利用者:Lpa32xqn/sandbox

制御理論において、スライディングモード制御(SMC: Sliding Mode Control)とは、不連続な制御入力により制御対象の動特性を切り換え、状態変数が状態空間中の所定の平面上を｢滑る｣ように遷移させる非線形制御則の一種である。 スライディングモード制御は切り換えのある状態フィードバックにより実現される可変構造制御(VSS)であり、状態空間を領域ごとに分割し、状態変数が現在どの領域に存在するかに応じて制御則を切り換える。各領域ごとの制御則は、状態変数が常に他の領域へと遷移するように設計されるため、状態変数がある領域に留まり続けることはない。そのかわりに、領域間の境界面上を滑るように移動する。この動作をスライディングモードと呼び、状態変数が滑る平面を滑り(超)曲面という。現代制御理論の文脈では、スライディングモード制御を含む可変構造系は、連続的なダイナミクスと離散的な制御モードの切り換えが混在するハイブリッドシステムである。

概要

図1:スライディングモード制御により安定化されたシステムの位相平面図。状態変数が滑り曲面$s=0$に 到達したのち、滑り曲面に沿って遷移している。滑り曲面の選定は、滑り曲面に拘束された動特性が好ましい性質を持つように設計される。この例では、スライディングモード状態のシステムは滑り曲面$s=x_{1}+{\dot {x}}_{1}=0$により指数安定な原点をもつ1次のLTIシステム ${\dot {x}}_{1}=-x_{1}$に従う。

状態変数が滑り曲面に達すると、有限時間でスライディングモードが発生する。理論的には、一度スライディングモードを生ずると、状態変数は滑り曲面上に拘束され、滑り曲面上における動特性のみを考慮すればよい。たとえば図1の位相平面において、制御対象の状態変数が滑り曲面$s={\dot {\mathbf {x} }}_{1}+\mathbf {x} _{1}=0$上に拘束されるとする。このとき、たとえ制御対象が非線形システムであっても、その動特性は指数安定な原点を持つLTIシステム${\dot {\mathbf {x} }}_{1}(t)=-\mathbf {x} _{1}(t)$に従う。ただし現実的には、スライディングモードは高速かつ(一般に)不規則なスイッチング入力により状態変数を滑り曲面近傍でチャタリングさせることで近似的に実現される。

直感的に、制御対象の状態変数を滑り曲面上に拘束し続けるためには無限大の制御ゲインが必要となる。滑り曲面は状態変数をその上に拘束した際に好ましい動特性(たとえば滑り曲面上を移動して所望の平衡点に達するなど)が得られるように設計される。スライディングモード制御の主たる利点はそのロバスト性である。スライディングモード制御は、最も簡単には2種類の制御状態(たとえば｢ONとOFF｣、｢前進と後退｣など)を切り換えることができれば精確な入力が必要なく、パラメータ変動への感度も高くないことがその理由である。加えて制御則が不連続であることから、連続的な漸近ではなく、有限時間でスライディングモードを発生させることができる。特定の状況下においてはバンバン制御が最適となる場合があり、従ってスライディングモード制御は広い範囲の動的システムに対する最適制御則を与える。

スライディングモード制御の応用例として、スイッチングコンバータによる電気機器の制御が挙げられる 。スイッチングコンバータは不連続な制御モードをもつため、連続的な制御則をPWMや類似の技術により不連続に変換するよりも、本質的に不連続なスライディングモード制御のほうが自然な実装が可能である。スライディングモード制御はロボティクスにおいても多くの応用例がある。特に、擬似的に再現された荒波の中での無人船(USV)の追従制御において大きな成功を収めている。

スライディングモード制御は、比較的穏やかな制御を行う他の非線形制御則に比べてより慎重に実装しなければならない。特にアクチュエータの遅延やその他の不完全性により、字義通りのスライディングモード制御はチャタリング、エネルギー損失、制御対象へのダメージ、非モデル化動特性の励起といった問題を引き起こす可能性がある 。連続的な設計法によりこれらの問題を緩和し、スライディングモード制御に類似した動作を実現する方法もある。

制御方式

以下の非線形動的システムを考える。

${\dot {\mathbf {x} }}(t)=f(\mathbf {x} ,t)+B(\mathbf {x} ,t)\,\mathbf {u} (t)$ (1)

ここに、

$\mathbf {x} (t)\triangleq {\begin{bmatrix}x_{1}(t)\\x_{2}(t)\\\vdots \\x_{n-1}(t)\\x_{n}(t)\end{bmatrix}}\in \mathbb {R} ^{n}$

はn次元の状態変数ベクトル、

$\mathbf {u} (t)\triangleq {\begin{bmatrix}u_{1}(t)\\u_{2}(t)\\\vdots \\u_{m-1}(t)\\u_{m}(t)\end{bmatrix}}\in \mathbb {R} ^{m}$

はm次元の状態フィードバック制御入力である。Picard–Lindelöf theoremにより解$\mathbf {x} (t)$の存在性と一意性を保証するため、関数$f:\mathbb {R} ^{n}\times \mathbb {R} \to \mathbb {R} ^{n}$および$B:\mathbb {R} ^{n}\times \mathbb {R} \to \mathbb {R} ^{n\times m}$は連続かつ十分に滑らかであると仮定する。

状態フィードバック制御則$\mathbf {u} (\mathbf {x} (t))$の設計(すなわち、現在の状態$\mathbf {x} (t)$から時刻tにおける制御入力$\mathbf {u}$への写像の決定)典型的な問題は、式(1)で記述される動的システムを原点$\mathbf {x} =[0,0,\ldots ,0]^{\text{T}}$の近傍で安定化することである。このような制御則のもとでは、初期状態が原点から外れていても、システムは原点に復帰する。たとえば、状態変数ベクトル$\mathbf {x}$の一要素 $x_{1}$がある既知の信号(例えば正弦波状の目標値)とプラント出力の偏差に関連しているとする。このとき制御入力$\mathbf {u}$が$x_{1}$を迅速に$x_{1}=0$とできるのであれば、プラント出力は目標値の正弦波に追従する。スライディングモード制御において設計者は、システムをその配位空間中の特定の部分空間に拘束することで、望ましい動作(たとえば安定な平衡点をもつなど)が実現できることがわかってる。スライディングモード制御はシステムの解軌道をこの部分空間に移動させ、その上を滑らせる。この部分空間を滑り(超)曲面といい、フィードバック制御により状態が滑り曲面を滑っている状態をスライディングモードという。滑り曲面に沿った解軌道はLTIシステムの固有ベクトル(すなわちモード)に関連付けることができるが、スライディングモードはハイゲインフィードバックによりベクトル場を折りたたむことで補強されている。小石が亀裂に沿って転がるように、システムの解軌道はスライディングモードに拘束される。

スライディングモード制御の設計手順は以下の通りである。

1. 滑り曲面となる超曲面や多様体を、システムをその上に拘束したときの挙動が好ましくなるように選ぶ。
2. 選択した多様体をシステムの解軌道が横切り、その上に留まるようなフィードバックゲインを求める。

スライディングモード制御則は非連続であるため、解軌道を有限時間でスライディングモードに導くことができる(つまり、滑り曲面の安定性は漸近的な制御よりも良好となる)。しかし、一度解軌道が滑り曲面に到達すれば、システムはスライディングモードの特性に従う(たとえば、システムが原点 $\mathbf {x} =\mathbf {0}$においてのみ漸近安定となるなど)。

設計者は切換関数$\sigma :\mathbb {R} ^{n}\to \mathbb {R} ^{m}$を選択する。この関数は状態変数$\mathbf {x}$と滑り曲面との｢距離｣を表し、以下の値を取る。

• 状態変数$\mathbf {x}$が滑り曲面上にないとき$\sigma (\mathbf {x} )\neq 0$.
• 状態変数が滑り曲面上にあるとき$\sigma (\mathbf {x} )=0$.

スライディングモード制御則はこの切換関数の符号により制御モードを切り換える。従ってスライディングモード制御は状態変数を$\sigma (\mathbf {x} )=0$となる滑り曲面上へ移動させようとする。望ましい$\mathbf {x} (t)$の軌道は滑り曲面に近づくこと、また制御則が不連続である(すなわち状態変数が滑り曲面をまたいだときに制御モードが切り換わる)ため、状態変数は有限時間で滑り曲面上に到達する。一度滑り曲面上に到達した状態は、その上を滑り、たとえば原点$\mathbf {x} =\mathbf {0}$へ向かう。つまり、切換関数は等高線図のように、解軌道がどの方向に向かうかを表している。

滑り(超)曲面は$n\times m$次元の曲面であり、各制御入力$1\leq k\leq m$に対し、下式で表される$n\times 1$次元の滑り曲面が存在する。

$\left\{\mathbf {x} \in \mathbb {R} ^{n}:\sigma _{k}(\mathbf {x} )=0\right\}$ (2)

スライディングモード制御の設計において重要なのは、スライディングモードが存在し、かつシステムの解軌道が滑り曲面(すなわち$\sigma (\mathbf {x} )=\mathbf {0}$で与えられる曲面)に到達するような制御則を求めることである。スライディングモード制御の原理は適切な制御入力により所望の特性をもつ滑り曲面上にシステムを拘束することにある。このとき、制御対象の動特性は(2)で得られる低次元化されたシステムに従う。

状態変数 $\mathbf {x}$が$\sigma (\mathbf {x} )=\mathbf {0}$を満たすよう拘束できるための条件は以下の通りである。

1. 初期状態によらずシステムが$\sigma (\mathbf {x} )=\mathbf {0}$に到達できること。
2. 一度$\sigma (\mathbf {x} )=\mathbf {0}$に到達したシステムを維持できること。

閉ループ解の存在性

スライディングモード制御は不連続な制御則であるため、明らかに局所的にLipschitz連続ではない。このため、閉ループシステムの解の存在性と一意性をPicard–Lindelöf theoremにより保証することができない。このため、解はFilippov の意味で解析せねばならない。大まかに言えば、$\sigma (\mathbf {x} )=\mathbf {0}$上に拘束された状態変数の挙動は滑らかな動特性${\dot {\sigma }}(\mathbf {x} )=\mathbf {0}$により近似することができる。しかしながら、この滑らかな挙動は実際には実現できないかもしれない。同様に、高速なPWMやΔΣ変調の出力は2つの値しか取り得ないが、連続的な出力を行うのに有効である。これらの煩雑さは、連続的な制御器を実現する他の非線形制御則を用いることで解消することができることがある。

Theoretical foundation

The following theorems form the foundation of variable structure control.

Theorem 1: Existence of sliding mode

Consider a Lyapunov function candidate

$V(\sigma (\mathbf {x} ))={\frac {1}{2}}\sigma ^{\text{T}}(\mathbf {x} )\sigma (\mathbf {x} )={\frac {1}{2}}\|\sigma (\mathbf {x} )\|_{2}^{2}$ (3)

where $\|{\mathord {\cdot }}\|$ is the Euclidean norm (i.e., $\|\sigma (\mathbf {x} )\|_{2}$ is the distance away from the sliding manifold where $\sigma (\mathbf {x} )=\mathbf {0}$). For the system given by Equation (1) and the sliding surface given by Equation (2), a sufficient condition for the existence of a sliding mode is that

$\underbrace {\overbrace {\sigma ^{\text{T}}} ^{\tfrac {\partial V}{\partial \sigma }}\overbrace {\dot {\sigma }} ^{\tfrac {\operatorname {d} \sigma }{\operatorname {d} t}}} _{\tfrac {\operatorname {d} V}{\operatorname {d} t}}<0\qquad {\text{(i.e., }}{\tfrac {\operatorname {d} V}{\operatorname {d} t}}<0{\text{)}}$

in a neighborhood of the surface given by $\sigma (\mathbf {x} )=0$.

Roughly speaking (i.e., for the scalar control case when $m=1$), to achieve $\sigma ^{\text{T}}{\dot {\sigma }}<0$, the feedback control law $u(\mathbf {x} )$ is picked so that $\sigma$ and ${\dot {\sigma }}$ have opposite signs. That is,

• $u(\mathbf {x} )$ makes ${\dot {\sigma }}(\mathbf {x} )$ negative when $\sigma (\mathbf {x} )$ is positive.
• $u(\mathbf {x} )$ makes ${\dot {\sigma }}(\mathbf {x} )$ positive when $\sigma (\mathbf {x} )$ is negative.

Note that

${\dot {\sigma }}={\frac {\partial \sigma }{\partial \mathbf {x} }}\overbrace {\dot {\mathbf {x} }} ^{\tfrac {\operatorname {d} \mathbf {x} }{\operatorname {d} t}}={\frac {\partial \sigma }{\partial \mathbf {x} }}\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)\mathbf {u} \right)} ^{\dot {\mathbf {x} }}$

and so the feedback control law $\mathbf {u} (\mathbf {x} )$ has a direct impact on ${\dot {\sigma }}$.

Reachability: Attaining sliding manifold in finite time

To ensure that the sliding mode $\sigma (\mathbf {x} )=\mathbf {0}$ is attained in finite time, $\operatorname {d} V/{\operatorname {d} t}$ must be more strongly bounded away from zero. That is, if it vanishes too quickly, the attraction to the sliding mode will only be asymptotic. To ensure that the sliding mode is entered in finite time,

${\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ({\sqrt {V}})^{\alpha }$

where $\mu >0$ and $0<\alpha \leq 1$ are constants.

Explanation by comparison lemma

This condition ensures that for the neighborhood of the sliding mode $V\in [0,1]$,

${\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ({\sqrt {V}})^{\alpha }\leq -\mu {\sqrt {V}}.$

So, for $V\in (0,1]$,

${\frac {1}{\sqrt {V}}}{\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ,$

which, by the chain rule (i.e., $\operatorname {d} W/{\operatorname {d} t}$ with $W\triangleq 2{\sqrt {V}}$), means

${\mathord {\underbrace {D^{+}{\Bigl (}{\mathord {\underbrace {2{\mathord {\overbrace {\sqrt {V}} ^{{}\propto \|\sigma \|_{2}}}}} _{W}}}{\Bigr )}} _{D^{+}W\,\triangleq \,{\mathord {{\text{Upper right-hand }}{\dot {W}}}}}}}={\frac {1}{\sqrt {V}}}{\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu$

where $D^{+}$ is the upper right-hand derivative of $2{\sqrt {V}}$ and the symbol $\propto$ denotes proportionality. So, by comparison to the curve $z(t)=z_{0}-\mu t$ which is represented by differential equation ${\dot {z}}=-\mu$ with initial condition $z(0)=z_{0}$, it must be the case that $2{\sqrt {V(t)}}\leq V_{0}-\mu t$ for all t. Moreover, because ${\sqrt {V}}\geq 0$, ${\sqrt {V}}$ must reach ${\sqrt {V}}=0$ in finite time, which means that V must reach $V=0$ (i.e., the system enters the sliding mode) in finite time. Because ${\sqrt {V}}$ is proportional to the Euclidean norm $\|{\mathord {\cdot }}\|_{2}$ of the switching function $\sigma$, this result implies that the rate of approach to the sliding mode must be firmly bounded away from zero.

Consequences for sliding mode control

In the context of sliding mode control, this condition means that

$\underbrace {\overbrace {\sigma ^{\text{T}}} ^{\tfrac {\partial V}{\partial \sigma }}\overbrace {\dot {\sigma }} ^{\tfrac {\operatorname {d} \sigma }{\operatorname {d} t}}} _{\tfrac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ({\mathord {\overbrace {\|\sigma \|_{2}} ^{\sqrt {V}}}})^{\alpha }$

where $\|{\mathord {\cdot }}\|$ is the Euclidean norm. For the case when switching function $\sigma$ is scalar valued, the sufficient condition becomes

$\sigma {\dot {\sigma }}\leq -\mu |\sigma |^{\alpha }$.

Taking $\alpha =1$, the scalar sufficient condition becomes

$\operatorname {sgn} (\sigma ){\dot {\sigma }}\leq -\mu$

which is equivalent to the condition that

$\operatorname {sgn} (\sigma )\neq \operatorname {sgn} ({\dot {\sigma }})\qquad {\text{and}}\qquad |{\dot {\sigma }}|\geq \mu >0$.

That is, the system should always be moving toward the switching surface $\sigma =0$, and its speed $|{\dot {\sigma }}|$ toward the switching surface should have a non-zero lower bound. So, even though $\sigma$ may become vanishingly small as $\mathbf {x}$ approaches the $\sigma (\mathbf {x} )=\mathbf {0}$ surface, ${\dot {\sigma }}$ must always be bounded firmly away from zero. To ensure this condition, sliding mode controllers are discontinuous across the $\sigma =0$ manifold; they switch from one non-zero value to another as trajectories cross the manifold.

Theorem 2: Region of attraction

For the system given by Equation (1) and sliding surface given by Equation (2), the subspace for which the $\{\mathbf {x} \in \mathbb {R} ^{n}:\sigma (\mathbf {x} )=\mathbf {0} \}$ surface is reachable is given by

$\{\mathbf {x} \in \mathbb {R} ^{n}:\sigma ^{\text{T}}(\mathbf {x} ){\dot {\sigma }}(\mathbf {x} )<0\}$

That is, when initial conditions come entirely from this space, the Lyapunov function candidate $V(\sigma )$ is a Lyapunov function and $\mathbf {x}$ trajectories are sure to move toward the sliding mode surface where $\sigma (\mathbf {x} )=\mathbf {0}$. Moreover, if the reachability conditions from Theorem 1 are satisfied, the sliding mode will enter the region where ${\dot {V}}$ is more strongly bounded away from zero in finite time. Hence, the sliding mode $\sigma =0$ will be attained in finite time.

Theorem 3: Sliding motion

Let

${\frac {\partial \sigma }{\partial {\mathbf {x} }}}B(\mathbf {x} ,t)$

be nonsingular. That is, the system has a kind of controllability that ensures that there is always a control that can move a trajectory to move closer to the sliding mode. Then, once the sliding mode where $\sigma (\mathbf {x} )=\mathbf {0}$ is achieved, the system will stay on that sliding mode. Along sliding mode trajectories, $\sigma (\mathbf {x} )$ is constant, and so sliding mode trajectories are described by the differential equation

${\dot {\sigma }}=\mathbf {0}$.

If an $\mathbf {x}$-equilibrium is stable with respect to this differential equation, then the system will slide along the sliding mode surface toward the equilibrium.

The equivalent control law on the sliding mode can be found by solving

${\dot {\sigma }}(\mathbf {x} )=0$

for the equivalent control law $\mathbf {u} (\mathbf {x} )$. That is,

${\frac {\partial \sigma }{\partial \mathbf {x} }}\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)\mathbf {u} \right)} ^{\dot {\mathbf {x} }}=\mathbf {0}$

and so the equivalent control

$\mathbf {u} =-\left({\frac {\partial \sigma }{\partial \mathbf {x} }}B(\mathbf {x} ,t)\right)^{-1}{\frac {\partial \sigma }{\partial \mathbf {x} }}f(\mathbf {x} ,t)$

That is, even though the actual control $\mathbf {u}$ is not continuous, the rapid switching across the sliding mode where $\sigma (\mathbf {x} )=\mathbf {0}$ forces the system to act as if it were driven by this continuous control.

Likewise, the system trajectories on the sliding mode behave as if

${\dot {\mathbf {x} }}=\overbrace {f(\mathbf {x} ,t)-B(\mathbf {x} ,t)\left({\frac {\partial \sigma }{\partial \mathbf {x} }}B(\mathbf {x} ,t)\right)^{-1}{\frac {\partial \sigma }{\partial \mathbf {x} }}f(\mathbf {x} ,t)} ^{f(\mathbf {x} ,t)+B(\mathbf {x} ,t)u}=f(\mathbf {x} ,t)\left(\mathbf {I} -B(\mathbf {x} ,t)\left({\frac {\partial \sigma }{\partial \mathbf {x} }}B(\mathbf {x} ,t)\right)^{-1}{\frac {\partial \sigma }{\partial \mathbf {x} }}\right)$

The resulting system matches the sliding mode differential equation

${\dot {\sigma }}(\mathbf {x} )=\mathbf {0}$

and so as long as the sliding mode surface where $\sigma (\mathbf {x} )=\mathbf {0}$ is stable (in the sense of Lyapunov), the trajectory conditions from the reaching phase now reduce to the above derived simpler condition. Hence, the system can be assumed to follow the simpler ${\dot {\sigma }}=0$ condition after some initial transient during the period while the system finds the sliding mode. The same motion is approximately maintained when the equality $\sigma (\mathbf {x} )=\mathbf {0}$ only approximately holds.

It follows from these theorems that the sliding motion is invariant (i.e., insensitive) to sufficiently small disturbances entering the system through the control channel. That is, as long as the control is large enough to ensure that $\sigma ^{\text{T}}{\dot {\sigma }}<0$ and ${\dot {\sigma }}$ is uniformly bounded away from zero, the sliding mode will be maintained as if there was no disturbance. The invariance property of sliding mode control to certain disturbances and model uncertainties is its most attractive feature; it is strongly robust.

As discussed in an example below, a sliding mode control law can keep the constraint

${\dot {x}}+x=0$

in order to asymptotically stabilize any system of the form

${\ddot {x}}=a(t,x,{\dot {x}})+u$

when $a(\cdot )$ has a finite upper bound. In this case, the sliding mode is where

${\dot {x}}=-x$

(i.e., where ${\dot {x}}+x=0$). That is, when the system is constrained this way, it behaves like a simple stable linear system, and so it has a globally exponentially stable equilibrium at the $(x,{\dot {x}})=(0,0)$ origin.

Control design examples

• Consider a plant described by Equation (1) with single input u (i.e., $m=1$). The switching function is picked to be the linear combination

$\sigma (\mathbf {x} )\triangleq s_{1}x_{1}+s_{2}x_{2}+\cdots +s_{n-1}x_{n-1}+s_{n}x_{n}$ (4)

where the weight $s_{i}>0$ for all $1\leq i\leq n$. The sliding surface is the simplex where $\sigma (\mathbf {x} )=0$. When trajectories are forced to slide along this surface,

${\dot {\sigma }}(\mathbf {x} )=0$

and so

$s_{1}{\dot {x}}_{1}+s_{2}{\dot {x}}_{2}+\cdots +s_{n-1}{\dot {x}}_{n-1}+s_{n}{\dot {x}}_{n}=0$

which is a reduced-order system (i.e., the new system is of order $n-1$ because the system is constrained to this $(n-1)$-dimensional sliding mode simplex). This surface may have favorable properties (e.g., when the plant dynamics are forced to slide along this surface, they move toward the origin $\mathbf {x} =\mathbf {0}$). Taking the derivative of the Lyapunov function in Equation (3), we have

${\dot {V}}(\sigma (\mathbf {x} ))=\overbrace {\sigma (\mathbf {x} )^{\text{T}}} ^{\tfrac {\partial \sigma }{\partial \mathbf {x} }}\overbrace {{\dot {\sigma }}(\mathbf {x} )} ^{\tfrac {\operatorname {d} \sigma }{\operatorname {d} t}}$

To ensure ${\dot {V}}$ is a negative-definite function (i.e., ${\dot {V}}<0$ for Lyapunov stability of the surface $\mathbf {\sigma } =0$), the feedback control law $u(\mathbf {x} )$ must be chosen so that

${\begin{cases}{\dot {\sigma }}<0&{\text{if }}\sigma >0\\{\dot {\sigma }}>0&{\text{if }}\sigma <0\end{cases}}$

Hence, the product $\sigma {\dot {\sigma }}<0$ because it is the product of a negative and a positive number. Note that

${\dot {\sigma }}(\mathbf {x} )=\overbrace {{\frac {\partial {\sigma (\mathbf {x} )}}{\partial {\mathbf {x} }}}{\dot {\mathbf {x} }}} ^{{\dot {\sigma }}(\mathbf {x} )}={\frac {\partial {\sigma (\mathbf {x} )}}{\partial {\mathbf {x} }}}\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)u\right)} ^{\dot {\mathbf {x} }}=\overbrace {[s_{1},s_{2},\ldots ,s_{n}]} ^{\frac {\partial {\sigma (\mathbf {x} )}}{\partial {\mathbf {x} }}}\underbrace {\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)u\right)} ^{\dot {\mathbf {x} }}} _{{\text{( i.e., an }}n\times 1{\text{ vector )}}}$ (5)

The control law $u(\mathbf {x} )$ is chosen so that

$u(\mathbf {x} )={\begin{cases}u^{+}(\mathbf {x} )&{\text{if }}\sigma (\mathbf {x} )>0\\u^{-}(\mathbf {x} )&{\text{if }}\sigma (\mathbf {x} )<0\end{cases}}$

where

• $u^{+}(\mathbf {x} )$ is some control (e.g., possibly extreme, like "on" or "forward") that ensures Equation (5) (i.e., ${\dot {\sigma }}$) is negative at $\mathbf {x}$
• $u^{-}(\mathbf {x} )$ is some control (e.g., possibly extreme, like "off" or "reverse") that ensures Equation (5) (i.e., ${\dot {\sigma }}$) is positive at $\mathbf {x}$

The resulting trajectory should move toward the sliding surface where $\sigma (\mathbf {x} )=0$. Because real systems have delay, sliding mode trajectories often chatter back and forth along this sliding surface (i.e., the true trajectory may not smoothly follow $\sigma (\mathbf {x} )=0$, but it will always return to the sliding mode after leaving it).

• Consider the dynamic system

${\ddot {x}}=a(t,x,{\dot {x}})+u$

which can be expressed in a 2-dimensional state space (with $x_{1}=x$ and $x_{2}={\dot {x}}$) as

${\begin{cases}{\dot {x}}_{1}=x_{2}\\{\dot {x}}_{2}=a(t,x_{1},x_{2})+u\end{cases}}$

Also assume that $\sup\{|a(\cdot )|\}\leq k$ (i.e., $|a|$ has a finite upper bound k that is known). For this system, choose the switching function

$\sigma (x_{1},x_{2})=x_{1}+x_{2}=x+{\dot {x}}$

By the previous example, we must choose the feedback control law $u(x,{\dot {x}})$ so that $\sigma {\dot {\sigma }}<0$. Here,

${\dot {\sigma }}={\dot {x}}_{1}+{\dot {x}}_{2}={\dot {x}}+{\ddot {x}}={\dot {x}}\,+\,\overbrace {a(t,x,{\dot {x}})+u} ^{\ddot {x}}$

• When $x+{\dot {x}}<0$ (i.e., when $\sigma <0$), to make ${\dot {\sigma }}>0$, the control law should be picked so that $u>|{\dot {x}}+a(t,x,{\dot {x}})|$
• When $x+{\dot {x}}>0$ (i.e., when $\sigma >0$), to make ${\dot {\sigma }}<0$, the control law should be picked so that $u<-|{\dot {x}}+a(t,x,{\dot {x}})|$

However, by the triangle inequality,

$|{\dot {x}}|+|a(t,x,{\dot {x}})|\geq |{\dot {x}}+a(t,x,{\dot {x}})|$

and by the assumption about $|a|$,

$|{\dot {x}}|+k+1>|{\dot {x}}|+|a(t,x,{\dot {x}})|$

So the system can be feedback stabilized (to return to the sliding mode) by means of the control law

$u(x,{\dot {x}})={\begin{cases}|{\dot {x}}|+k+1&{\text{if }}\underbrace {x+{\dot {x}}} <0,\\-\left(|{\dot {x}}|+k+1\right)&{\text{if }}\overbrace {x+{\dot {x}}} ^{\sigma }>0\end{cases}}$

which can be expressed in closed form as

$u(x,{\dot {x}})=-(|{\dot {x}}|+k+1)\underbrace {\operatorname {sgn} (\overbrace {{\dot {x}}+x} ^{\sigma })} _{{\text{(i.e., tests }}\sigma >0{\text{)}}}$

Assuming that the system trajectories are forced to move so that $\sigma (\mathbf {x} )=0$, then

${\dot {x}}=-x\qquad {\text{(i.e., }}\sigma (x,{\dot {x}})=x+{\dot {x}}=0{\text{)}}$

So once the system reaches the sliding mode, the system's 2-dimensional dynamics behave like this 1-dimensional system, which has a globally exponentially stable equilibrium at $(x,{\dot {x}})=(0,0)$.

Automated design solutions

Although various theories exist for sliding mode control system design, there is a lack of a highly effective design methodology due to practical difficulties encountered in analytical and numerical methods. A reusable computing paradigm such as a genetic algorithm can, however, be utilized to transform a 'unsolvable problem' of optimal design into a practically solvable 'non-deterministic polynomial problem'. This results in computer-automated designs for sliding model control.

Sliding mode observer

Sliding mode control can be used in the design of state observers. These non-linear high-gain observers have the ability to bring coordinates of the estimator error dynamics to zero in finite time. Additionally, switched-mode observers have attractive measurement noise resilience that is similar to a Kalman filter. For simplicity, the example here uses a traditional sliding mode modification of a Luenberger observer for an LTI system. In these sliding mode observers, the order of the observer dynamics are reduced by one when the system enters the sliding mode. In this particular example, the estimator error for a single estimated state is brought to zero in finite time, and after that time the other estimator errors decay exponentially to zero. However, as first described by Drakunov, a sliding mode observer for non-linear systems can be built that brings the estimation error for all estimated states to zero in a finite (and arbitrarily small) time.

Here, consider the LTI system

${\begin{cases}{\dot {\mathbf {x} }}=A\mathbf {x} +B\mathbf {u} \\y={\begin{bmatrix}1&0&0&\cdots &\end{bmatrix}}\mathbf {x} =x_{1}\end{cases}}$

where state vector $\mathbf {x} \triangleq (x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}$, $\mathbf {u} \triangleq (u_{1},u_{2},\dots ,u_{r})\in \mathbb {R} ^{r}$ is a vector of inputs, and output y is a scalar equal to the first state of the $\mathbf {x}$ state vector. Let

$A\triangleq {\begin{bmatrix}a_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}}$

where

• $a_{11}$ is a scalar representing the influence of the first state $x_{1}$ on itself,
• $A_{21}\in \mathbb {R} ^{(n-1)}$ is a column vector representing the influence of the other states on the first state,
• $A_{22}\in \mathbb {R} ^{(n-1)\times (n-1)}$ is a matrix representing the influence of the other states on themselves, and
• $A_{12}\in \mathbb {R} ^{1\times (n-1)}$ is a row vector corresponding to the influence of the first state on the other states.

The goal is to design a high-gain state observer that estimates the state vector $\mathbf {x}$ using only information from the measurement $y=x_{1}$. Hence, let the vector ${\hat {\mathbf {x} }}=({\hat {x}}_{1},{\hat {x}}_{2},\dots ,{\hat {x}}_{n})\in \mathbb {R} ^{n}$ be the estimates of the n states. The observer takes the form

${\dot {\hat {\mathbf {x} }}}=A{\hat {\mathbf {x} }}+B\mathbf {u} +Lv({\hat {x}}_{1}-x_{1})$

where $v:\mathbb {R} \to \mathbb {R}$ is a nonlinear function of the error between estimated state ${\hat {x}}_{1}$ and the output $y=x_{1}$, and $L\in \mathbb {R} ^{n}$ is an observer gain vector that serves a similar purpose as in the typical linear Luenberger observer. Likewise, let

$L={\begin{bmatrix}-1\\L_{2}\end{bmatrix}}$

where $L_{2}\in \mathbb {R} ^{(n-1)}$ is a column vector. Additionally, let $\mathbf {e} =(e_{1},e_{2},\dots ,e_{n})\in \mathbb {R} ^{n}$ be the state estimator error. That is, $\mathbf {e} ={\hat {\mathbf {x} }}-\mathbf {x}$. The error dynamics are then

${\begin{aligned}{\dot {\mathbf {e} }}&={\dot {\hat {\mathbf {x} }}}-{\dot {\mathbf {x} }}\\&=A{\hat {\mathbf {x} }}+B\mathbf {u} +Lv({\hat {x}}_{1}-x_{1})-A\mathbf {x} -B\mathbf {u} \\&=A({\hat {\mathbf {x} }}-\mathbf {x} )+Lv({\hat {x}}_{1}-x_{1})\\&=A\mathbf {e} +Lv(e_{1})\end{aligned}}$

where $e_{1}={\hat {x}}_{1}-x_{1}$ is the estimator error for the first state estimate. The nonlinear control law v can be designed to enforce the sliding manifold

$0={\hat {x}}_{1}-x_{1}$

so that estimate ${\hat {x}}_{1}$ tracks the real state $x_{1}$ after some finite time (i.e., ${\hat {x}}_{1}=x_{1}$). Hence, the sliding mode control switching function

$\sigma ({\hat {x}}_{1},{\hat {x}})\triangleq e_{1}={\hat {x}}_{1}-x_{1}.$

To attain the sliding manifold, ${\dot {\sigma }}$ and $\sigma$ must always have opposite signs (i.e., $\sigma {\dot {\sigma }}<0$ for essentially all $\mathbf {x}$). However,

${\dot {\sigma }}={\dot {e}}_{1}=a_{11}e_{1}+A_{12}\mathbf {e} _{2}-v(e_{1})=a_{11}e_{1}+A_{12}\mathbf {e} _{2}-v(\sigma )$

where $\mathbf {e} _{2}\triangleq (e_{2},e_{3},\ldots ,e_{n})\in \mathbb {R} ^{(n-1)}$ is the collection of the estimator errors for all of the unmeasured states. To ensure that $\sigma {\dot {\sigma }}<0$, let

$v(\sigma )=M\operatorname {sgn} (\sigma )$

where

$M>\max\{|a_{11}e_{1}+A_{12}\mathbf {e} _{2}|\}.$

That is, positive constant M must be greater than a scaled version of the maximum possible estimator errors for the system (i.e., the initial errors, which are assumed to be bounded so that M can be picked large enough; al). If M is sufficiently large, it can be assumed that the system achieves $e_{1}=0$ (i.e., ${\hat {x}}_{1}=x_{1}$). Because $e_{1}$ is constant (i.e., 0) along this manifold, ${\dot {e}}_{1}=0$ as well. Hence, the discontinuous control $v(\sigma )$ may be replaced with the equivalent continuous control $v_{\text{eq}}$ where

$0={\dot {\sigma }}=a_{11}{\mathord {\overbrace {e_{1}} ^{{}=0}}}+A_{12}\mathbf {e} _{2}-{\mathord {\overbrace {v_{\text{eq}}} ^{v(\sigma )}}}=A_{12}\mathbf {e} _{2}-v_{\text{eq}}.$

So

${\mathord {\underbrace {v_{\text{eq}}} _{\text{scalar}}}}={\mathord {\underbrace {A_{12}} _{1\times (n-1) \atop {\text{ vector}}}}}{\mathord {\underbrace {\mathbf {e} _{2}} _{(n-1)\times 1 \atop {\text{ vector}}}}}.$

This equivalent control $v_{\text{eq}}$ represents the contribution from the other $(n-1)$ states to the trajectory of the output state $x_{1}$. In particular, the row $A_{12}$ acts like an output vector for the error subsystem

${\mathord {\overbrace {\begin{bmatrix}{\dot {e}}_{2}\\{\dot {e}}_{3}\\\vdots \\{\dot {e}}_{n}\end{bmatrix}} ^{{\dot {\mathbf {e} }}_{2}}}}=A_{2}{\mathord {\overbrace {\begin{bmatrix}e_{2}\\e_{3}\\\vdots \\e_{n}\end{bmatrix}} ^{\mathbf {e} _{2}}}}+L_{2}v(e_{1})=A_{2}\mathbf {e} _{2}+L_{2}v_{\text{eq}}=A_{2}\mathbf {e} _{2}+L_{2}A_{12}\mathbf {e} _{2}=(A_{2}+L_{2}A_{12})\mathbf {e} _{2}.$

So, to ensure the estimator error $\mathbf {e} _{2}$ for the unmeasured states converges to zero, the $(n-1)\times 1$ vector $L_{2}$ must be chosen so that the $(n-1)\times (n-1)$ matrix $(A_{2}+L_{2}A_{12})$ is Hurwitz (i.e., the real part of each of its eigenvalues must be negative). Hence, provided that it is observable, this $\mathbf {e} _{2}$ system can be stabilized in exactly the same way as a typical linear state observer when $A_{12}$ is viewed as the output matrix (i.e., "C"). That is, the $v_{\text{eq}}$ equivalent control provides measurement information about the unmeasured states that can continually move their estimates asymptotically closer to them. Meanwhile, the discontinuous control $v=M\operatorname {sgn} ({\hat {x}}_{1}-x)$ forces the estimate of the measured state to have zero error in finite time. Additionally, white zero-mean symmetric measurement noise (e.g., Gaussian noise) only affects the switching frequency of the control v, and hence the noise will have little effect on the equivalent sliding mode control $v_{\text{eq}}$. Hence, the sliding mode observer has Kalman filter–like features.

The final version of the observer is thus

${\begin{aligned}{\dot {\hat {\mathbf {x} }}}&=A{\hat {\mathbf {x} }}+B\mathbf {u} +LM\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\\&=A{\hat {\mathbf {x} }}+B\mathbf {u} +{\begin{bmatrix}-1\\L_{2}\end{bmatrix}}M\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\\&=A{\hat {\mathbf {x} }}+B\mathbf {u} +{\begin{bmatrix}-M\\L_{2}M\end{bmatrix}}\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\\&=A{\hat {\mathbf {x} }}+{\begin{bmatrix}B&{\begin{bmatrix}-M\\L_{2}M\end{bmatrix}}\end{bmatrix}}{\begin{bmatrix}\mathbf {u} \\\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\end{bmatrix}}\\&=A_{\text{obs}}{\hat {\mathbf {x} }}+B_{\text{obs}}\mathbf {u} _{\text{obs}}\end{aligned}}$

where

• $A_{\text{obs}}\triangleq A,$
• $B_{\text{obs}}\triangleq {\begin{bmatrix}B&{\begin{bmatrix}-M\\L_{2}M\end{bmatrix}}\end{bmatrix}},$ and
• $u_{\text{obs}}\triangleq {\begin{bmatrix}\mathbf {u} \\\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\end{bmatrix}}.$

That is, by augmenting the control vector $\mathbf {u}$ with the switching function $\operatorname {sgn} ({\hat {x}}_{1}-x_{1})$, the sliding mode observer can be implemented as an LTI system. That is, the discontinuous signal $\operatorname {sgn} ({\hat {x}}_{1}-x_{1})$ is viewed as a control input to the 2-input LTI system.

For simplicity, this example assumes that the sliding mode observer has access to a measurement of a single state (i.e., output $y=x_{1}$). However, a similar procedure can be used to design a sliding mode observer for a vector of weighted combinations of states (i.e., when output $\mathbf {y} =C\mathbf {x}$ uses a generic matrix C). In each case, the sliding mode will be the manifold where the estimated output ${\hat {\mathbf {y} }}$ follows the measured output $\mathbf {y}$ with zero error (i.e., the manifold where $\sigma (\mathbf {x} )\triangleq {\hat {\mathbf {y} }}-\mathbf {y} =\mathbf {0}$).

See also

• Variable structure control
• Variable structure system
• Hybrid system
• Nonlinear control
• Robust control
• Optimal control
• Bang–bang control – Sliding mode control is often implemented as a bang–bang control. In some cases, such control is necessary for optimality.
• H-bridge – A topology that combines four switches forming the four legs of an "H". Can be used to drive a motor (or other electrical device) forward or backward when only a single supply is available. Often used in actuator in sliding-mode controlled systems.
• Switching amplifier – Uses switching-mode control to drive continuous outputs
• Delta-sigma modulation – Another (feedback) method of encoding a continuous range of values in a signal that rapidly switches between two states (i.e., a kind of specialized sliding-mode control)
• Pulse-density modulation – A generalized form of delta-sigma modulation.
• Pulse-width modulation – Another modulation scheme that produces continuous motion through discontinuous switching.

Further reading

• Acary, V.; Brogliato, B. (2008). Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. Heidelberg: Springer-Verlag, LNACM 35. ISBN 978-3-540-75391-9 
• Drakunov S.V., Utkin V.I.. (1992). “Sliding mode control in dynamic systems”. International Journal of Control 55 (4): 1029–1037. doi:10.1080/00207179208934270. http://www.tandfonline.com/doi/abs/10.1080/00207179208934270?journalCode=tcon20#.UbOFHlF_mf4. 
• Edwards, Cristopher; Fossas Colet, Enric; Fridman, Leonid, eds (2006). Advances in Variable Structure and Sliding Mode Control. Lecture Notes in Control and Information Sciences. vol 334. Berlin: Springer-Verlag. ISBN 978-3-540-32800-1 
• Edwards, C.; Spurgeon, S. (1998). Sliding Mode Control: Theory and Applications. London: Taylor and Francis. ISBN 0-7484-0601-8 
• Utkin, V.I. (1992). Sliding Modes in Control and Optimization. Springer-Verlag. ISBN 978-0-387-53516-6 
• Zinober, Alan S.I., ed (1994). Variable Structure and Lyapunov Control. London: Springer-Verlag. doi:10.1007/BFb0033675. ISBN 978-3-540-19869-7