Advanced Control: The Augmented SDRE Technique

Abstract

This chapter introduces the augmented state-dependent Riccati equation (SDRE) technique for the advanced control of the tandem cold metal rolling process. The advantages of the SDRE method when compared to conventional control are presented. The characteristics of several other advanced control methods are included for evaluation in light of the augmented SDRE method. A brief introductory discussion on linear quadratic control concepts also is included as background in preparation for the presentation of the SDRE technique. The results of simulations using the augmented SDRE method as applied to typical tandem cold rolling applications show the effectiveness of this technique for control of the tandem cold rolling process.

Appendix

The sections of this appendix provide material that supplements what is presented earlier in this chapter. Provided are some applicable definitions and theorems, detailed methods for the computation of gradients, derivations of various relationships relating to functions of the state variables, and a derivation of the necessary conditions for the optimality of the SDRE controller.

5.1.1 Definitions and Theorems

The definitions and theorems which follow are related to the material presented in Section 5.3.1. The definitions are based on [37] and [15]. Theorems are based on [15]. A more detailed theoretical treatment is found in [12] and the references cited therein.

Definition 1. A system is considered an autonomous system if the function f does not depend explicitly on t, i.e.

$$ \dot{x} = f(x). $$

(5.59)

Definitions 2 through 8 are based on an autonomous system, where $f:D \to {R^n}$ is a locally Lipshitz map from D into ${R^n}$.

Definition 2. The point $\tilde{x}$ is an equilibrium point of (5.59) if

$$ f(\tilde{x}) = 0. $$

(5.60)

Definition 3. Taking $\tilde{x} = 0$ for convenience and without loss of generality, the equilibrium point of (5.60) is stable if for each $\it\varepsilon > 0$ there is a $\it\delta$ such that

$$ \left\| {x(0)} \right\| < \delta \Rightarrow \left\| {x(t)} \right\| < \varepsilon, \,\,\,\forall t \ge 0. $$

(5.61)

Definition 4. The equilibrium point of (5.60) is unstable if it is not stable.

Definition 5. The equilibrium point $\tilde{x} = 0$ is asymptotically stable if it is stable and a $\delta$ can be chosen such that

$$ \left\| {x(0)} \right\| < \delta \Rightarrow \mathop {{\lim }}\limits_{t \to \infty } x(t) = 0. $$

(5.62)

Definition 6. Let $\phi (x;t)$ be the solution of (5.59) that starts at time t = 0 and at an initial state x ₀ , with $\tilde{x} = 0$. Then the region of attraction is the set of all points x such that

$$ \mathop {{\lim }}\limits_{t \to \infty } \,\,\phi (x;t) = 0. $$

(5.63)

Definition 7. The equilibrium point $\tilde{x} = \it 0$ is locally asymptotically stable if it is asymptotically stable and its region of attraction is some neighborhood of the origin.

Definition 8. The equilibrium point $\tilde{x} = 0$ is globally asymptotically stable if

$$ \mathop {{\lim }}\limits_{t \to \infty } \phi (x;t) = 0, $$

(5.64)

no matter how large $\left\| x \right\|$ is.

Definition 9. {C(x), A(x)} is a pointwise observable parameterization of the nonlinear system in a region $\Omega$ if the pair {C(x), A(x)} is pointwise observable (in the linear sense) for all $x \in \Omega$.

Definition 10. {C(x), A(x)} is a pointwise detectable parameterization of the nonlinear system in a region $\Omega$ if the pair {C(x), A(x)} is pointwise detectable⁵ (in the linear sense) for all $x \in \Omega$.

Definition 11. {A(x), B(x)} is a pointwise controllable parameterization of the nonlinear system in a region $\Omega$ if the pair {A(x), B(x)} is pointwise controllable (in the linear sense) for all $x \in \Omega$.

Definition 12. {A(x), B(x)} is a pointwise stabilizable parameterization of the nonlinear system in a region $\Omega$ if the pair {A(x), B(x)} is pointwise stabilizable⁶ (in the linear sense) for all $x \in \Omega$.

Theorem 1. In addition to a(x), b(x), R(x), Q(x), Q(x) $\in {C^k}$, $k \ge 1$, assume that A(x) is smooth (i.e. $A(x) \in {C^k}$) and that A(x) is both a stabilizable and detectable coefficient parameterization of the nonlinear system. Then the state-dependent Riccati equation method produces a closed-loop solution which is locally asymptotically stable.

Proof: The proof is provided in [15].

Theorem 2. Assume that the functions A(x), b(x), K(x), Q(x), and R(x), and their gradients⁷ ${ \nabla_x}A{\hbox{\it (}}x{\hbox{\it )}},\,\,{\nabla_x}b{\hbox{\it (}}x{\hbox{\it )}},\,\,{\nabla_x}K{\hbox{\it (}}x{\hbox{\it )}}$, and ${\nabla_x}A{\hbox{\it (}}x{\hbox{\it )}}$ are bounded along trajectories. Then, under stability, as the state x is driven to zero, the necessary condition for optimality is asymptotically satisfied at a quadratic rate.

Proof: The proof is provided in [15].

5.1.2 Computation of Gradients

With $x \in {R^n},\,\,Q'(x) = Q(x) \in {R^{nxn}}$, and $Q(x) \in {C^1}$, and using matrix differentiation formulae as given in [39],

$$ {\nabla_x}(x'Q(x)x) = 2Q(x)x + x'{\nabla_x}Q(x)x, $$

(5.65)

where

$$ {\nabla_x}(x'Q(x)x) = \left[ \begin{array}{ll} x'{\nabla_{x1}}Q(x)x \hfill \\x'{\nabla_{x2}}Q(x)x \hfill \\ \vdots \hfill \\x'{\nabla_{xn}}Q(x)x \hfill \\\end{array} \right]\!\!, $$

(5.66)

and

$$ {\nabla_{xi}}Q(x) = \left[ \begin{array}{lll} \frac{{\partial {q_{11}}(x)}}{{\partial {x_i}}} \frac{{\partial {q_{12}}(x)}}{{\partial {x_i}}} \cdots \frac{{\partial {q_{1n}}(x)}}{{\partial {x_i}}}\\\vdots \vdots \\\frac{{\partial {q_{n1}}(x)}}{{\partial {x_i}}} \cdots \ \ \cdots \cdots \frac{{\partial {q_{nn}}(x)}}{{\partial {x_i}}} \hfill \\\end{array} \right],\,\,\,i = 1,2,\, \ldots n. $$

(5.67)

Equation 5.65 can be verified by first computing $x'Q(x)x$ on an element-by-element basis, and then computing ${\nabla_x}(x'Q(x)x)$.

Example 1

Computation of ${\nabla_x}(x'Q(x)x)$, for $x \in {R^2},\ Q'(x) = Q(x) \in {R^{2x2}},\ Q(x) \in {C^1}$:

Using (5.65) and (5.66), and not showing function arguments,

$$ {\nabla_x}(x'Qx) = 2\left[ \begin{array}{lll} {q_{11}}\,\,\,\,\,{q_{12}} \hfill \\{q_{21}}\,\,\,\,{q_{22}} \hfill \\\end{array} \right]\,\,\left[ \begin{array}{ll} {x_1} \hfill \\{x_2} \hfill \\\end{array} \right] + \left[ \begin{array}{ll} x'{\nabla_{x1}}Qx \hfill \\x'{\nabla_{x2}}Qx \hfill \\\end{array} \right], $$

(5.68)

and then using (5.67),

$$ x'{\nabla_{x1}}Qx = \left[ {{x_1}\,\,\,\,\,{x_2}} \right]\,\,\left[ \begin{array}{lll} \frac{{\partial {q_{11}}(x)}}{{\partial {x_1}}}\,\,\,\,\,\,\,\,\frac{{\partial {q_{12}}(x)}}{{\partial {x_1}}} \hfill \\\frac{{\partial {q_{21}}(x)}}{{\partial {x_1}}}\,\,\,\,\,\,\,\,\frac{{\partial {q_{22}}(x)}}{{\partial {x_1}}} \hfill \\\end{array} \right]\,\,\left[ \begin{array}{ll} {x_1} \hfill \\{x_2} \hfill \\\end{array} \right], $$

(5.69)

$$ x'{\nabla_{x2}}Qx = \left[ {{x_1}\,\,\,\,\,{x_2}} \right]\,\,\left[ \begin{array}{lll} \frac{{\partial {q_{11}}(x)}}{{\partial {x_2}}}\,\,\,\,\,\,\,\,\frac{{\partial {q_{12}}(x)}}{{\partial {x_2}}} \hfill \\\frac{{\partial {q_{21}}(x)}}{{\partial {x_2}}}\,\,\,\,\,\,\,\,\frac{{\partial {q_{22}}(x)}}{{\partial {x_2}}} \hfill \\\end{array} \right]\,\,\left[ \begin{array}{ll} {x_1} \hfill \\{x_2} \hfill \\\end{array} \right]. $$

(5.70)

Performing the multiplications and substituting into (5.68), and noting that q ₁₂ = q ₂₁,

$$ {\nabla_x}(x'Qx) = \left[ \begin{array}{ll} 2{x_1}{q_{11}} + 2{x_2}{q_{12}} + x_1^2\frac{{\partial {q_{11}}}}{{\partial {x_1}}} + 2{x_1}{x_2}\frac{{\partial {q_{12}}}}{{\partial {x_1}}} + x_2^2\frac{{\partial {q_{22}}}}{{\partial {x_1}}} \hfill \\2{x_1}{q_{12}} + 2{x_2}{q_{22}} + x_1^2\frac{{\partial {q_{11}}}}{{\partial {x_2}}} + 2{x_1}{x_2}\frac{{\partial {q_{12}}}}{{\partial {x_2}}} + x_2^2\frac{{\partial {q_{22}}}}{{\partial {x_2}}} \hfill \\\end{array} \right], $$

(5.71)

Computing $x'Qx$ on an element-by-element basis and then computing ${\nabla_x}{\hbox {\it (}}x'Qx{\hbox {\it )}}$ verifies the result obtained in (5.71).

Example 2

Computation of ${\nabla_x}(\lambda 'A(x)x)$, where $x \in {R^2},\,\,A(x) \in {R^{2x2}},\,\,A(x) \in {C^1},\,\lambda \in {R^2}$, for all x:

Using matrix differentiation formulae, and not showing function arguments,

$${\nabla_x}(\lambda 'Ax) = (x'{\nabla_x}A' + A')\lambda, $$

(5.72)

where

$$ x'{\nabla_x}A' = \left[ \begin{array}{ll} x'{\nabla_{x1}}A' \hfill \\x'{\nabla_{x2}}A' \hfill \\\end{array} \right], $$

(5.73)

and

$$ x'{\nabla_{x1}}A = \left[ {{x_1}\,\,\,\,{x_2}} \right]\,\,\left[ \begin{array}{lll} \frac{{\partial {a_{11}}}}{{\partial {x_1}}}\,\,\,\,\,\,\frac{{\partial {a_{12}}}}{{\partial {x_1}}} \hfill \\\frac{{\partial {a_{21}}}}{{\partial {x_1}}}\,\,\,\,\,\frac{{\partial {a_{22}}}}{{\partial {x_1}}} \hfill \\\end{array} \right], $$

(5.74)

$$ x'{\nabla_{x2}}A = \left[ {{x_1}\,\,\,\,{x_2}} \right]\,\,\left[ \begin{array}{ll} \frac{{\partial {a_{11}}}}{{\partial {x_2}}}\,\,\,\,\,\,\frac{{\partial {a_{12}}}}{{\partial {x_2}}} \hfill \\\frac{{\partial {a_{21}}}}{{\partial {x_2}}}\,\,\,\,\,\frac{{\partial {a_{22}}}}{{\partial {x_2}}} \hfill \\\end{array} \right]. $$

(5.75)

It is straightforward to use (5.74) and (5.75) and substitute into (5.72) and (5.73) to obtain the result.

5.1.3 Derivation of Relationships as Functions of the State Variables

The material which follows relates closely to the determination of the state-dependent elements of the A(x) matrix as noted in Section 5.3.3.

5.1.3.1 Relationships for Output Thickness and Specific Roll Force

During each scan of the controller, $\xi$ and $\alpha$ are computed at a number of equally spaced points in a predetermined neighborhood of h _out0 as

$$ \xi = \frac{{\mu \sqrt {{R'({h_{in}} - {h_{out}}}}) }}{{\bar{h}}}, $$

(5.76)

$$ \alpha = \sqrt {{\frac{{{h_{out}}}}{{{h_{in}}}}}} \exp (\xi ) - 1, $$

(5.77)

where h _in for stand 1 is the input thickness to the mill, and for stands 2,3,4,5 is the output thickness of the previous stand delayed by the appropriate interstand time delay,

$$ {h_{in,i + 1}}(t) = {h_{out,i}}(t - {t_{d,i,i + 1}}),\,\,\,i = 1,2,3,4, $$

(5.78)

where μ and $R'$are the friction coefficient and the deformed work roll radius, and t _d,i,i+1 is the time delay between stands i and i + 1.

During the same controller scan, using the relationship (2.12) for the specific roll force and noting that F = PW, the total roll force is computed (at each point) as

$$ F = (\bar{k} - \bar{\sigma })\sqrt {{R'\delta }} (1 + 0.4\alpha )W, $$

(5.79)

where W is the strip width and other variables are as denoted in Chapter 2. In the neighborhood of h _out0, F then is approximated by a linear fit which is reasonable because the neighborhood is not large,

$$ F = {c_1}{h_{out}} + {c_2}, $$

(5.80)

where c ₁ and c ₂ are constants. Using (2.26) and (5.80), h _out is then

$$ {h_{out}} = \frac{{M(S + {S_0})}}{{(M - {c_1})}}, $$

(5.81)

and the specific roll force is

$$ P = \frac{{M({h_{out}} - (S + {S_0}))}}{W}, $$

(5.82)

and thus for stands 2,3,4,5, h _out and P depend on the state variables.

5.1.3.2 Relationships for Entry and Exit Strip Speeds

Using (2.19) the strip speed at the exit of the roll bite can be written as

$$ {V_{out}} = V(\,f + 1), $$

(5.83)

where the forward slip f as given in (2.23) ultimately depends on the state variables. By conservation of mass flow across the roll bite,

$$ {V_{in.i + 1}} = \frac{{{V_{out,i + 1}}{h_{out.i + 1}}}}{{{h_{in,i + 1}}}} = \frac{{{V_{i + 1}}(\,{f_{i + 1}} + \it 1){h_{out,i + {\rm 1}}}}}{{{h_{out,i}}(t - {t_{d.i,i + 1}})}}, $$

(5.84)

and thus $({V_{in,i + 1}} - {V_{out,i}})$ also depends on the state variables.

5.1.4 Derivation of the Necessary Conditions for Optimality

What is presented in this section supports the material discussed in Section 5.3.1.

From the cost function (5.17) and the nonlinear constraint (5.15), the Hamiltonian function is formed as:

$$ H(x,u,\lambda ) = \tfrac{1}{2}(x'Q(x)x + u'R(x)u) + \lambda '(A(x)x + Bu), $$

(5.85)

where $\lambda \in {R^n}$ is a Lagrange multiplier. The necessary conditions for optimality of a nonlinear controller are:

$$ {\nabla_\lambda }\,H = \dot{x}, $$

(5.86)

$$ {\nabla_x}\,H = - \dot{\lambda }, $$

(5.87)

$$ {\nabla_u}H = 0. $$

(5.88)

Using (5.85) and the control law (5.19),

$$ u = - {R^{ - 1}}(x)B'K(x)x, $$

(5.89)

$$ {\nabla_u}H = R(x)u + B'\lambda, $$

(5.90)

$$ {\nabla_u}H = R(x)( - {R^{ - 1}}(x)B'K(x)x) + B'\lambda, $$

(5.91)

$$ {\nabla_u}H = B'(\lambda - K(x)x). $$

(5.92)

${\nabla_u}H$ will be zero if λ is chosen so that

$$ \lambda = K(x)x. $$

(5.93)

Differentiating with respect to time results in

$$ \dot{\lambda } = \dot{K}(x)x + K(x)\dot{x}. $$

(5.94)

Using (5.85) and (5.87),

$$ \dot{\lambda } = - Q(x)x - \tfrac{1}{2}(x'{\nabla_x}Q(x)x + u'{\nabla_x}R(x)u) - (x'{\nabla_x}A'(x) + A'(x))\lambda. $$

(5.95)

Equating (5.94) and (5.95), and using the nonlinear constraint (5.15), (5.86), (5.89), and (5.93),

$$ \begin{array}{lll}b \dot{K}(x)x + K(x)(A(x)x - B{R^{ - 1}}(x)B'K(x)x) = \hfill \\ - Q(x)x - \tfrac{1}{2}(x'{\nabla_x}Q(x)x + u'{\nabla_x}R(x)u) - (x'{\nabla_x}A'(x) + A'(x))K(x)x. \hfill \\\end{array} $$

(5.96)

Rearranging and grouping terms,

$$ \begin{array}{ll} \dot{K}(x)x + \tfrac{1}{2}(x'{\nabla_x}Q(x)x + u'{\nabla_x}R(x)u) + x'{\nabla_x}A'(x)K(x)x \hfill \cr + (A'(x)K(x) + K(x)A(x) - K(x)B{R^{ - 1}}(x)B'K(x) + Q(x))x = 0. \hfill \\\end{array} $$

(5.97)

From the state-dependent algebraic Riccati equation, the expression (A′(x) K(x)+K(x) A(x) − K(x) B R ⁻¹ (x) B′ K(x) + Q(x)) is equal to zero, and substituting for u (5.89) gives the necessary condition for the closed-loop solution to be near-optimal

$$ \begin{array}{ll} \dot{K}(x)x + \tfrac{1}{2}(x'{\nabla_x}Q(x)x + x'K(x)B{R^{ - 1}}(x){\nabla_x}R(x){R^{ - 1}}(x)B'K(x)x) \hfill \\ + x'{\nabla_x}A'(x)K(x)x = 0. \hfill \\\end{array} $$

(5.98)