# DT State-Space State Feedback Control

Suppose we are given the following discrete-time system:

$$$$\begin{bmatrix} x_1(k+1)\\ x_2(k+1) \end{bmatrix} = \begin{bmatrix} 0 && 1\\ -2 && 0 \end{bmatrix}\begin{bmatrix} x_1(k)\\ x_2(k) \end{bmatrix} + \begin{bmatrix} 0\\ 1 \end{bmatrix}u(k)$$$$

$$$$y_1 = \begin{bmatrix} 1 && 1 \end{bmatrix}\begin{bmatrix} x_1\\ x_2 \end{bmatrix}$$$$

For this system we can construct a state feedback in the form

$$u(k) = v(k) + \begin{bmatrix} k_1 && k_2 \end{bmatrix}\begin{bmatrix} x_1(k)\\ x_2(k) \end{bmatrix}$$

for which the system has a double pole at some $$\beta$$. I can design such a state feedback.

Now, suppose this desgined feedback can be active for one time step and the systems returns to its original form (no feedback) until the the next re-activation of the feedback. This re-activation takes 2 time steps. I want to find the value $$\beta$$ to maintain the stability. I'm confused at this question since the stability is considered as $$k$$ goes $$\infty$$.

• This will be a typical switched system with two subsystems. You may try to design a control law that stabilizes the system for any switching sequence, but that will be restrictive. Otherwise, you may consider certain dwell-time constraints and try to stabilize the system under those dwell-time constraints. Typical dwell-times are fixed, constant, minimum, persistent, and range dwell-times. There is an abundant literature on the topic.
– KBS
Commented Sep 30, 2022 at 10:15
• Do you have any idea of the activation pattern of the controller?
– KBS
Commented Sep 30, 2022 at 10:52
• @KBS, I'm not familiar with this concept. I studied state feedback and pole placement. Can you suggest a source (textbook) for switched systems?
– eet
Commented Sep 30, 2022 at 10:58
• Please answer the question regarding how the controller activates.
– KBS
Commented Sep 30, 2022 at 11:06
• @KBS, Activation pattern repeats itself perdiodically of 3 time steps. 1 step with feedback and 2 steps with no feedback. Then repeat.
– eet
Commented Sep 30, 2022 at 11:37

Based on the comment, this is a periodic system that can be written as

$$x(3(k+1))=A^2(A+BK)x(3k).$$

Asymptotic stability is ensured provided that the eigenvalues of $$A^2(A+BK)$$ are inside the unit disc. A way to ensure that is through the consideration of a quadratic Lyapunov function of the form $$V(x)=x^TPx$$ where $$P$$ symmetric positive definite (i.e. $$P\succ0$$). This yields the following Linear Matrix Inequality

$$\begin{bmatrix} -Q & A^3Q+A^2BU\\(A^3Q+A^2BU)^T & -Q \end{bmatrix}\prec0$$

where $$U$$ and $$Q\succ0)$$ are matrices to be computed and the inequality sign means that the above expression is negative definite. A suitable gain $$K$$ can be constructed using the expression $$K=UQ^{-1}$$.

This can be slightly modified by considering a scalar $$\beta\in(0,1)$$ and solving for the condition

$$\begin{bmatrix} -\beta^2 Q & A^3Q+A^2BU\\(A^3Q+A^2BU)^T & -Q \end{bmatrix}\prec0,$$

which is equivalent to saying that the spectral radius of $$A^2(A+BK)$$ is less than $$\beta$$.

A more direct approach (see Kwin's comment) is to just design a state-feedback controller for the system $$(A^3,A^2B)$$. It is interesting to note that this system may not be controllable even if $$(A,B)$$ is. A simple example is the integrator $$\dot{x}=u$$.

• Lyapunov equation for discrete time systems is that there exists a positive definite $P$ which is the solution of the equation $A^TPA - P < 0$ for the exponential stability (or asy. stability for lti). I dont understand the the matrix you constructed (which is negative definite) in the answer. What's that?
– eet
Commented Sep 30, 2022 at 12:22
• @eet I considered the Lyapunov condition, performed a Schur complement, then a congruence transformation, and finally I used the change of variables to get a condition which is affine in both $Q$ and $U$.
– KBS
Commented Sep 30, 2022 at 12:42
• One can also just use $(A^3,A^2B)$ instead of $(A,B)$ and design a state feedback gain $K$ for that, for example using pole placement. In that case one doesn't need to involve Lyapunov functions. Commented Sep 30, 2022 at 16:15
• @KwinvanderVeen Yes, this is correct. The thing is that I started to write a different answer for dwell-time switching and ended up writing this one. So, I kept the LMI formulation.
– KBS
Commented Sep 30, 2022 at 19:55
• @KBS, If you can provide Kwin's approach when you have time, that would be very helpful since I'm not familiar with Schur complement approach.
– eet
Commented Oct 1, 2022 at 7:24