Chapter Summary and Problems

Summary¶

OLOC directly optimizes a control trajectory and is valuable for ideal performance benchmarks, qualitative insight, and plant-design studies. It can also assume future information unavailable to a real controller. Controller-parameter optimization instead designs an implementable feedback policy. Feedback-controller CCD combines physical and policy decisions and evaluates them under measurements and realized disturbances. MPC bridges these views by repeatedly solving a short open-loop problem using updated information.

Key terms¶

Open-loop optimal control; control trajectory; controller parameterization; feedback controller; feedback-controller co-design; future information; information availability; measured state; estimated state; forecast; implementable controller; model predictive control; receding horizon; prediction horizon; benchmark solution; realizability; complete horizon; instantaneous information; limited horizon; digital controller realization; prediction horizon as a design variable; nested CCD; covariance matrix adaptation evolution strategy (CMA-ES); hybrid Kalman filter.

Problems¶

1. Pontryagin analysis with a plant variable. Minimize $J=\tfrac12q_fx(T)^2+\int_0^T\tfrac12(qx^2+ru^2+\gamma p^2)dt$ subject to $\dot x=-p x+u$, $x(0)=x_0$, and $p\in[p_-,p_+]$. Derive the costate, optimal control, boundary conditions, and stationarity condition determining an interior optimal $p$.

2. LQR control co-design. For $\dot x=A(p)x+B(p)u$ and infinite-horizon cost $\int_0^\infty(x^TQ(p)x+u^TRu)dt+C(p)$, derive a gradient of the optimized value with respect to $p$ using differentiated Riccati and Lyapunov equations without differentiating the feedback gain explicitly.

3. Static output-feedback co-design. The controller is $u=-Ky$ with $y=C(s)x$, where sensor placement $s$ changes $C$. Formulate an $H_2$ plant--sensor--controller co-design problem and derive first-order stationarity conditions using the closed-loop controllability and observability Gramians.

4. Stabilizing model predictive control. For $x_{k+1}=A(p)x_k+B(p)u_k$ with polyhedral state and input constraints, formulate a finite-horizon MPC law with terminal cost and terminal set, then derive a sufficient decrease inequality proving recursive feasibility and asymptotic stability for every admissible plant design.

5. Value of preview. A disturbance-preview controller knows $w_{k:k+N_p}$ for $x_{k+1}=Ax_k+Bu_k+Ew_k$. Derive the finite-horizon optimal control law and an explicit expression for the reduction in quadratic cost relative to the no-preview law as a function of $N_p$.

6. Information-horizon monotonicity. Let $J_N^*(p)$ be the optimal expected cost when a controller has an $N$-step information horizon. Prove that $J_{N+1}^*(p)\le J_N^*(p)$ under nested policy classes and construct a counterexample showing that the corresponding optimal plant design need not vary monotonically with $N$.

7. Open-loop to closed-loop performance gap. For $\dot x=ax+bu+w$ with bounded disturbance $|w|\le\bar w$, derive the worst-case tracking-cost gap between a nominal open-loop control optimized for $w=0$ and a stabilizing linear feedback law, including actuator saturation in the bound.

8. Estimator-aware CCD. For $\dot x=A(p)x+Bu+Gw$ and $y=C(s)x+v$, formulate a joint plant--sensor--LQG design and derive the coupled Riccati sensitivities showing how plant and sensor decisions affect both regulation and estimation cost.

9. Actuator sizing under saturation. The scalar system $\dot x=-ax+bu$ uses $u=\operatorname{sat}(-Kx,U)$, and actuator mass is $m_a(U)=m_0+\alpha U$. Determine the globally optimal rating $U$ for a prescribed initial-state distribution and quadratic regulation objective by deriving a one-dimensional optimality condition that accounts for switching between saturated and unsaturated arcs.

10. Robust closed-loop CCD. The uncertain plant is $x_{k+1}=(A(p)+H\Delta E)x_k+B(p)u_k$ with $\Delta^T\Delta\preceq I$. Formulate a min--max state-feedback CCD problem and derive an LMI-based sufficient condition that jointly certifies robust stability, an upper bound on worst-case quadratic cost, and admissible plant bounds.

References and further reading¶

1. Betts, J. T. (2010). Practical methods for optimal control and estimation using nonlinear programming (2nd ed.). Society for Industrial and Applied Mathematics. DOI: 10.1137/1.9780898718577

2. Bryson, A. E., Jr., & Ho, Y.-C. (1975). Applied optimal control: Optimization, estimation, and control. Hemisphere Publishing Corporation.

3. Allison, J. T., & Herber, D. R. (2014). Multidisciplinary design optimization of dynamic engineering systems. AIAA Journal, 52(4), 691–710. DOI: 10.2514/1.J052182

4. Herber, D. R., & Allison, J. T. (2019). Nested and simultaneous solution strategies for general combined plant and control design problems. Journal of Mechanical Design, 141(1), Article 011402. DOI: 10.1115/1.4040705

5. Garcia-Sanz, M. (2019). Control co-design: An engineering game changer. Advanced Control for Applications: Engineering and Industrial Systems, 1(1), Article e18. DOI: 10.1002/adc2.18

6. Mayne, D. Q., Rawlings, J. B., Rao, C. V., & Scokaert, P. O. M. (2000). Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789–814. DOI: 10.1016/S0005-1098(99)00214-9

7. Deshmukh, A. P., Herber, D. R., & Allison, J. T. (2015). Bridging the gap between open-loop and closed-loop control in co-design: A framework for complete optimal plant and control architecture design. In 2015 American Control Conference (ACC) (pp. 4916-4922).

8. Bayat, S., & Allison, J. T. (2026). Control co-design with varying available information applied to vehicle suspensions. ASME Journal of Dynamic Systems, Measurement, and Control, 148(1), Article 011013. DOI: 10.1115/1.4069918