Adaptive Neuron-Like Control of Time-Delay Systems Enhanced with Feedforward and Supervisory Strategies

Abstract

Tracking control of nonlinear systems with significant delay effects has been the focus of intensive research. In this paper, we propose an effective supervised adaptive control scheme to tackle the problem. The scheme is composed of an adaptive control part of two neuron-like models with delay effects and a supervisory control part to enhance robustness against disturbance and model uncertainties. A design methodology based on the Lyapunov analysis is presented. Experimental results obtained from a practical temperature control system show that not only is the design procedure conceptually simple but also the control performance is also excellent when compared with the traditional PD controller. Also, the feedforward term is able to provide extra improvement in the regulation performance.

1. Introduction

The study of stability and stabilization for time-delay systems has received considerable attention in recent years [1–4] since delay is a major source of instability in many important engineering systems [5, 6]. For instance, Hopf bifurcation caused by time delay is extensively investigated in [7–10].

The applicability of neural-network-based techniques in nonlinear control systems has been successively demonstrated in [11–13] because of their unique modeling capability and adaptability [14, 15]. However, delay effects are not effectively considered in most of the proposed schemes and modeling error is ignored, which may be a potential source of instability [16]. As neural networks have superior capability in the construction of models of complex nonlinear systems, [17–20] use a feed-forward neural network for model-based predictive control. However, only simulation results are demonstrated in most of these researches. Reference [21] also applied an indirectly derived feedforward term in a simulation study, but the approach is based on predictability of disturbances.

In this paper, a particular class of adaptive neural controller is proposed based on a time-delay neural model. Inspired by [22], the model-based adaptation law has two auto-tuning neurons in which both delay effects and feed-forward terms are explicitly included, which are not considered in the original contribution. Robustness and stability conditions are derived in the sense of Lyapunov for the design of the proposed adaptation scheme, and performance of the proposed scheme is demonstrated by experimental results of a temperature control system.

2. Problem Formulation

The plant under control is assumed to be nonlinear with system dynamics being represented as 𝑥^(𝑛)(𝑡)=𝑓𝑋,𝑋_𝑑,𝑤_𝑚(𝑡,𝜏)+𝑔(𝑋)𝑢(𝑡)+𝑑₀(𝑡), (1) where 𝑢(𝑡) is the control input, 𝑑₀ is unknown but bounded disturbance with |𝑑₀(𝑡)|≤𝑑_𝑈, and 𝑤_𝑚(𝑡,𝜏) is measurable or predictable disturbance with delay 𝜏. Furthermore, 𝑋=[𝜉(𝑡),𝜉̇(𝑡),…,𝜉^(𝑛−1)(𝑡)]^𝑇=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇∈ℜ^𝑛×1 is the state vector of the nonlinear system with 𝑥(𝑡) being the state of the system and 𝑥_𝑖(𝑡),𝑖=1∼𝑛 being its derivatives, and 𝑋_𝑑=[𝜉(𝑡−𝜏_𝑑),𝜉̇(𝑡−𝜏_𝑑),…,𝜉^(𝑛−1)(𝑡−𝜏_𝑑)]^𝑇=[𝑥₁(𝑡−𝜏_𝑑),𝑥₂(𝑡−𝜏_𝑑),…,𝑥_𝑛(𝑡−𝜏_𝑑)]^𝑇 is the delayed state vector with delay 𝜏_𝑑; 𝑓(⋅) is the nonlinear system function in 𝑋, 𝑋_𝑑, and 𝑤_𝑚. Note that the nonlinear input gain 𝑔(⋅) is assumed to be a function in 𝑋 only.

3. Controller Design

Firstly, we define the desired output as 𝑦_𝑟 and tracking error as 𝑒=𝑦_𝑟−𝜉. Then we may define 𝑌_𝑟=[𝑦_𝑟,𝑦̇_𝑟,…,𝑦_𝑟^(𝑛−1)]^𝑇 and 𝐸=[𝑒,𝑒,̇…,𝑒^(𝑛−1)]^𝑇. Suppose that we choose a gain vector 𝐾=[𝑘₀,𝑘₁,…,𝑘_𝑛−1]^𝑇such that all roots of 𝑠^𝑛+𝑘_𝑛−1𝑠^𝑛−1+⋯+𝑘₁𝑠+𝑘₀=0 are in the open left-half complex plane. The proposed control law 𝑢(𝑡) is given by 𝑢(𝑡)=𝑢_𝐶𝜃_𝑓,𝜃_𝑔+𝑢_𝑆, (2) where 𝑢_𝐶(𝜃_𝑓,𝜃_𝑔) is an adaptive control law and 𝑢_𝑆 is a supervisory control law which enhances the robustness of the closed-loop system and improves transient performance by keeping system states stay in some prespecified region. The adaptive control law is defined as 𝑢_𝐶𝜃_𝑓,𝜃_𝑔=𝑔⁻¹𝜃_𝑔−𝑓𝜃_𝑓+𝑦_𝑟^(𝑛)+𝐾^𝑇𝐸, (3) where 𝑓(𝜃_𝑓) and 𝑔(𝜃_𝑔) are two neuron-like models: 𝑔𝜃_𝑔=2𝑎_𝑔1+exp−𝑏_𝑔⋅−ℎ_𝑔+∑^𝑛_𝑖=1𝜇_𝑖𝑥_𝑖(𝑡,𝑓𝜃_𝑓)=𝑎_𝑓⋅×1−exp−𝑏_𝑓⋅ℎ_𝑓+^𝐽𝑛_𝑗=0𝑖=1𝜇_𝑖𝑗𝑥_𝑖+𝑡−𝑗⋅𝜏_𝐿𝑙=0𝐾_𝐹𝑙𝑤_𝑚()××𝑡−𝑙⋅𝜏1+exp−𝑏_𝑓⋅ℎ_𝑓+^𝐽𝑛_𝑗=0𝑖=1𝜇_𝑖𝑗𝑥_𝑖+𝑡−𝑗⋅𝜏_𝐿𝑙=0𝐾_𝐹𝑙𝑤_𝑚(𝑡−𝑙⋅𝜏)⁻¹, (4)(5) where 𝜃_𝑓=[𝑎_𝑓,𝑏_𝑓,ℎ_𝑓,𝐾_𝐹0,𝐾_𝐹1,…,𝐾_𝐹𝐿,𝜇₁₀,𝜇₂₀,…]^𝑇 and 𝜃_𝑔=[𝑎_𝑔,𝑏_𝑔,ℎ_𝑔,𝜇₁,𝜇₂,…]^𝑇, with 𝑎_𝑔, 𝑏_𝑔, and ℎ_𝑔 being positive, are adjustable parameters, 𝑤_𝑚(𝑡) is a feed-forward term, and 𝐾_𝐹0,𝐾_𝐹1,…,𝐾_𝐹𝐿 are the corresponding feed-forward gains. In this study, two adaptation laws with projection mapping are defined as follows: 𝜃_𝑓̇=𝜑_𝑓,if𝜃_𝑓<𝑀_𝑓or𝜃_𝑓=𝑀_𝑓,−𝜃^𝑇_𝑓𝜑_𝑓,𝜑≥0_𝑓−𝜃_𝑓𝜃_𝑓²𝜃^𝑇_𝑓𝜑_𝑓𝜃̇,otherwise,_𝑔=𝜑_𝑔,if𝜃_𝑔<𝑀_𝑔or𝜃_𝑔=𝑀_𝑔,−𝜃^𝑇_𝑔𝜑_𝑔,𝜑≥0_𝑔−𝜃_𝑔𝜃_𝑔²𝜃^𝑇_𝑔𝜑_𝑔,otherwise, (6)(7) where 𝜑_𝑓=−𝛾_𝑓𝐸^𝑇𝑃⋅𝐵_𝑐𝜕𝑓𝜕𝜃_𝑓,𝛾_𝑓𝜑>0,_𝑔=−𝛾_𝑔𝐸^𝑇𝑃⋅𝐵_𝑐𝜕𝑔𝜕𝜃_𝑔𝑢_𝐶,𝛾_𝑔𝜕𝑓>0;𝜕𝜃_𝑓=𝜕𝑓𝜕𝑎_𝑓,𝜕𝑓𝜕𝑏_𝑓,𝜕𝑓𝜕ℎ_𝑓,𝜕𝑓𝜕𝐾_𝐹0,𝜕𝑓𝜕𝐾_𝐹1𝜕𝑓,…,𝜕𝐾_𝐹𝐿,𝜕𝑓𝜕𝜇₁₀,𝜕𝑓𝜕𝜇₂₀,…^𝑇,𝜕𝑓𝜕𝑎_𝑓=𝑓𝑎_𝑓,𝜕𝑔𝜕ℎ_𝑔=−𝑏_𝑔𝑔𝑔⋅1−2𝑎_𝑔,𝜕𝑓𝜕𝑏_𝑓=𝑎_𝑓ℎ_𝑓+∑^𝐽_𝑗=0∑^𝑛_𝑖=1𝜇_𝑖𝑗𝑥_𝑖𝑡−𝑗⋅𝜏2𝑓1+𝑎_𝑓×𝑓1−𝑎_𝑓,𝜕𝑓𝜕ℎ_𝑓=𝑎_𝑓𝑏_𝑓2𝑓1+𝑎_𝑓𝑓1−𝑎_𝑓,𝜕𝑓𝜕𝜇_𝑖𝑗=𝑎_𝑓𝑏_𝑓𝑥_𝑖𝑡−𝑗⋅𝜏2𝑓1+𝑎_𝑓𝑓1−𝑎_𝑓,𝜕𝑓𝜕𝐾_𝐹𝑙=𝑎_𝑓𝑏_𝑓𝑑_𝑓𝑓(𝑡−𝑙⋅𝜏)2𝑓1+𝑎_𝑓𝑓1−𝑎_𝑓,𝜕𝑔𝜕𝑎_𝑔=𝑔𝑎_𝑔,𝜕𝑔𝜕𝜃_𝑔=𝜕𝑔𝜕𝑎_𝑔,𝜕𝑔𝜕𝑏_𝑔,𝜕𝑔𝜕ℎ_𝑔,𝜕𝑔𝜕𝜇₁,𝜕𝑔𝜕𝜇₂,…^𝑇,𝜕𝑔𝜕𝑏_𝑔=−ℎ_𝑔+^𝑛_𝑖=1𝜇_𝑖𝑥_𝑖(𝑡𝑔)𝑔⋅1−2𝑎_𝑔,𝜕𝑔𝜕𝜇_𝑖=𝑏_𝑔𝑥_𝑖(𝑡)𝑔𝑔⋅1−2𝑎_𝑔. (8) The adaptation law, (4) and (5), is designed to ensure the boundedness of 𝜃_𝑓 and 𝜃_𝑔. Substituting (2) into (1), we have 𝑥^(𝑛)=𝑓+𝑔⋅𝑢_𝐶+𝑢_𝑆+𝑑₀=𝑦_𝑟^(𝑛)+𝐾^𝑇𝐸+𝑓−𝑓+𝑔−𝑔𝑢_𝐶+𝑔⋅𝑢_𝑆+𝑑₀. (9) This implies that 𝑒^(𝑛)=−𝐾^𝑇𝐸+𝑓−𝑓+𝑔−𝑔𝑢_𝐶−𝑔⋅𝑢_𝑆−𝑑₀. (10) Let 𝐴_𝑐=^{0(𝑛−1)×1,𝐼(𝑛−1)×(𝑛−1)}_−𝐾𝑇 and 𝐵_𝑐=[0_{1×(𝑛−1)},1]^𝑇 be a companion form pair; we may rewrite (10) as 𝐸̇=𝐴_𝑐𝐸+𝐵_𝑐⋅𝑓−𝑓+𝑔−𝑔𝑢_𝐶−𝑔⋅𝑢_𝑆−𝑑₀. (11) Now consider a Lyapunov function candidate 𝑉_𝐸=2⁻¹𝐸^𝑇𝑃⋅𝐸, (12) where 𝑃>0 which satisfies the Lyapunov equation 𝐴^𝑇_𝑐𝑃+𝑃⋅𝐴_𝑐=−𝑄, (13) with 𝑄 being a positive definite symmetric matrix. In the subsequent derivation, we will choose 𝑄 such that

𝜆_min(𝑄)>1 with 𝜆_min(𝑄) being the minimum eigenvalue of 𝑄. Define 𝑉_𝑀=2⁻¹𝜆_min(𝑃)𝑀_𝑋−𝑌_𝑟∞², (14) where 𝑀_𝑥 is a positive constant and 𝑌_𝑟=[𝑦_𝑟,𝑦̇_𝑟,…,𝑦_𝑟^(𝑛−1)]^𝑇. Note that, if ‖𝑥‖≥𝑀_𝑥≥‖𝑌_𝑟(𝑡)‖_∞=sup_𝑡≥0‖𝑌_𝑟(𝑡)‖, from (14), we have 𝑉_𝐸≥2⁻¹⋅𝜆_min(𝑃)⋅‖𝐸‖²≥2⁻¹⋅𝜆_min(𝑃)⋅‖𝑥‖−𝑌_𝑟(𝑡)²≥2⁻¹⋅𝜆_min(𝑃)⋅𝑀_𝑥−𝑌_𝑟(𝑡)_∞²=𝑉_𝑀. (15) Hence, if 𝑉_𝐸<𝑉_𝑀, we have that ‖𝑥‖<𝑀_𝑥. Moreover, the derivative of 𝑉_𝐸 along the trajectories of the closed-loop system (11) satisfies 𝑉̇_𝐸=2⁻¹𝐸^𝑇𝐴^𝑇_𝑐𝑃+𝑃𝐴_𝑐𝐸+𝐸^𝑇𝑃𝐵_𝑐⋅𝑓−𝑓+𝑔−𝑔𝑢_𝐶−𝑔⋅𝑢_𝑆−𝑑₀≤−2⁻¹𝐸^𝑇𝑄𝐸+𝐸^𝑇𝑃𝐵_𝑐×𝑓+𝑓+𝑔𝑢_𝐶+𝑔𝑢_𝐶+𝑑_𝑈−𝐸^𝑇𝑃𝐵_𝑐𝑔𝑢_𝑆. (16) From (16), if there exists a supervisory control law with𝑢_𝑆=𝐼^∗sgn𝐸^𝑇𝑃𝐵_𝑐𝑔_𝐿⁻¹𝑓_𝑈+𝑔_𝑈𝑢_𝐶+𝑓+𝑑_𝑈+𝑔𝑢_𝐶, (17) where 𝑓_𝑈, 𝑔_𝐿, 𝑔_𝑈 are boundary functions for 𝑓 and 𝑔 such that 0≤|𝑓|≤𝑓_𝑈 and 0<𝑔_𝐿≤𝑔≤𝑔_𝑈, then we can guarantee that𝑉̇_𝐸<0,where𝐼^∗=1,for𝑉_𝐸≥𝑉_𝑀,0,otherwise. (18) For the following deviation, we define the modeling error 𝑒_mod=𝑓𝜃^∗_𝑓−𝑓+𝑔𝜃^∗_𝑔−𝑔𝑢_𝐶−𝑑₀, (19) where 𝜃^∗_𝑓 and 𝜃^∗_𝑔 are the optimal parameters. Then (11) can be rewritten using Taylor series expansions as 𝐸̇=𝐴_𝑐𝐸−𝐵_𝑐𝑔𝑢_𝑆+𝐵_𝑐𝑓𝜃_𝑓−𝑓𝜃^∗_𝑓+𝑔𝜃_𝑔−𝑔𝜃^∗_𝑔𝑢_𝐶+𝑒_mod. (20) We have 𝑓𝜃_𝑓−𝑓𝜃^∗_𝑓=𝜃_𝑓−𝜃^∗_𝑓^𝑇𝜕𝑓𝜃_𝑓𝜕𝜃_𝑓+𝜀_𝑓,𝑔𝜃_𝑔−𝑔𝜃^∗_𝑔=𝜃_𝑔−𝜃^∗_𝑔^𝑇𝜕𝑔𝜃_𝑔𝜕𝜃_𝑔+𝜀_𝑔, (21) with 𝜀_𝑓 and 𝜀_𝑔 being the approximation errors of higher order terms. Now consider another Lyapunov function candidate 𝑉_𝐺 given by 𝑉_𝐺=2⁻¹𝐸^𝑇𝑃⋅𝐸+2𝛾_𝑓⁻¹𝜃_𝑓−𝜃^∗_𝑓^𝑇𝜃_𝑓−𝜃^∗_𝑓+2𝛾_𝑔⁻¹𝜃_𝑔−𝜃^∗_𝑔^𝑇𝜃_𝑔−𝜃^∗_𝑔. (22) Using (20)–(22), we have 𝑉̇_𝐺=−2⁻¹𝐸^𝑇𝑄⋅𝐸+𝛾_𝑓⁻¹⋅𝜃_𝑓−𝜃^∗_𝑓^𝑇×𝜃̇_𝑓+𝛾_𝑓𝐸^𝑇𝑃⋅𝐵_𝑐𝜕𝑓𝜃_𝑓𝜕𝜃_𝑓+𝛾_𝑔⁻¹⋅𝜃_𝑔−𝜃^∗_𝑔^𝑇𝜃̇_𝑔+𝛾_𝑔𝐸^𝑇𝑃⋅𝐵_𝑐𝜕𝑔𝜃_𝑔𝜕𝜃_𝑔𝑢_𝐶−𝑔⋅𝐸^𝑇𝑃⋅𝐵_𝑐𝑢_𝑆+𝐸^𝑇𝑃⋅𝐵_𝑐⋅𝑒_mod+𝜀_𝑓+𝜀_𝑔𝑢_𝐶. (23)

Furthermore, as 𝐸(0), 𝑒_mod(0), 𝑢_𝐶(0), 𝑑₀, 𝜀_𝑓 and 𝜀_𝑔 are bounded, and the projection method of the adaptation laws ensures the boundedness of 𝑢_𝐶, there exists a boundary function 𝑒̃ such that 𝐸^𝑇𝑃⋅𝐵_𝑐⋅𝑒_mod+𝜀_𝑓+𝜀_𝑔𝑢_𝐶≤̃𝑉̇𝑒,_𝐺≤−2⁻¹𝐸^𝑇𝑄𝐸+𝐸^𝑇𝑃𝐵_𝑐𝑒_mod+𝜀_𝑓+𝜀_𝑔𝑢_𝐶≤−2⁻¹𝜆_min(𝑄)‖𝐸‖²+𝑒.̃ (24)

This guarantees that 𝑉̇_𝐺<0, if ‖𝐸‖>(√𝑒/̃2⁻¹𝜆_min(𝑄)). From the two adaptation laws (6) and (7), we obtain that if (24) is satisfied, the system (1) is uniformly ultimately bounded (UUB) stable.

4. Experimental Study

Temperature control systems are among the nonlinear systems with significant delay effects.

The proposed control scheme has been implemented on a prototype temperature control system. The system includes a water tank, a water pump, a resistor heater which serves as disturbance, and four thermal couples, as shown in Figure 1. The 40 cm diameter tank is filled with water to a depth of 60 cm, the pump is driven by a 370 W frequency inverter, and the heater is driven by a SSR power IC. The control objective is to maintain the water temperature around the desired value 𝑦_𝑟=30^∘C.

(a)

(b)

Figure 1 

(a) Structure of the experimental setup. (b) A close view of the installation of heater, cooling pipe, and thermocouples in the water tank (with the cover being opened for observation). A 60 mm thick LDPE (low density polyethylene) foam insulates the tank and forms its cover.

The function and gains of the supervisory control law (17) are given as 𝑓_𝑈(𝑋)=1+0.004𝑥²₂, 𝑔_𝐿=2, 𝑔_𝑈=2.2. Moreover, we choose 𝑘₀=0.3 and 𝑘₁=0.01 so that the roots of 𝑠²+𝑘₁𝑠+𝑘₀=0 are in the left-half complex plane. For (11), we have 𝐴_𝑐=⁰¹_{−0.3−0.01} and 𝐵_𝑐=⁰₁. Besides, with 𝑄=²⁰⁰₀₂₀ for (13), we have 𝑃=^{1.3003×10333.3333}_{33.33334.3333×103}.

In designing the supervisory control law, we choose 𝑀_𝑋=0.25 for (14) and 𝑀_𝑓=4, 𝑀_𝑔=4, 𝛾_𝑓=1.6×10⁻³, and 𝛾_𝑔=0.8×10⁻⁶ for the adaptation laws (6) and (7).

For comparison, three different control schemes were implemented based on similar initial states around 𝑥₁(0)=𝜉(0)=30^∘C and 𝑥₂(0)=𝜉̇(0)=0:(1)the PD controller, 𝑢(𝑡)=𝐾_𝑃𝑒(𝑡)+𝐾_𝐷𝑒(̇𝑡), which was exhaustively tuned for best tracking performance to be with 𝐾_𝑃=0.3 and 𝐾_𝐷=0.05,(2)the proposed control scheme without the feed-forwarded disturbance term 𝑤_𝑚(𝑡). That is, 𝐾_𝐹0=⋯=𝐾_𝐹𝐿=0 in   (5),(3)the proposed control scheme.

Experimental results, shown in Figure 2, demonstrate that, in the face of disturbances, the output fluctuation was within 30 ± 0.42°C for (1), within 30 ± 0.25°C for (2), and within 30 ± 0.15°C using the proposed control scheme. It is clear that the proposed scheme was able to achieve accurate tracking performance in the face of measurable or predictable disturbance. Furthermore, under the condition of immeasurable disturbance, temperature of the adaptively controlled system, the control scheme of (2), suffered from larger fluctuation but is still better than that of the PD controlled system, demonstrating effectiveness of the adaptation for the nonlinear and delayed temperature control system.

(a)

(b)

Figure 2 

Experimental records of temperature when the system was controlled by four different control schemes.

5. Conclusion

We proposed a simple yet effective adaptive neural control scheme for delayed nonlinear systems. Experimental results validate its effectiveness and show that the feed-forward of disturbance, if available, can achieve further improvements. It is clear that the proposed scheme has an excellent regulation performance when compared with PD control law, and the feedforward term can achieve further improvements.

Acknowledgments

This paper was sponsored in part by Chang Gung University and the National Science Council, Taiwan, under Contract nos. NSC 100-2221-E-182-008 and NSC 101-2221-E-182-006.