Prescribed Performance Control 具有预设性能的控制

xiaoxiao2021-02-28  4

最近控制理论兴起了一种具有预设性能的控制方法,称之为prescribed performance control (PPC)。近些年来在机械系统的控制,比如无人艇(USV),水下潜器(UUV)和无人机(UAV)等领域得到了一些研究。其主要思路如下

A. Prescribed Performance Function

Definition: A smooth function ρ : R + → R + \rho: \mathfrak{R_+ \to R_+} ρ:R+R+ will be called a performance function if:\

ρ ( t ) \rho (t) ρ(t) is positive and decreasing l i m t → ∞ ρ ( t ) = ρ ∞ > 0 lim_{t \to \infty} \rho(t) =\rho_\infty >0 limtρ(t)=ρ>0

We can choose the ρ ( t ) \rho(t) ρ(t) as:

ρ ( t ) = ( ρ 0 − ρ ∞ ) e − α t + ρ ∞ \rho(t)=(\rho_0-\rho_\infty)e^{-\alpha t}+\rho_\infty ρ(t)=(ρ0ρ)eαt+ρ

Where ρ 0 , ρ ∞ \rho_0,\rho_\infty ρ0,ρ are both positive constants. If we want to achieve the prescribed transient and steady state behavioral bounds on the tracking errors e i ( t ) = x i ( t ) − x d i ( t ) e_i(t)=x_i(t)-x_{d_i}(t) ei(t)=xi(t)xdi(t), then guaranteeing the objective is equivalent to:

− δ i ρ i ( t ) < e i ( t ) < ρ i ( t ) i f e i ( 0 ) > 0 -\delta_i\rho_i(t)<e_i(t)<\rho_i(t)\quad\quad if \quad e_i(0)>0 δiρi(t)<ei(t)<ρi(t)ifei(0)>0 − ρ i ( t ) < e i ( t ) < δ i ρ i ( t ) i f e i ( 0 ) < 0 -\rho_i(t)<e_i(t)<\delta_i\rho_i(t)\quad\quad if \quad e_i(0)<0 ρi(t)<ei(t)<δiρi(t)ifei(0)<0

for all t ≥ 0 t\geq 0 t0 , where 0 ≤ δ i ≤ 1 0\leq \delta_i\leq 1 0δi1. Then provided that 0 < e i ( 0 ) < ρ i ( 0 ) 0<e_i(0)< \rho_i(0) 0<ei(0)<ρi(0), the constant represents the maximum allowable size of the tracking error at the steady state. Furthermore, the decreasing rate of ρ i ( t ) \rho_i(t) ρi(t) introduces a lower bound on the required speed of convergence of e i ( t ) e_i(t) ei(t), while the maximum overshoot is prescribed less than δ i ρ i ( t ) \delta_i\rho_i(t) δiρi(t) which may even become zero by setting δ i = 0 \delta_i=0 δi=0. Thus, the appropriate selection of the performance functions ρ i ( t ) \rho_i(t) ρi(t), as well as the design constants, imposes behavioral bounds on the system output trajectories.

B. Error Transformation

The aforementioned statements impose constraints on the errors equivalently. Then we propose an error transformation capable of transforming the original nonlinear system, with the constrained tracking error behavior, into an equivalent unconstrained one. Define: ε i ( t ) = T i ( e i ( t ) ρ i ( t ) ) \varepsilon_i(t)=T_i\Big(\frac{e_i(t)}{\rho_i(t)}\Big) εi(t)=Ti(ρi(t)ei(t)) Where ε i ( t ) \varepsilon_i(t) εi(t) is the transformed error and T i ( ⋅ ) T_i(\cdot) Ti() is a smooth strictly increasing function which define a mapping: KaTeX parse error: Undefined control sequence: \mbox at position 66: …ty,\infty), & \̲m̲b̲o̲x̲{if }e_i(0)\geq… In general ,the T i ( ⋅ ) T_i(\cdot) Ti() can be chose as T i ( e i ( t ) ρ i ( t ) ) = ln ⁡ ( δ i + e i ( t ) / ρ i ( t ) 1 − e i ( t ) / ρ i ( t ) ) if e i ( 0 ) ≥ 0 T_i(\frac{e_i(t)}{\rho_i(t)})=\ln \Big(\frac{\delta_i+e_i(t)/\rho_i(t)}{1-e_i(t)/\rho_i(t)}\Big) \quad \quad \text{if} \quad e_i(0)\geq 0 Ti(ρi(t)ei(t))=ln(1ei(t)/ρi(t)δi+ei(t)/ρi(t))ifei(0)0 T i ( e i ( t ) ρ i ( t ) ) = ln ⁡ ( 1 + e i ( t ) / ρ i ( t ) δ i − e i ( t ) / ρ i ( t ) ) if e i ( 0 ) ≤ 0 T_i(\frac{e_i(t)}{\rho_i(t)})=\ln \Big(\frac{1+e_i(t)/\rho_i(t)}{\delta_i-e_i(t)/\rho_i(t)}\Big) \quad \quad \text{if} \quad e_i(0)\leq 0 Ti(ρi(t)ei(t))=ln(δiei(t)/ρi(t)1+ei(t)/ρi(t))ifei(0)0 T i − 1 ( ε i ) = S ( ε i ) = { e ε i − δ i e − ε i e ε i + e − ε i , if e i ( 0 ) ≥ 0 δ i e ε i − e − ε i e ε i + e − ε i , if e i ( 0 ) ≤ 0 T_i^{-1}(\varepsilon_i)=S(\varepsilon_i)=\begin{cases} \frac{e^{\varepsilon_i}-\delta_i e^{-\varepsilon_i}}{e^{\varepsilon_i}+e^{-\varepsilon_i}}, & \text{if} \quad e_i(0)\geq 0\\ \frac{\delta_i e^{\varepsilon_i}-e^{-\varepsilon_i}}{e^{\varepsilon_i}+e^{-\varepsilon_i}}, & \text{if} \quad e_i(0)\leq 0\\ \end{cases} Ti1(εi)=S(εi)={eεi+eεieεiδieεieεi+eεiδieεieεiifei(0)0ifei(0)0

At the beginning, ε ( 0 ) \varepsilon(0) ε(0) is well defined, and if ∣ e ( 0 ) ∣ < ρ ( 0 ) |e(0)|<\rho(0) e(0)<ρ(0), then ε ( t ) \varepsilon(t) ε(t) can be well defined for t ≥ 0 t\geq 0 t0 with appropriate control law. And hence, the error transient and steady state can be guaranteed within the given function ρ ( t ) \rho(t) ρ(t)

Reference

Bechlioulis, C. P., & Rovithakis, G. A. (2008). Robust adaptive control of feedback linearizable mimo nonlinear systems with prescribed performance. IEEE Transactions on Automatic Control, 53(9), 2090-2099. Jia, Z., Hu, Z., & Zhang, W. (2019). Adaptive output-feedback control with prescribed performance for trajectory tracking of underactuated surface vessels. ISA transactions, 95, 18-26.

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