Subsections

• 10.1 Generalized LQR problem
• 10.2 Explicit solutions of generalized Riccati equations
• 10.3 LQR design techniques

10 LQR design techniques

For linear finite-dimensional systems, LQR theory plays a special role among various approaches because an optimal gain can be easily calculated by solving ARE and the corresponding control (if it exists) stabilizes the closed-loop system. The corresponding algorithms are realized in Control System Toolbox [8].

For systems with delays theoretical aspects of LQR theory have been also well developed in different directions and it was shown that the coefficients of optimal control are solutions of some specific system of algebraic matrix equation, ordinary and partial differential equations, the so called generalized Riccati equations - GREs (see below equations (10.5) - (10.9)). Solution of GREs is the main obstacle for effective realization of LQR algorithms for systems with delays.

In Time-Delay System Toolbox we realized the procedure of designing optimal LQR controller based on the explicit form solutions of GREs [21]. The approach is based on utilization of a generalized quadratic cost functional [33]. Let us briefly discuss this procedure.

10.1 Generalized LQR problem

Consider the linear system with discrete state delay ⁴

$$\dot{x} \tau \tau$$ (10.1)

where A, A_$\tau$, B are n×n, n×n, n×r constant matrices. The problem is to minimize the generalized quadratic cost functional [33]

J = $$\int\limits_{0}^{\infty}$$$$\Bigl\{$$x'(t)$$\Phi_{0}^{}$$ x(t) + 2 x'(t)$$\int\limits_{-\tau}^{0}$$$$\Phi_{1}^{}$$(s) x(t + s) ds +

$$\int\limits_{-\tau}^{0} \int\limits_{-\tau}^{0} \Phi_{2}^{} \nu \nu \nu \Bigr\}$$ (10.2)

on trajectories of system (10.1). Here $\Phi_{0}^{}$ is a constant n×n matrix, $\Phi_{1}^{}$(s) is n×n matrix with piece-wise continuous elements on [- $\tau$, 0], $\Phi_{2}^{}$(s,$\nu$) is n×n matrix with piece-wise continuous elements on [- $\tau$, 0]×[- $\tau$, 0], N is r×r symmetric positive definite matrix.

Remark 10.1. Note, the state weight functional in (10.2) is the quadratic functional

$$\Phi_{0}^{} \int\limits_{-\tau}^{0} \Phi_{1}^{} \int\limits_{-\tau}^{0} \int\limits_{-\tau}^{0} \Phi_{2}^{} \nu \nu \nu$$ (10.3)

defined on H = $\bf R^{n}_{}$×Q[- $\tau$, 0), but not only the quadratic form x'$\Phi_{0}^{}$ x. $\Box$

Optimal control for problem (10.1), (10.2) has the form

$$\Bigl[ \int\limits_{-\tau}^{0} \Bigr]$$ (10.4)

where n×n matrices P, D(s) and R(s,$\nu$) are solutions of the following system of matrix equations (Generalized Riccati Equations)

$$\Phi_{0}^{}$$ (10.5)

$${\frac{d D(s)}{d s}} \Bigl[ \Bigr] \Phi_{1}^{}$$ (10.6)

$${\frac{\partial R(s,\nu)}{\partial s}} {\frac{\partial R(s,\nu)}{\partial \nu}} \nu \Phi_{2}^{} \nu$$ (10.7)

with boundary conditions

$$\tau \tau$$ (10.8)

$$\tau \tau$$ (10.9)

for - $\tau$$\le$s$\le$ 0, - $\tau$$\le$$\nu$$\le$ 0. Here K = B N^-1 B'.
Note that P is the symmetric positive semi-definite matrix, and R(s,$\nu$) = R'($\nu$, s).

The optimal value of the cost functional J is

W[x, y( ^. )] = x'P x + 2 x' $$\int\limits_{-\tau}^{0}$$D(s) y(s) ds +

$$\int\limits_{-\tau}^{0} \int\limits_{-\tau}^{0} \nu \nu \nu$$ (10.10)

Let us formulate a proposition concerning the optimal property of the corresponding feedback control.

Theorem 10.1. Let P, D(s) and R(s,$\nu$) are solutions of system (10.5) - (10.9). If linear control (10.4) stabilizes system (10.1) then this control minimizes the quadratic cost functional (10.2) and optimal value of the cost functional is (10.10). $\Box$

10.2 Explicit solutions of generalized Riccati equations

On the basis of suitable choices of matrices $\Phi_{0}^{}$, $\Phi_{1}^{}$(s) and $\Phi_{2}^{}$(s,$\nu$) one can simplify equations (10.5) - (10.9) and find their solutions in explicit forms.

Let us take matrices of the cost functional (10.2) as

$$\Phi_{0}^{}$$ = M - $$\Bigl($$D(0) + D'(0)$$\Bigr)$$ ,  $$\Phi_{1}^{}$$(s) = - R(0, s) ,

$$\Phi_{2}^{} \nu \nu$$ (10.11)

where n×n matrices P, D(s) and R(s,$\nu$) are solutions of the following system of special generalized Riccati equations

PA + A'P + M = PKP , (10.12)

$${\frac{d D(s)}{d s}} \Bigl[ \Bigr]$$ (10.13)

$${\frac{\partial R(s,\nu)}{\partial s}} {\frac{\partial R(s,\nu)}{\partial \nu}}$$ (10.14)

with boundary conditions (10.8) - (10.9), M is a positive definite symmetric n×n matrix.

One can see, in system (10.12) - (10.14) matrices D(s) and R(s,$\nu$) are separated (these matrices are interconnected only through boundary condition (10.9)). Equation (10.12) is the classic algebraic Riccati equation (ARE). So we can find the solution of this system in explicit form.

Theorem 10.2. Let P be the solution of ARE (10.12). Then matrices D(s) and R(s,$\nu$) (that are solutions of system (10.13), (10.14), (10.8), (10.9) ) have the forms

$$\tau \tau$$ (10.15)

$$\nu \left\{\vphantom{ \begin{array}{ccc} Q(s) D(\nu) & \mbox{for} ... ... D'(s) Q'(\nu) & \mbox{for} & (s,\nu) \in \Omega_2 ,\end{array}}\right. \begin{array}{ccc} Q(s) D(\nu) & \mbox{for} & (s,\nu) \in \Ome... ... , \\ D'(s) Q'(\nu) & \mbox{for} & (s,\nu) \in \Omega_2 ,\end{array}$$ (10.16)

where Q(s) = A_$\tau$'e^{[PK - A'](s + $\tau$)}, and $\Omega_{1}^{}$ = $\Bigl\{$(s,$\nu$) $\in$ [- $\tau$, 0]×[- $\tau$, 0] : s - $\nu$ < 0$\Bigr\}$, $\Omega_{2}^{}$ = $\Bigl\{$(s,$\nu$) $\in$ [- $\tau$, 0]×[- $\tau$, 0] : s - $\nu$ > 0$\Bigr\}$. $\Box$

Corollary 10.1. Let P be the solution of ARE (10.12). Then matrices D(s) and R(s,$\nu$), defined by (10.15) and (10.16), are solutions of GREs (10.5) - (10.9) with matrices $\Phi_{0}^{}$, $\Phi_{1}^{}$(s), $\Phi_{2}^{}$(s,$\nu$) defined by (10.11). $\Box$

Theorem 10.3. Let P, D(s) and R(s,$\nu$) be the solutions of system (10.12) - (10.14), (10.8), (10.9) (hence these matrices have the forms (10.15), (10.16)). If linear control (10.4) stabilizes system (10.1) then this control minimizes the quadratic cost functional (10.2) with state weight matrices (10.11), and optimal value of the cost functional is (10.10). $\Box$

Remark 10.2. Remember, in case of ODE if a solution of ARE exists then the corresponding linear feedback control stabilizes the ODE system. However, not every linear time-invariant system with delays can be stabilizable by linear feedback control. So, after finding explicit solutions of GRE it is necessary to verify the stabilizing properties of the corresponding control (10.4). $\Box$

10.3 LQR design techniques

One can use the following procedure of designing and testing LQ-regulator.

1.

Using commands gre and lqdelay to calculate matrices C, D₀, D₁, D₂, related with the optimal control

u⁰(x, y( ^. )) = C x + D₀$$\int\limits_{-\tau}^{0}$$e^D₁sD₂ y(s) ds .

2.

Using the function clso to form the corresponding closed-loop system and to simulate it with respect to arbitrary initial functions.

3.

Command test allows to simulate the optimal closed-loop system with respect to the basic initial functions. Using test one can check stability of the closed-loop system with respect to initial function space spanned on the corresponding basic functions.

Besides, Toolbox offers some additional functions. For example, costfun calculates the approximate value of the cost functional of LQR problem. Using isdef one can check the definiteness of a matrix that is the finite dimensional approximation of the optimal value quadratic functional.