# Adaptive MIMO Supervisory Control Design Using Modeling Error

Authors: Zhi-Ren Tsai; Yau-Zen Chang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper proposes an adaptive control scheme for nonlinear systems with significant nonminimum phase dynamics. The scheme is composed of an inner-level adaptive fuzzy PD control law and an outer-level supervisory control law. Importantly, the inner-level controller of the two-level scheme is designed based on a fuzzy model, which takes nonminimum phase phenomenon and modeling error explicitly into account. The scheme is both much simpler in design and more applicable to general nonlinear systems when compared with most existing nonlinear controllers. Effectiveness of the proposed control strategy is demonstrated by numerical simulation of the control of a five-degree-of-freedom aircraft system in the face of bursting disturbances.


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## Body

## 1. Introduction

Many critical dynamic systems, such as aircraft, are nonminimum phase, MIMO, and highly nonlinear, which undergo significant disturbances and parameter variation during operation. To control these systems, robust control [1], optimal tuning of fuzzy controllers with output sensitivity function [2], adaptive control [3–5], and feedback linearization with discrete sliding-mode control [6] have attracted much attention from both academic and industrial communities due to their robustness to uncertainties. Recently, many interests have been focused on applying these techniques to flight control systems, such as [7, 8]. However, for systems with significant nonminimum phase phenomenon, direct application of these approaches tends to introduce unstable zero dynamics.For instance, in [9], the nonminimum phase plants are approximated by minimum phase models. The research [10] applied the output regulation theory to solve the output tracking problem, but a set of partial differential equations must be solved. The control scheme of [11] is based on decomposing the aircraft dynamics into a minimum phase part and a nonminimum phase part. Inversion is used on the minimum phase part to obtain asymptotic tracking, while a robust linear control approach is used to stabilize the nonminimum phase part, which is linearized at equilibrium. As this strategy is based on local linearization of the nonminimum phase part, the result can only apply to simplified models.By estimating parameters online, adaptive control can adapt to a controlled system with varying or unknown parameters. Nevertheless, in spite of the prosperous literature of adaptive control, practical application of these control strategies on MIMO systems has been restricted by the lack of assurance in closed-loop stability. Among them, the adaptive neural controller of [3] is too complex to implement, while the adaptive fuzzy terminal sliding-mode controller of [4, 5] is applicable only to robotic manipulators.The proposed adaptive control scheme is inspired by [12], which was developed for SISO nonlinear systems based on the feedback linearization technique, with the distinction that the scheme is extended to nonminimum phase MIMO control systems.The scheme is composed of an inner-level tracking control law and an outer-level supervisory control law. The design procedure is hence divided into two parts. First, an adaptive fuzzy-model-based PD control scheme is designed at the inner level to achieve robust output tracking. Special care is taken for the nonminimum phase fuzzy subsets in the control law by restricting parameter magnitudes in the singular-value decomposition operation. Next, a supervisory controller is employed at the outer level to minimize both the approximation error between the fuzzy model and the plant and the effects of external disturbance. Effectiveness of the adaptive control scheme is demonstrated by simulation results of the fight control of a complete 5-DOF aircraft model.


## 2. Problem Formulation

System dynamics of the plant are firstly represented in a general MIMO state-space representation as(1)x˙=Fx+Gx·u+w0,y=H·x,where x∈Rn×1 is the state vector, u∈Rm×1 is the control vector, w0∈Rn×1 is the disturbance vector, y∈RN×1 is the output vector, and F, G are corresponding nonlinear matrices in state vectors with H being a constant matrix, all of compatible dimensions.Equation (1) can be further represented in output vector y as(2)y˙=H·Fx+H·Gx·u+H·w0=fx+gx·u+w=hf·Af+∑i=1Lhi·Bi·u+fx-hf·Af+gx-∑i=1Lhi·Bi·u+w=hf·Af+∑i=1Lhi·Bi·u+e-mod≜hf·Af+Bc·u+e-mod,where (3)Bc≜∑i=1Lhi·Bi,hf=hf10000hf20000⋱0000hfn,Af=Af1Af2⋯AfnT,f1=∑i=1ph1i·A1i=hf1·Af1,f2=∑i=1ph2i·A2i=hf2·Af2,…,fn=∑i=1phni·Ani=hfn·Afn,e-mod=fx-hf·Af+gx-∑i=1Lhi·Bi·u+w,and the external disturbance w=H·w0.In the last representation, it is assumed thatg(x) is bounded and is away from singularity in a compact set. Furthermore, f(x) and g(x) are identified in fuzzy form as hf(y)·Af(t) and ∑j=1Lhj(y)·Bj, respectively, where the fuzzy logic systems are universal approximations which can uniformly approximate nonlinear continuous functions to arbitrary accuracy [13–15].


## 3. Controller Design for the Nonminimum Phase Dynamics

Firstly, the tracking error is defined as(4)et=-e-,where e-=yr-y(yr is reference input); we have that(5)e˙t=hf·Af+∑i=1Lhi·Bi·u+e-mod-y˙r=hf·Af+Bc·u+e-mod-y˙r,where u is a combination of two signals [16]:(6)u=uF+uS,with(7)uF=1-I∗·Bc-1·-hfy·Aft+y˙r+KP·et+KD·e˙t,(8)uS=-I∗·sgn⁡Bc·P·et·Bc-1·hfy·Aft-y˙r+eU.In (7), the proportional gain of the inner fuzzy control law is designed as(9)KP=α-R-1·P.The switching variable in both (7) and (8) is defined as(10)I∗=0,ife-mod≤e-U,I∗=1,otherwise,with(11)Bc-1·e-mod≤eU,α=-α10000-α20000⋱0000-αn,P>0,R>0.A complete control scheme of the two-level architecture is shown in Figure1.Figure 1
The proposed two-level switching control scheme.To avoid encountering singularity of the control law, the singular-value decomposition of the matrixBc is introduced as follows:(12)Bc=U·S·VT,where(13)S=σ10000σ20000⋱0000σnand σi is replaced by ε if σi≤ε, where ε is a small value.Substituting the adaptive fuzzy PD controlleruF (7) into (5), we have(14)e˙t=KP·et+hfy·A~ft+emod,after algebra manipulations, where (15)A~ft=Af∗-Aft,and the modeling error (16)emod=fx-hfy·Af∗+gx-∑i=1Lhiy·Bi·u+w+KD·e˙t.In the following derivation, we need the following condition to be satisfied [17–20]:(17)J≤e0T·P·e0+γf-1·traceA~f0T·A~f0,where (18)J=∫0tfetT·Q+PT·R-T·P·et-ρ2·emodT·emod·dt,the weighting factor γf>0, ρ2·I≥R, the matrix Q>0, and XF=trace(XT·X) is the Frobenius norm of matrix X. Derivation of this condition, (17), is given in the Appendix.Hence, we have that(19)A~˙ft=-A˙f=-γf·hfyT·P·et,αT·P+P·α+Q-PT·R-T·P=-ρ-2·PT·P.Furthermore, to guarantee boundedness ofAf, the parameter update laws must be modified as follows:(20)A˙f=γf·hfyT·P·et,ifAf<MforAf=Mf,A˙fT·Af≤0,Fγf·hfyT·P·et,otherwise,where Mf is a positive design parameter and the projection function F(·) is defined as(21)Fγf·hfyT·P·et=γf·hfyT·P·et-γf·Af·AfT·hfyT·P·etAf2.Next, the supervisor control law of (8) is designed by the following Lyapunov candidate:(22)V=etT·P·et.Its time derivative, V˙, can be obtained as(23)V˙=hf·Af+Bc·uS+e-mod-y˙rT·P·et+etT·P·hf·Af+Bc·uS+e-mod-y˙r=2etT·P·Bc·uS+2etT·P·hf·Af+e-mod-y˙r.Substituting (8) into (23) yields(24)V˙≤2etT·P·Bc·uS+2etT·P·Bc·Bc-1·hf·Af+e-mod-y˙r≤0.Hence, we can infer that if the supervisory control signal (8) is injected into fuzzy system (2), time derivative of the Lyapunov candidate V˙≤0 and system (2) is UUB stable.


## 4. Numerical Simulation

In this section, the proposed control strategy is applied on a five-degree-of-freedom aircraft system described in [21] for performance evaluation. We consider the angle of attack α and the roll angle ϕ as outputs to be tracked. Tracking of angle of attack is directly related to tracking of normal acceleration [21], which plays an important role in many practical maneuvers.Letb=3 be the reference length (m), c-=2 the mean aerodynamic chord (m), g=9.8 the gravitational acceleration (m/s2), I=50 the moment of inertia (kg-m2), p the roll angle rate, q the pitch angle rate, r the yaw angle rate, Q=80 the dynamic pressure (kg/m2), S=5 the reference wing area (m2), V=100 the aircraft velocity (m/s), θ the pitch angle, δa=0 the aileron deflection, δe the elevator deflection, δr the rudder deflection, and m=100 the mass of aircraft (kg); the aircraft dynamics can be written as [21](25)p˙q˙r˙=fpfqfr+Lδa0Lδr0Mδe0Nδa0Nδr·δaδeδr,α˙β˙φ˙θ˙=fαfβ00+-tβ·cα1-tβ·sαsα0-cα1tθ·sφtθ·cφ0cφ-sφ·pqr+w,where(26)ϕ=y1,α=y2,Lδa=I·Q·S·b·Clδa,Lδr=I·Q·S·b·Clδr,Mδe=I·Q·S·c-·Cmδe,Nδa=I·Q·S·b·Cnδa,Nδr=I·Q·S·b·Cnδr,fp=Q·S·I·b2·Clp·p2V,fq=Q·S·I·c-2·Cmq·q2V,fr=Q·S·I·b2·Cnr·r2V,fα=-Q·S·CLα·α+m·g·cθ·cϕ·cα+sθ·sαm·V·cβ,fβ=Q·S·CYβ·β+m·g·sθ·cα·sβ+cθ·sϕ·cβ-cθ·cϕ·sα·sβm·V,tβ=tan⁡β,sβ=sin⁡β,cβ=cos⁡β,sθ=sin⁡θ,cθ=cos⁡θ,tθ=tan⁡θ,cα=cos⁡α,sα=sin⁡α,sϕ=sin⁡ϕ,cϕ=cos⁡ϕ.In the following simulation, we assume(27)Clδa=-10-4,Clδr=10-2,Cmδe=-1.6×10-4,Cnδa=10-2,Cnδr=-10-4,Clp=-3.8×10-2,Cmq=-0.9×10-2,Cnr=-4.5×10-3,CLα=2.8×10-1,CYβ=-2.8.For the inner fuzzy control law, we select the following membership functions:(28)μF11y1=exp⁡-12y1-c11d112,μF12y1=exp⁡-12y1-c12d122,μF13y1=exp⁡-12y1-c13d132,μF14y1=exp⁡-12y1-c14d142,μF21y2=exp⁡-12y2-c21d212,μF22y2=exp⁡-12y2-c22d222,μF23y2=exp⁡-12y2-c23d232,μF24y2=exp⁡-12y2-c24d242,where(29)c11=c21=0,c12=c22=0.4,c13=c23=0.8,c14=c24=1.2,d11=d21=d12=d22=d13=d23=d14=d24=0.4.Furthermore, 8 fuzzy rules of the following form comprise the fuzzy rule base:R(1): if y1 is F1j, then f1=A1j for j=1,2,3,4 and l=1,2,3,4.R(2): if y2 is F2j, then f2=A2j for j=1,2,3,4 and l=5,6,7,8.Then, we obtain the following initial system parameters:(30)A11=-0.1338,A12=0.5183,A13=-1.0891,A14=1.546,A21=-1.0328,A22=1.669,A23=-0.0957,A24=-0.3342,Bc=16.5381-0.44291.14390.4489.Finally, we design the following control gains:(31)KP=-0.5200-51,KD=-0.0500-4.9.The tracking performances ofα and ϕ, together with the reference (or command), are presented in Figures 2 and 3. These figures show the responses with several step reference inputs. The disturbance is w=[1,1,1,1]T·δ(t-2), a burst at t=2 s.Figure 2
Time history of system outputy1 (the angle of attack, α), reference yr1, and control input u1.Figure 3
Time history of system outputy2 (the roll angle, ϕ), reference yr2, and control input u2.From the simulation results, it is clear that the output tracks the desired command asymptotically with small transient errors, and the zero dynamics remain stable for all the simulated interval.


## 5. Conclusion

We propose a two-level adaptive control scheme for nonlinear systems, such as the aircraft, which are MIMO and suffer from nonminimum phase phenomena. The control scheme is composed of an inner-level adaptive fuzzy PD control law and an outer-level supervisory control law. Importantly, the outer-level controller of the two-level scheme is designed based on a fuzzy model taking nonminimum phase phenomena and modeling error explicitly into account. Special care is taken of the nonminimum phase fuzzy subsets by restricting the magnitude of parameters in the singular-value decomposition operation.The control strategy is much simpler and applicable to general MIMO, nonlinear, and nonminimum phase systems when compared with [3–5]. Simulation results of the application of the proposed control scheme on a five-degree-of-freedom nonlinear aircraft model verify its effectiveness.



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*Source: 645168-2015-11-16.xml*