the acceleration of gravity, y represents the contractile length. The position of y = 0 corresponds to the fully deflated position. K(P ) = K0 + K1 P is the spring coefficient, B(P ) = B0 + B1 P is the damping coefficient, and F (P ) = F0 + F1 P is the effective force provided by the contractile element. Meanwhile, P is the air pressure, which acts as the input of the PMA system.
Applying the Gaussian Process Regression, the predictive mean and variance of the ESGP can be described as: uj∗ = φ(t∗ )T A−1 Ψ˜ yj (10) σ∗2 = σ 2 φ(t∗ )T A−1 φ(t∗ )
(11)
where A = ΨΨT + σ 2 INk ∈ (N +K)×(N +K)
2.2 Echo State Network The ESN contains K inputs, N neurons in the dynamic reservoir, and L neurons in the output layer. The state transformation and the outputs of ESN can be desribed as: x(k + 1) = f [Win u(k + 1) + Wx(k) + Wback y ˆ(k)] (2) y ˆ(k + 1) = Wout [x(k + 1); u(k + 1)]
(3)
where f (·) is a nonlinear activation function, often f (x) = tanh(x); Win ∈ N ×K , W ∈ N ×N , Wback ∈ N ×L and Wout ∈ L×(N +K) are the input, internal, output and feedback weight matrices; x(k) = [x1 (k), ..., xN (k)] ∈ N , u(k) = [u1 (k), ..., uK (k)] ∈ K and y ˆ(k + 1) = [ˆ y1 (k + 1), . . . , yˆL (k + 1)] ∈ L represent the reservoir states, network inputs and network outputs respectively. By updating the dynamic reservoir states and the outputs in accordance with equations (2), (3), the evolution of an ESN can be done.
and y˜j = yˆj + ε , ε ∼ N (0, σ 2 ) denotes that the outputs of ESN superimpose on an independent white Gaussian M noise signal. y ˜j = [˜ yj (tτ )]M means the outputs τ =1 ∈ of ESN at historical time points t1 , ..., tM . According to the inference theory of Bayesian rules, y ˜j is a set of certain values rather than random variables. Therefore, the historical states of ESN together with its inputs and the corresponding outputs serve as the training set, which ˜j (ti ))|i = 1, ..., M }. can be represented as D = {(φ(ti ), y Then, φ(t∗ ) acts as the testing input at time point t∗ . By combining the training set and the testing input, the predicted output uj∗ at time point t∗ can be attained. 3. ECHO STATE GAUSSIAN PROCESSES BASED PREDICTIVE CONTROL Fig. 1 shows the architecture for the control system. Due to the unknown PMA model, its dynamic behaviours are approximated by the ESGP, which utilizes the histories
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yi (k)(i = 1, 2, 3, 4) are the inputs of SNN, which can be described by: y1 (k) = yr (k + p) − yc (k + p) y2 (k) = yr (k + p)
y3 (k) = yr (k + p) − yr (k + p − 1) y4 (k) = y1 (k) − y1 (k − 1)
where yr (k + p) means the reference output of the PMA at time k + p. Fig. 1. The architecture of the nonlinear control system of the PMA plant to obtain the distribution of outputs. In this way, t∗ and φ(t∗ ) can be regarded as a future time point and the correspoding testing input respectively, refering to t∗ = k + i and φ(t∗ ) = φ(k + i). Also the expression of the ESGP predicted mean uj∗ can be indicated as the predicted output of the system ym (k + i) ˆ out = at time point k + i where i = 1, 2, ..., p. Let W −1 T (A Ψ˜ yj ) , the predictive output of the ESGP can be obtained as: ˆ out φ(k + i) ym (k + i) = W (12) The p step prediction of outputs ym (k + 1), . . . , ym (k + p) of the PMA plant can be calculated by employing the historical outputs of the plant [y(tτ )]kτ=k−M +1 and the control signal uf (The system control signal is the first element of the ESGP’s input). At the same time, we can easily find that the mean function of predicted distribution can be divided into two parts from equation (10). The first part is the testing input referring to φ(t∗ ) = φ(k + p) = [x(k + p); u(k)], in which the control variables change once for p step in accordance with the single-value predictive control. Depends on the ESGP predictive model, we can compute x(k + p) by applying (2), (12) iteratively. The second part is the training set, referring to the equation A−1 Ψ˜ yj . Once the training data set is obtained from the histories of the system, the equation will be fixed during p step prediction. In other words, the training part is necessary to be calculated only once. In order to overcome errors between the predicted mean function and the plant’s actual outputs, we first compute the one-step prediction ym (k), and obtain the compensation term y(k) − ym (k). By combining the compensation with p step prediction, the actual prediction of p step can be expressed as: yc (k + p) = ym (k + p) + (y(k) − ym (k)) (13) As soon as the p step actual prediction is attained, the signal feeds back to the SNN controller. Fig 1 shows that a single-layer neural network controller consists of four input weights and one output neuron. The output can be obtained as: 4 uf (k + 1) = λh ku vi (k)yi (k) − ξ (14)
For the tracking problem, the actual predicted output yc (k + i)(i = 1, ..., p) is able to approximate the reference yr (k + i)(i = 1, ..., p) on purpose. In order to attain the proper parameters of SNN and satisfy the real-time requirement, without loss any of generality, we define the cost function as: 1 J = (yr (k + p) − yc (k + p))2 (16) 2 Then, the gradient descent algorithm is utilized to update the weights of SNN and generate the iterative rule. The function can be expressed as follow: vi (k + 1) = vi (k) − ηi
1 1 + e−x
(15)
(17)
The reference yr (k + p) can be seen as a known constant value, and we let e(k + p) = yr (k + p) − yc (k + p)
(18)
thus ∂yc (k + p) ∂J = − e(k + p) ∂vi (k) ∂vi (k) ∂y(k) ∂ym (k) ∂ym (k + p) + − ) = −e(k + p)( ∂vi (k) ∂vi (k) ∂vi (k) ∂ym (k + p) ∂uf (k + 1) = −e(k + p)( (19) ∂uf (k + 1) ∂vi (k) ∂ym (k) ∂uf (k + 1) ∂y(k) ∂uf (k + 1) − ) + ∂uf (k + 1) ∂vi (k) ∂uf (k + 1) ∂vi (k) ∂ym (k + p) ∂uf (k + 1) = −e(k + p) ∂uf (k + 1) ∂vi (k) due to ∂ym (k) ∂y(k) = 0, =0 ∂uf (k + 1) ∂uf (k + 1) According to equation (14)(15), we can get ∂uf (k + 1) = λku h(ku fu )[1 − h(ku fu )]yi (k) ∂vi (k)
(20)
where fu =
i=1
h(x) =
∂J ∂vi (k)
4 i=1
vi (k)yi (k) − ξ
(21)
On the other hand, we can derive the following equation:
where vi (k) represents the weights of SNN, h(·) means the activation function, λ, ku , ξ are the coefficients, and 1990
∂ym (k + p) ∂φ(k + p)T ∂ym (k + p) = ∂uf (k + 1) ∂uf (k + 1) ∂φ(k + p)
(22)
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Yu Cao et al. / IFAC PapersOnLine 50-1 (2017) 1952–1957
and
According to equation (10), we have T
−1
∂(φ(k + p) A Ψ˜ yj ) ∂ym (k + p) = = A−1 Ψ˜ yj ∂φ(k + p) ∂φ(k + p) Meanwhile ∂φ(k + p)T ∂x(k + p)T ∂φ(k + p)T = ∂uf (k + 1) ∂uf (k + 1) ∂x(k + p)
∂x(k + i)T ∂x(k + i − 1) ∂xN (k + i) ∂x1 (k + i) ∂x1 (k + i − 1) . . . ∂x1 (k + i − 1) .. .. .. = . . . ∂x1 (k + i) ∂xN (k + i) ··· ∂xN (k + i − 1) ∂xN (k + i − 1)
(23)
(24)
Since the input vector of the ESGP can be expressed as the following form: φ(k +p) = [x1 (k +p), · · · , xN (k +p),u1 (k + T p), · · · ,uK (k + p)] The derive term is given by: 1 ... 0 ... 0 T ∂φ(k + p) = ... . . . ... . . . ... ∂x(k + p) 0 ··· 1 ··· 0 IN ×(N +K) ∈ N ×(N +K) = ( IN ×N 0N ×K ) = ¯
(25)
n=1
+
L
j=1
∂xm (k + i) ∂xn (k + i − 1) L back out = (wmn + wmj w ˆjn )f (gn (k + i − 1))
According to (31) and (33), we can attain the equation: ∂x(k + i)T ¯ +W ¯ )Λ(k + i − 1) = (W ∂x(k + i − 1) where
L
(28)
1j
j=1
back yˆi (k) wji
0
in Λ(k) = WU
(26), (27)
(29)
back out wN ˆj1 j w
.. .. . . L back out ··· wN j w ˆjN ...
j=1
j=1
. . . wN 1 . .. . .. · · · wN N
Next, we can get the derivative term from (30), (34)
f (gN (k))
p
∂x(k + p)T ¯ +W ¯ )Λ(k + i − 1) = (W ∂x(k + 1) i=2
Λ(k) = diag{f (g1 (k)), · · · , f (gN (k))} in in T in T [WU ] = w11 , · · · , wN 1
Since we have ∂x(k + 2)T ∂x(k + p)T ∂x(k + p)T = · .. · ∂x(k + 1) ∂x(k + 1) ∂x(k + p − 1) p ∂x(k + i)T = ∂x(k + i − 1) i=2
L
(34)
and Λ(k + i − 1) = diag{f (g1 (k + i − 1)), · · · , f (gN (k + i − 1))}
where
jN
w11 .. ¯ W= . w1N
i=1
in , · · · , wN 1 f (gN (k)) ··· 0 .. .. . .
back out w ˆj1 w1j
j=1 .. ¯ W = . L ˆ out wback w
wji xi (k)
i=1
Then, we put equation (28) to in ∂x(k + 1)T = w11 f (g1 (k)) ∂uf (k + 1) f (g1 (k)) in .. in = w11 · · · wN 1 .
(33)
where m, n = 1, ..., N
where
i=1
(32)
j=1
(26)
where the control signal uf (k+1) is the first element of the ESN’s input. Rewriting the equation (2), we can express a new function as: g1 (k) (27) x(k + 1) = f ... gN (k) N
(31)
Then
∂x(k + 1)T ∂x(k + p)T ∂x(k + p)T = ∂uf (k + 1) ∂uf (k + 1) ∂x(k + 1)
in wji ui (k + 1) +
xm (k + i) N L back out = f( [(wmn + wmj w ˆjn )xn (k + i − 1)])
In addition, we have
K
Let equation (12) bring back to equation (2) , we can get the relationship between x(k + i − 1) and x(k + i)
where IN ×N is the identity matrix with N dimension, and 0N ×K is the zero matrix with N × K dimension.
gj (k) =
1955
(35)
Combining (22), (23), (24), (25), (26),(29), (35), we have ∂φ(k + p)T ∂ym (k + p) ∂ym (k + p) = ∂uf (k + 1) ∂uf (k + 1) ∂φ(k + p) p in ¯ +W ¯ )Λ(k + i − 1)} Λ(k){ (W = WU i=2
yj ·¯ IN ×(N +K) A−1 Ψ˜
(30)
Finally, it follows that 1991
(36)
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∂ym (k + p) ∂J = −e(k + p) ∂vi (k) ∂vi (k) = −λku h(ku fu )[1 − h(ku fu )]yi (k)e(k + p) p in ¯ +W ¯ )Λ(k + i − 1)} Λ(k){ (W ·WU
(37)
i=2
yj ·¯ IN ×(N +K) A−1 Ψ˜
4. CONVERGENCE ANALYSIS So as to demonstrate the convergence of the system conveniently, the discrete Lyapunov function can be defined as: 1 V (k) = e2 (k + p) (38) 2
Let the sum of the higher order terms O(k + p) equals to O(t) for short. Theorem 1. The closed-loop system is asymptotically stable when the sum of the higher order terms of the Taylor expansion is bounded and should be depicted as follow: O(t) + (1−QT (t)ηQ(t))e(t) (44) 2 < (1 − (QT (t)ηQ(t)) )e2 (t) and QT (t)ηQ(t) < 1 where
Q(t) = Let
where e(k +p) has been described above which denotes the modelling error of the ESGP referring to (18). The increasement of the Lyapunov function ∆V (k) can be obtained by: ∆V (k) = V (k + 1) − V (k) 1 = (e2 (k + p + 1) − e2 (k + p)) 2
(39)
By applying Taylor series expansion, the error can be decomposed into a known affine term plus a higher order term(Han et al. (2016))
p(t) = QT (t)ηQ(t)
(45)
Proof. Combining (41), (42), (43), (45), we can get 1 ∆V = −p(t)(1 − p(t))e2 (t) 2 1 +(1 − p(t))e(t)O(t) + O2 (t) (46) 2 < −p(t)(1 − p(t))e2 (t) 1 +(1 − p(t))e(t)O(t) + O2 (t) 2 Using (44),(45) and (46), it follows that
∂e(k + p) T e(k + p + 1) = e(k+p) + [ ] ∆v(k) + O(k+p)(40) ∂v(k) Then, we have 1 ∆V (k) = (e2 (k + p + 1) − e2 (k + p)) 2 1 = (e(k + p + 1) − e(k + p))(e(k + p + 1) + e(k + p)) 2 (41) 1 ∂e(k + p) T = ([ ] ∆v(k) + O(k+p)) 2 ∂v(k) ∂e(k + p) T ·(2e(k + p) + [ ] ∆v(k) + O(k+p)) ∂v(k)
∂ym (k + p) , e(t)=e(k + p) ∂v(k)
∆V < −p(t)(1 −p(t))e2 (t) +(1 − p(t))e(t)( (1 − p2 (t))e2 (t) − (1−p(t))e(t)) (47) 1 + ( (1 − p2 (t))e2 (t) − (1−p(t))e(t))2 2 =0 5. EXPERIMENT STRUDY The PMA system experiment platform that is comprised of the PMA plant and related devices is shown in Fig 2.
According to (13) and (18), it follows that ∂(yr (k + p) − yc (k + p)) ∂e(k + p) = ∂v(k) ∂v(k) ∂yc (k + p) ∂ym (k + p) =− =− ∂v(k) ∂v(k)
(42)
Also, the errors of SNN’s weights can be expressed in the following form: ∂J ∆v(k) = v(k + 1) − v(k) = −η ∂v(k) (43) ∂ym (k + p) ∂yc (k + p) =e(k + p)η = e(k + p)η ∂v(k) ∂v(k) T
where v(k) = [ v1 (k) v2 (k) v3 (k) v4 (k) ] , T ∂J ∂J ∂J ∂J ∂J = ∂v1 (k) ∂v2 (k) ∂v3 (k) ∂v4 (k) ∂v(k) and η = diag { η1 η2 η3 η4 } means the step size of the gradient descent algorithm.
Fig. 2. PMA system experiment platform An air compressor provides air pressure via the electromagnetic proportion valve. The measured displacement and force of the PMA are sampled and delivered to the computer through the data acquisition card. After that, the computer calculates control signals using SNN-ESGP strategy and feeds them into the electromagnetic proportion valve, which is able to regulate the input pressure.
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network. By analyzing the coefficients of SNN, we derived the iteration rules. Also, the Lyapunov function was put forward to illustrate the stability of the closed-loop system. The experiments were implemented to compare the performance of different methods. The results demonstrate that the SNN-ESGP can estimate the behaviours of the PMA plant and it is more accurate than PID as well as SMC.
0.04 Reference SNN-ESGP PID SMC
displacement (m)
0.035 0.03
1957
0.025 0.02 0.015 0.01
REFERENCES 4
6
8
10 time (s)
12
14
16
Fig. 3. Trajectory tracking control results of the model 3
× 10 -3 SNN-ESGP PID SMC
2
error (m)
1 0 -1 -2 -3
4
6
8
10 time (s)
12
14
16
Fig. 4. Comparison of the tracking error using SNN-ESGP, PID and SMC In our experiments, the MATLAB toolbox xPC Target, which provides an environment to construct the real-time system, is applied for the control strategies of the PMA plant. With the MATLAB/Simulink, C code is generated and downloaded into target computer for running. The PMA system is modeled by nonlinear autoregressive exogenous model y(k + 1) = f (u(k), y(k), y(k − 1)). Combining [u(k), y(k), y(k − 1)] and the dynamic reservoir states, the input of the predictive model using ESGP can be built and y(k+1) represents the predicted output of the ESGP. Together with the gradient descent algorithm, the SNN-ESGP approach has been implemented to evaluate the control performance. Since it is difficult to obtain optimal value of coefficients and weights, we apply PSO for the proper value. Then, the coefficients of the SNN are chosen as λ = 369598, ξ = 1.235, ku = 1.2. The position tracking result of the proposed control strategy is shown in Fig 3, and it clearly shows that it can track the reference with high-precision performance. Fig 4 gives the comparison of tracking errors of control strategies. It shows that the SMC is unable to get a satisfactory result, since it suffers from the inaccuracy of the model, and errors of the model and parameters are inevitable in identifying physical system. As a result, it is difficult for the SMC to get perferable performance. On the other hand, even though PID can achieve tracking the reference, it cannot meet the requirement of highprecision trajectory tracking, and the average error of SNN-ESGP is much smaller than the PID controller. With the application of the SNN-ESGP, it can satisfy the requirements of controlling rehabilitation robots. 6. DISCUSSION AND CONCLUSION In this paper, we proposed a predictive control based on an echo state Gaussian process with a single layer neural
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