# On steering the state of an LTI system to a desired state and keeping it there

For sake of simplicity, consider the super classic LTI system $$\begin{equation*} \begin{cases} & \dot{{x}}(t) =A {x}(t)+B {u}(t) , \quad \quad \text{for } 0\le t \le t_f \\ & x(0)=x_i \end{cases} \end{equation*}$$

I interpret this setting as a certain known dynamic $$\dot{{x}}(t) =A {x}(t)$$ describing a real process that we want to control by injecting the control term $$B {u}(t)$$. Now, assume that the system is controllable and say that we want to steer it from $$x(0)=x_i$$ to $$x(t_f)=t_f$$ with a control $$u(t)$$ that acts from $$t=0$$ to $$t=t_f$$.

By definition, since the system is controllable we can always find such a control $$u$$ but my question is: what happens after $$t_f$$? Since I interpret $$\dot{{x}}(t) =A {x}(t)$$ as a dynamical model that describes a real process, I also assume that such a dynamic will persist after $$t_f$$. If $$x_f$$ is not an equilibrium point of the dynamic, the system would revert to an equilibrium (assuming that $$A$$ is stable) after $$t_f$$. As you can notice, I am assuming $$u(t)=0$$ for $$t>t_f$$.

In other words: I bring the system to $$x(t_f)=x_f$$ with the control $$u$$ but after $$t_f$$ it seems that I am left with the system $$\begin{equation*} \begin{cases} & \dot{{x}}(t) =A {x}(t), \quad \quad \text{for } t > t_f \\ & x(t_f)=x_f \end{cases} \end{equation*}$$ which would converge to an equilibrium, possibily different from $$x_f$$. Was the effect of my control policy to bring the system to $$x_f$$ for an instant only? Should one consider an infinite-horizon in this case?

• You are right. But the answer to your question depends on the answer to "what is the desired behavior of the underlying real process"? Jun 1, 2020 at 13:41
• Well, I guess one might want to keep a system at $x_f$, or sufficiently close to it, for a certain period of time. Jun 1, 2020 at 13:55
• How about solving the linear system $B \bar{u} = - A x_f$ for $\bar{u}$? Jun 3, 2020 at 12:19
• Then you cannot put $u=0$. Typically, we say that $u(t)=u_0+u_1(t)$, where $u_0$ is the steady-state constant control that makes $x_f$ an equilibrium, and $u_1(t)$ is the control signal that drives $x$ to $x_f$. Then you can change coordinates as $e=x-x_f$ that yields $\dot{e} = Ae + Bu_1 + Ax_f + Bu_0$ and under the condition $Ax_f + Bu_0=0$ you have $\dot{e}=Ae+Bu_1$, which you want to stabilize at $e=0$. Jun 3, 2020 at 12:20
• How can I practically design $u_0$? I guess for $u_1(t)$ I could use, for example, the minimum energy control that drives the system to $x_f$. Jun 3, 2020 at 12:31

To summarize, controllability is very important as a concept, but it is not directly used for control signal calculations. And it isn't interested in what happens after $$t_f$$.
• For single input systems such a controller is unique and you can use the so-called Ackermann's formula (en.wikipedia.org/wiki/Ackermann%27s_formula). For multi input systems there are many different approaches but simplest one is the dyadic approach, where you can select a fan-out vector $f$ and convert the system to single input such as $(A,Bf)$. In general this problem is called eigenstructure assignment. Jun 3, 2020 at 12:33