Stability of sampled-data systems using Lyapunov functions

Question

For continuous systems, Lyapunov functions provide a general technique to establish stability. For example, the simple system $x' = -x$, a Lyapunov function is $V(x) = \frac{1}{2}x^2$. It is easy to see that

1. $V(0) = 0$
2. $V(x) > 0$ for $x \neq 0$
3. $V(x)' = \frac{dV(x)}{dx}\frac{dx}{dt} = -x^2 < 0$ for $x\neq 0$

Is there an analogous technique for sampled-data systems?

For example, suppose I take the simple continuous system $x' = -x$ and naively turn it into a sampled-data system by introducing a zero-order hold. In other words, every $\delta$ time units, $x'$ is set to $-x$ and held at that value for the next $\delta$ time units. More precisely,

$$\forall k \in N, ~~~~ \forall t \in [k*\delta, (k+1)*\delta], ~~~~ x'(t) = -x(k*\delta)$$

Looking at the solutions to this system shows that it is asymptotically stable for $\delta < 2$. However, looking at the solutions of a system is only possible for simple systems. Is there instead a way of establishing asymptotic stability of this system using some sort of Lyapunov function?

Answer 1

For linear systems, you can consider the corresponding discrete-time system. In your example since $$\dot{x}(t)=-x(k\delta)\quad\forall t\in[k\delta,(k+1)\delta)$$ if we integrate over $[k\delta,(k+1)\delta)$ we obtain $$x((k+1)\delta)=x(k\delta)-\delta x(k\delta)=(1-\delta)x(k\delta)$$ which is asymptotically stable iff $$|1-\delta|<1$$ i.e. if $\delta<2$. This approach can be generalized to arbitrary linear systems. Applying a sampled-data control $$u(t)=u(k\delta)\qquad \forall t\in[k\delta,(k+1)\delta)$$ to a linear system of the form $$\dot{x}=Ax+Bu$$ creates the discrete-time system $$x((k+1)\delta)=A_d x(k\delta)+B_du(k\delta)$$ with $$A_d:=e^{A\delta}\\B_d:=\int_0^{\delta}{e^{A(\delta-s)}ds}\cdot B$$ If the pair $(A_d,B_d)$ is controllable then there exists some $K$ such that the eigenvalues of the matrix $A_d+B_dK$ lie in the interior of the unit circle. Selecting then the sampled-data state-feedback control law $$u(k\delta)=K\cdot x(k\delta)$$ we obtain the stable closed-loop discrete-time system $$x((k+1)\delta)=(A_d+B_d K) x(k\delta)$$ with $\lim_{k\rightarrow\infty}x(k\delta)=0$. For the state vector within the sampled-data interval it holds true that $$x(t)=\left[ e^{A(t-k\delta)}+\int_{k\delta}^t{e^{A(t-s)}ds}\cdot BK\right]x(k\delta)\qquad \forall t\in[k\delta,(k+1)\delta)$$ From the above equation, asymptotic stability can be proved now for the continuous time system ($\lim_{t\rightarrow\infty}x(t)=0$) using the fact that there exist some $C_1,C_2>0$ such that $$\sup_{t\in[k\delta,(k+1)\delta)}\|e^{A(t-k\delta)}\|=\sup_{\tau\in[0,\delta)}\|e^{A\tau}\|\leq C_1\\ \sup_{t\in[k\delta,(k+1)\delta)}\|\int_{k\delta}^t{e^{A(t-s)}ds}\cdot BK\|=\sup_{\tau\in[0,\delta)}\|\int_{k\delta}^{\tau+k\delta}{e^{A(\tau+k\delta-s)}ds}\cdot BK\|\\=\sup_{\tau\in[0,\delta)}\|\int_{0}^{\tau}{e^{A(\tau-w)}dw}\cdot BK\|\leq C_2$$ and $\lim_{k\rightarrow\infty}x(k\delta)=0$.

Edit: For varying-length sample time intervals $\delta_i$ ($i=1,2,\cdots$) let us define the sample times $$t_k:=\sum_{i=1}^k{\delta_i}$$ and the sampled and hold control law $$u(t)=K_k x\left(t_k\right)\quad t\in\left[t_k,t_{k+1}\right)$$ Then, the matrices describing the discrete-time system $A_d,B_d$ are not constant i.e. $$x(t_{k+1})=A_{d,k}x(t_k)+B_{d,k}u(t_k)$$ with $$A_{d,i}=e^{A\delta_{i+1}}\\ B_{d,i}:=\int_0^{\delta_{i+1}}{e^{A(\delta_{i+1}-s)}ds}\cdot B$$ The closed-loop system takes the form $$x(t_{k+1})=(A_{d,k}+B_{d,k}K_k)x(t_k)$$ If $$\sup_{k\in\mathbb{N}}\|A_{d,k}+B_{d,k}K_k\|=\epsilon<1$$ then $$\|x(t_{k+1})\|=\left\|\prod_{i=1}^k{(A_{d,i}+B_{d,i}K_i)} x(0) \right\|\leq \prod_{i=1}^k\|A_{d,i}+B_{d,i}K_i\| \|x(0)\|\leq \epsilon^k \|x(0)\|$$ and therefore $$\lim_{k\rightarrow\infty}x(t_k)=0$$ For the state vector within the sampled-data interval it holds true that $$x(t)=\left[ e^{A(t-t_k)}+\int_{t_k}^t{e^{A(t-s)}ds}\cdot BK_k\right]x(t_k)\qquad \forall t\in[t_k,t_{k+1})$$ Assuming an upper bound on the allowed time span between two samples i.e. $\delta_k\leq \bar{\delta}$ $\forall k\in\mathbb{N}$ for some $\bar{\delta}>0$ then there exist constants $C'_1,C'_2>0$ such that $$\sup_{k\in\mathbb{N}}\sup_{t\in[t_k,t_{k+1})}\|e^{A(t-t_k)}\|=\sup_{k\in\mathbb{N}}\sup_{\tau\in[0,\delta_{k+1})}\|e^{A\tau}\|\leq\sup_{\tau\in[0,\bar{\delta})}\|e^{A\tau}\|\leq C'_1\\ \sup_{k\in\mathbb{N}}\sup_{t\in[t_k,t_{k+1})}\|\int_{t_k}^t{e^{A(t-s)}ds}\cdot BK_k\|=\sup_{k\in\mathbb{N}}\sup_{\tau\in[0,\delta_{k+1})}\|\int_{t_k}^{\tau+t_k}{e^{A(\tau+t_k-s)}ds}\cdot BK_k\|\\=\sup_{k\in\mathbb{N}}\sup_{\tau\in[0,\delta_{k+1})}\|\int_{0}^{\tau}{e^{A(\tau-w)}dw}\cdot BK_k\|\leq \sup_{\tau\in[0,\bar{\delta})}\|\int_{0}^{\tau}{e^{A(\tau-w)}dw}\|\cdot \sup_{k\in\mathbb{N}}\|BK_k\|\leq C'_2$$ Now we can prove similarly that $\lim_{t\rightarrow\infty}x(t)=0$.

For nonlinear systems the problem is more difficult and the results are typically based on discrete-time approximations of the system. See for example the papers

Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations, D. Nesic, A.R. Teel, P.V. Kokotovic, Systems and Control Letters, 38, pp. 259-270, 1999

L. Grune, D. Nesić, Optimization based stabilization of sampled-data nonlinear systems via their approximate discrete-time models, SIAM Journal on Control and Optimisation, 42 (2003), pp. 98–122

for further details.