# Prescribed Performance Control 具有预设性能的控制

xiaoxiao2021-02-28  4

## A. Prescribed Performance Function

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

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

We can choose the ρ ( t ) \rho(t) as:

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

Where ρ 0 , ρ ∞ \rho_0,\rho_\infty 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) , 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 − 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

for all t ≥ 0 t\geq 0 , where 0 ≤ δ i ≤ 1 0\leq \delta_i\leq 1 . Then provided that 0 < e i ( 0 ) < ρ i ( 0 ) 0<e_i(0)< \rho_i(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) introduces a lower bound on the required speed of convergence of e i ( t ) e_i(t) , while the maximum overshoot is prescribed less than δ i ρ i ( t ) \delta_i\rho_i(t) which may even become zero by setting δ i = 0 \delta_i=0 . Thus, the appropriate selection of the performance functions ρ i ( t ) \rho_i(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) Where ε i ( t ) \varepsilon_i(t) is the transformed error and T i ( ⋅ ) T_i(\cdot) 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) 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 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 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}

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