Gain-Scheduled Worst-Case Control on Nonlinear Stochastic Systems Subject to Actuator Saturation and Unknown Information

Abstract

In this paper, we propose a method for designing continuous gain-scheduled worst-case controller for a class of stochastic nonlinear systems under actuator saturation and unknown information. The stochastic nonlinear system under study is governed by a finite-state Markov process, but with partially known jump rate from one mode to another. Initially, a gradient linearization procedure is applied to describe such nonlinear systems by several model-based linear systems. Next, by investigating a convex hull set, the actuator saturation is transferred into several linear controllers. Moreover, worst-case controllers are established for each linear model in terms of linear matrix inequalities. Finally, a continuous gain-scheduled approach is employed to design continuous nonlinear controllers for the whole nonlinear jump system. A numerical example is given to illustrate the effectiveness of the developed techniques.

Appendix

1.1 Proof of Proposition 3.1

Proof

The Lyapunov–Krasovskii function for system (2) is constructed by using symmetric positive definite matrices P _k (i):

$$V\bigl(x(t),i\bigr)=x^\mathrm{T}(t)P_{k}(i)x(t), \quad i\in \varLambda. $$

The time derivative of V(x(t),i) for system (2) is

Recalling Lemma 3.1, one can define that the system state belongs to a set Θ(H _k (i)), and ε(P _k (i),1)⊂Θ(H _k (i)), then σ(u(t)) can be described as $\sigma (u(t))=\mathit{co}\{[D_{t}K_{k}(i)+D_{t}^{-}H_{k}(i)]x(t) : t\in[1,2^{m}]\}$.

Obviously, a sufficient stabilizable condition for system (2) is that all the vertex of the convex hull satisfy the desired stable requirements.

One can define

Since $\sum_{j=1}^{N}\pi_{ij}=0$, one can obtain

$$\varPhi_{1k}(i)=\varPhi_{1k}(i)+\sum _{j=1}^{N}\pi_{ij}\bigl[\hat {A}^\mathrm{T}_{k}(i)P_{k}(i)+P_{k}(i) \hat{A}_{k}(i)\bigr], $$

where

$$\hat {A}_{k}(i)=A_{k}(i)+B_{k}(i) \bigl(D_{t}K_{k}(i)+D_{t}^{-}H_{k}(i) \bigr),\quad \forall t\in\bigl[1,2^{m}\bigr]. $$

Recalling Assumption 2.1, one can also obtain

$$\varPhi_{1k}(i)=\varPhi_{2k}(i)+\sum _{j\in \varLambda_{UK}^{i}}\pi_{ij}\bigl[\hat{A}^\mathrm {T}_{k}(i)P_{k}(i)+P_{k}(i)\hat{A}_{k}(i)+P_{k}(j) \bigr], $$

where

$$\varPhi_{2k}(i)=\biggl(1+\sum_{j\in \varLambda_{K}^{i}} \pi_{ij}\biggr) \bigl(\hat{A}^\mathrm {T}_{k}(i)P_{k}(i)+P_{k}(i) \hat{A}_{k}(i)\bigr)+\biggl(\sum_{j\in \varLambda_{K}^{i}} \pi_{ij}\biggr)P_{k}(j). $$

Subsequently, for i,j∈Λ, π _ij ≥0, i≠j, and $j\in\varLambda_{UK}^{i}$, if π _ii is known, the following condition (19) can be guaranteed both by (5), (6) under the condition (8).

$$ \varPhi_{1k}(i)<0, \quad\forall t\in\bigl[1,2^{m}\bigr]. $$

(19)

On the other hand, for $j\in\varLambda_{UK}^{i}$, if π _ii is unknown, Φ _1k (i) can be described as:

As we known, $\pi_{ii}=-\sum_{j=1,\,j\neq i}^{N}<0$, then, for dynamic system (2), condition (19) can be guaranteed both by (5), (6), (7) under the condition (8).

Obviously, condition (19) implies

$$\varGamma V\bigl(x(t),i\bigr)<0,\quad\forall t\in\bigl[1,2^{m}\bigr]. $$

Therefore, the dynamic MJLS (2) is stochastic stable with w(t)=0, and this concludes the proof. □

1.2 Proof of Theorem 3.1

Proof

Introduce the following cost function for system (2) as T>0

$$ J(T)=E\biggl\{\int_0^{T}z^\mathrm{T}(t)z(t)\,dt \biggr\}- \lambda^{2}E\biggl\{\int_0^{T}w^\mathrm{T}(t)w(t)\,dt \biggr\}. $$

(20)

Under zero initial condition, index J(T) can be rewritten as

(21)

Under condition (13), it follows that

(22)

where $\triangle_{k}(i)=D_{t}K_{k}(i)+D_{t}^{-}H_{k}(i)$.

Thus,

$$ J(T)\leq E\int_0^{T}\bigl\{S\cdot X_{1k}\cdot S^\mathrm{T}\bigr\}\,dt, $$

(23)

where S=[x ^T(t) w ^T(t)],

By using Schur complement for X ₁, one can obtain

$$X_{2k}=\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{}} M_{2} & P_{k}(i)B_{wk}(i) & (C_{k}(i)+D_{k}(i)\triangle_{k}(i))^\mathrm{T}\\[3pt] \ast& -\lambda^{2}I & D_{wk}^{\mathrm{T}}(i)\\[3pt] \ast& \ast& -I \end{array} \right ], $$

where $M_{2}=\hat{A}^{\mathrm{T}}_{k}(i)P_{k}(i)+P_{k}(i)\hat{A}_{k}(i)+\sum_{j=1}^{N}\pi_{ij}P_{k}(j)$.

Along the same line as in the proof of Proposition 3.1, one can obtain:

$$X_{2k}=\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{}} M_{3} & P_{k}(i)B_{wk}(i) & (C_{k}(i)+D_{k}(i)\triangle_{k}(i))^\mathrm{T}\\[3pt] \ast& -\lambda^{2}I & D_{wk}^{\mathrm{T}}(i)\\[3pt] \ast& \ast& -I \end{array} \right ]+\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{}} M_{4} & 0 & 0\\ \ast& 0 & 0\\ \ast& \ast& 0 \end{array} \right ], $$

where

Clearly, X _2k <0 can be reduced to inequality (5), by denoting w(t)=0, it can also be guaranteed by inequalities (10)–(12) under the condition (13), so the dynamic system (2) is stochastic stable in the proposed region. On the other hand, for T→∞, X _2k <0 results in J(∞)<−V(∞)<0, that is,

$$ E\biggl\{\int_0^{\infty}z^\mathrm{T}(t)z(t)\,dt \biggr\}\leq \lambda^{2}E\biggl\{\int_0^{\infty}w^\mathrm{T}(t)w(t)\,dt \biggr\}. $$

(24)

This completes the proof. □